PHYSICS BOOK PART-2

MECHANICAL PROPERTIES OF FLUIDS 271 8. Bernoulli’s principle states that as we move along a streamline, the sum of the pressure (P), the kinetic energy per unit volume (ρv2/2) and the potential energy per unit volume (ρgy) remains a constant. P + ρv2/2 + ρgy = constant The equation is basically the conservation of energy applied to non viscuss fluid motion in steady state. There is no fluid which have zero viscosity, so the above statement is true only approximately. The viscosity is like friction and converts the kinetic energy to heat energy. 9. Though shear strain in a fluid does not require shear stress, when a shear stress is applied to a fluid, the motion is generated which causes a shear strain growing with time. The ratio of the shear stress to the time rate of shearing strain is known as coefficient of viscosity, η. where symbols have their usual meaning and are defined in the text. 10. Stokes’ law states that the viscous drag force F on a sphere of radius a moving with velocity v through a fluid of viscosity is, F = 6πηav. 11. Surface tension is a force per unit length (or surface energy per unit area) acting in the plane of interface between the liquid and the bounding surface. It is the extra energy that the molecules at the interface have as compared to the interior. POINTS TO PONDER 1. Pressure is a scalar quantity. The definition of the pressure as “force per unit area” may give one false impression that pressure is a vector. The “force” in the numerator of the definition is the component of the force normal to the area upon which it is impressed. While describing fluids as a concept, shift from particle and rigid body mechanics is required. We are concerned with properties that vary from point to point in the fluid. 2. One should not think of pressure of a fluid as being exerted only on a solid like the walls of a container or a piece of solid matter immersed in the fluid. Pressure exists at all points in a fluid. An element of a fluid (such as the one shown in Fig. 10.2) is in equilibrium because the pressures exerted on the various faces are equal. 3. The expression for pressure P = Pa + ρgh holds true if fluid is incompressible. Practically speaking it holds for liquids, which are largely incompressible and hence is a constant with height. 4. The gauge pressure is the difference of the actual pressure and the atmospheric pressure. P – Pa = Pg Many pressure-measuring devices measure the gauge pressure. These include the tyre pressure gauge and the blood pressure gauge (sphygmomanometer). 5. A streamline is a map of fluid flow. In a steady flow two streamlines do not intersect as it means that the fluid particle will have two possible velocities at the point. 6. Bernoulli’s principle does not hold in presence of viscous drag on the fluid. The work done by this dissipative viscous force must be taken into account in this case, and P2 [Fig. 10.9] will be lower than the value given by Eq. (10.12). 7. As the temperature rises the atoms of the liquid become more mobile and the coefficient of viscosity, η falls. In a gas the temperature rise increases the random motion of atoms and η increases. 8. Surface tension arises due to excess potential energy of the molecules on the surface in comparison to their potential energy in the interior. Such a surface energy is present at the interface separating two substances at least one of which is a fluid. It is not the property of a single fluid alone. 2020-21

272 PHYSICS EXERCISES 10.1 Explain why 10.2 (a) The blood pressure in humans is greater at the feet than at the brain 10.3 (b) Atmospheric pressure at a height of about 6 km decreases to nearly half of 10.4 its value at the sea level, though the height of the atmosphere is more than 10.5 100 km (c) Hydrostatic pressure is a scalar quantity even though pressure is force divided by area. Explain why (a) The angle of contact of mercury with glass is obtuse, while that of water with glass is acute. (b) Water on a clean glass surface tends to spread out while mercury on the same surface tends to form drops. (Put differently, water wets glass while mercury does not.) (c) Surface tension of a liquid is independent of the area of the surface (d) Water with detergent disolved in it should have small angles of contact. (e) A drop of liquid under no external forces is always spherical in shape Fill in the blanks using the word(s) from the list appended with each statement: (a) Surface tension of liquids generally . . . with temperatures (increases / decreases) (b) Viscosity of gases . . . with temperature, whereas viscosity of liquids . . . with temperature (increases / decreases) (c) For solids with elastic modulus of rigidity, the shearing force is proportional to . . . , while for fluids it is proportional to . .. (shear strain / rate of shear strain) (d) For a fluid in a steady flow, the increase in flow speed at a constriction follows (conservation of mass / Bernoulli’s principle) (e) For the model of a plane in a wind tunnel, turbulence occurs at a ... speed for turbulence for an actual plane (greater / smaller) Explain why (a) To keep a piece of paper horizontal, you should blow over, not under, it (b) When we try to close a water tap with our fingers, fast jets of water gush through the openings between our fingers (c) The size of the needle of a syringe controls flow rate better than the thumb pressure exerted by a doctor while administering an injection (d) A fluid flowing out of a small hole in a vessel results in a backward thrust on the vessel (e) A spinning cricket ball in air does not follow a parabolic trajectory A 50 kg girl wearing high heel shoes balances on a single heel. The heel is circular with a diameter 1.0 cm. What is the pressure exerted by the heel on the horizontal floor ? 2020-21

MECHANICAL PROPERTIES OF FLUIDS 273 10.6 Toricelli’s barometer used mercury. Pascal duplicated it using French wine of density 10.7 984 kg m–3. Determine the height of the wine column for normal atmospheric 10.8 pressure. 10.9 A vertical off-shore structure is built to withstand a maximum stress of 109 Pa. Is 10.10 the structure suitable for putting up on top of an oil well in the ocean ? Take the 10.11 depth of the ocean to be roughly 3 km, and ignore ocean currents. 10.12 10.13 A hydraulic automobile lift is designed to lift cars with a maximum mass of 3000 kg. The area of cross-section of the piston carrying the load is 425 cm2. What 10.14 maximum pressure would the smaller piston have to bear ? 10.15 A U-tube contains water and methylated spirit separated by mercury. The mercury columns in the two arms are in level with 10.0 cm of water in one arm and 12.5 cm of spirit in the other. What is the specific gravity of spirit ? In the previous problem, if 15.0 cm of water and spirit each are further poured into the respective arms of the tube, what is the difference in the levels of mercury in the two arms ? (Specific gravity of mercury = 13.6) Can Bernoulli’s equation be used to describe the flow of water through a rapid in a river ? Explain. Does it matter if one uses gauge instead of absolute pressures in applying Bernoulli’s equation ? Explain. Glycerine flows steadily through a horizontal tube of length 1.5 m and radius 1.0 cm. If the amount of glycerine collected per second at one end is 4.0 × 10–3 kg s–1, what is the pressure difference between the two ends of the tube ? (Density of glycerine = 1.3 × 103 kg m–3 and viscosity of glycerine = 0.83 Pa s). [You may also like to check if the assumption of laminar flow in the tube is correct]. In a test experiment on a model aeroplane in a wind tunnel, the flow speeds on the upper and lower surfaces of the wing are 70 m s–1and 63 m s-1 respectively. What is the lift on the wing if its area is 2.5 m2 ? Take the density of air to be 1.3 kg m–3. Figures 10.23(a) and (b) refer to the steady flow of a (non-viscous) liquid. Which of the two figures is incorrect ? Why ? Fig. 10.23 10.16 The cylindrical tube of a spray pump has a cross-section of 8.0 cm2 one end of 10.17 which has 40 fine holes each of diameter 1.0 mm. If the liquid flow inside the tube 10.18 is 1.5 m min–1, what is the speed of ejection of the liquid through the holes ? A U-shaped wire is dipped in a soap solution, and removed. The thin soap film formed between the wire and the light slider supports a weight of 1.5 × 10–2 N (which includes the small weight of the slider). The length of the slider is 30 cm. What is the surface tension of the film ? Figure 10.24 (a) shows a thin liquid film supporting a small weight = 4.5 × 10–2 N. What is the weight supported by a film of the same liquid at the same temperature in Fig. (b) and (c) ? Explain your answer physically. 2020-21

274 PHYSICS Fig. 10.24 10.19 What is the pressure inside the drop of mercury of radius 3.00 mm at room 10.20 temperature ? Surface tension of mercury at that temperature (20 °C) is 4.65 × 10–1 N m–1. The atmospheric pressure is 1.01 × 105 Pa. Also give the excess pressure inside the drop. What is the excess pressure inside a bubble of soap solution of radius 5.00 mm, given that the surface tension of soap solution at the temperature (20 °C) is 2.50 × 10–2 N m–1 ? If an air bubble of the same dimension were formed at depth of 40.0 cm inside a container containing the soap solution (of relative density 1.20), what would be the pressure inside the bubble ? (1 atmospheric pressure is 1.01 × 105 Pa). Additional Exercises 10.21 A tank with a square base of area 1.0 m2 is divided by a vertical partition in the 10.22 middle. The bottom of the partition has a small-hinged door of area 20 cm2. The tank is filled with water in one compartment, and an acid (of relative density 1.7) in the other, both to a height of 4.0 m. compute the force necessary to keep the door close. A manometer reads the pressure of a gas in an enclosure as shown in Fig. 10.25 (a) When a pump removes some of the gas, the manometer reads as in Fig. 10.25 (b) The liquid used in the manometers is mercury and the atmospheric pressure is 76 cm of mercury. (a) Give the absolute and gauge pressure of the gas in the enclosure for cases (a) and (b), in units of cm of mercury. (b) How would the levels change in case (b) if 13.6 cm of water (immiscible with mercury) are poured into the right limb of the manometer ? (Ignore the small change in the volume of the gas). Fig. 10.25 10.23 Two vessels have the same base area but different shapes. The first vessel takes twice the volume of water that the second vessel requires to fill upto a particular common height. Is the force exerted by the water on the base of the vessel the same in the two cases ? If so, why do the vessels filled with water to that same height give different readings on a weighing scale ? 2020-21

MECHANICAL PROPERTIES OF FLUIDS 275 10.24 During blood transfusion the needle is inserted in a vein where the gauge pressure 10.25 is 2000 Pa. At what height must the blood container be placed so that blood may just enter the vein ? [Use the density of whole blood from Table 10.1]. 10.26 In deriving Bernoulli’s equation, we equated the work done on the fluid in the tube 10.27 to its change in the potential and kinetic energy. (a) What is the largest average 10.28 velocity of blood flow in an artery of diameter 2 × 10–3 m if the flow must remain 10.29 laminar ? (b) Do the dissipative forces become more important as the fluid velocity increases ? Discuss qualitatively. 10.30 (a) What is the largest average velocity of blood flow in an artery of radius 2×10–3m if the flow must remain lanimar? (b) What is the corresponding flow rate ? (Take viscosity of blood to be 2.084 × 10–3 Pa s). A plane is in level flight at constant speed and each of its two wings has an area of 25 m2. If the speed of the air is 180 km/h over the lower wing and 234 km/h over the upper wing surface, determine the plane’s mass. (Take air density to be 1 kg m–3). In Millikan’s oil drop experiment, what is the terminal speed of an uncharged drop of radius 2.0 × 10–5 m and density 1.2 × 103 kg m–3. Take the viscosity of air at the temperature of the experiment to be 1.8 × 10–5 Pa s. How much is the viscous force on the drop at that speed ? Neglect buoyancy of the drop due to air. Mercury has an angle of contact equal to 140° with soda lime glass. A narrow tube of radius 1.00 mm made of this glass is dipped in a trough containing mercury. By what amount does the mercury dip down in the tube relative to the liquid surface outside ? Surface tension of mercury at the temperature of the experiment is 0.465 N m–1. Density of mercury = 13.6 × 103 kg m–3. Two narrow bores of diameters 3.0 mm and 6.0 mm are joined together to form a U-tube open at both ends. If the U-tube contains water, what is the difference in its levels in the two limbs of the tube ? Surface tension of water at the temperature of the experiment is 7.3 × 10–2 N m–1. Take the angle of contact to be zero and density of water to be 1.0 × 103 kg m–3 (g = 9.8 m s–2) . Calculator/Computer – Based Problem 10.31 (a) It is known that density ρ of air decreases with height y as ρ = ρ0e −y/yo where ρ = 1.25 kg m–3 is the density at sea level, and y0 is a constant. This density 0 variation is called the law of atmospheres. Obtain this law assuming that the temperature of atmosphere remains a constant (isothermal conditions). Also assume that the value of g remains constant. (b) A large He balloon of volume 1425 m3 is used to lift a payload of 400 kg. Assume that the balloon maintains constant radius as it rises. How high does it rise ? [Take y0 = 8000 m and ρ = 0.18 kg m–3]. He 2020-21

276 PHYSICS APPENDIX 10.1 : WHAT IS BLOOD PRESSURE ? In evolutionary history there occurred a time when animals started spending a significant amount of time in the upright position. This placed a number of demands on the circulatory system. The venous system that returns blood from the lower extremities to the heart underwent changes. You will recall that veins are blood vessels through which blood returns to the heart. Humans and animals such as the giraffe have adapted to the problem of moving blood upward against gravity. But animals such as snakes, rats and rabbits will die if held upwards, since the blood remains in the lower extremities and the venous system is unable to move it towards the heart. Fig. 10.26 Schematic view of the gauge pressures in the arteries in various parts of the human body while standing or lying down. The pressures shown are averaged over a heart cycle. Figure 10.26 shows the average pressures observed in the arteries at various points in the human body. Since viscous effects are small, we can use Bernoulli’s equation, Eq. (10.13), P + 1 ρv2 + ρgy = Constant 2 to understand these pressure values. The kinetic energy term (ρ v2/2) can be ignored since the velocities in the three arteries are small (≈ 0.1 m s–1) and almost constant. Hence the gauge pressures at the brain PB, the heart PH, and the foot PF are related by PF = PH + ρ g hH = PB + ρ g hB (10.34) where ρ is the density of blood. Typical values of the heights to the heart and the brain are hH = 1.3 m and hB = 1.7 m. Taking ρ = 1.06 × 103 kg m–3 we obtain that PF = 26.8 kPa (kilopascals) and PB = 9.3 kPa given that PH = 13.3 kPa. Thus the pressures in the lower and upper parts of the body are so different when a person is standing, but are almost equal when he is lying down. As mentioned in the text the units for pressure more commonly employed in medicine and physiology are torr and mm of Hg. 1 mm of Hg = 1 torr = 0.133 kPa. Thus the average pressure at the heart is PH = 13.3 kPa = 100 mm of Hg. The human body is a marvel of nature. The veins in the lower extremities are equipped with valves, which open when blood flows towards the heart and close if it tends to drain down. Also, blood is returned at least partially by the pumping action associated with breathing and by the flexing of the skeletal muscles during walking. This explains why a soldier who is required to stand at attention may faint because of insufficient return of the blood to the heart. Once he is made to lie down, the pressures become equalized and he regains consciousness. An instrument called the sphygmomanometer usually measures the blood pressure of humans. It is a fast, painless and non-invasive technique and gives the doctor a reliable idea about the patient’s health. The measurement process is shown in Fig. 10.27. There are two reasons why the upper arm is used. First, it is at the same level as the heart and measurements here give values close to that at the heart. Secondly, the upper arm contains a single bone and makes the artery there (called the brachial artery) easy to compress. We have all measured pulse rates by placing our fingers over the wrist. Each pulse takes a little less than a second. During each pulse the pressure in the heart and the circulatory system goes through a 2020-21

MECHANICAL PROPERTIES OF FLUIDS 277 maximum as the blood is pumped by the heart (systolic pressure) and a minimum as the heart relaxes (diastolic pressure). The sphygmomanometer is a device, which measures these extreme pressures. It works on the principle that blood flow in the brachial (upper arm) artery can be made to go from laminar to turbulent by suitable compression. Turbulent flow is dissipative, and its sound can be picked up on the stethoscope. The gauge pressure in an air sack wrapped around the upper arm is measured using a manometer or a dial pressure gauge (Fig. 10.27). The pressure in the sack is first increased till the brachial artery is closed. The pressure in the sack is then slowly reduced while a stethoscope placed just below the sack is used to listen to noises arising in the brachial artery. When the pressure is just below the systolic (peak) pressure, the artery opens briefly. During this brief period, the blood velocity in the highly constricted artery is high and turbulent and hence noisy. The resulting noise is heard as a tapping sound on the stethoscope. When the pressure in the sack is lowered further, the artery remains open for a longer portion of the heart cycle. Nevertheless, it remains closed during the diastolic (minimum pressure) phase of the heartbeat. Thus the duration of the tapping sound is longer. When the pressure in the sack reaches the diastolic pressure the artery is open during the entire heart cycle. The flow is however, still turbulent and noisy. But instead of a Fig. 10.27 Blood pressure measurement using the tapping sound we hear a steady, continuous roar sphygmomanometer and stethoscope. on the stethoscope. The blood pressure of a patient is presented as the ratio of systolic/diastolic pressures. For a resting healthy adult it is typically 120/80 mm of Hg (120/80 torr). Pressures above 140/90 require medical attention and advice. High blood pressures may seriously damage the heart, kidney and other organs and must be controlled. 2020-21

CHAPTER ELEVEN THERMAL PROPERTIES OF MATTER 11.1 Introduction 11.1 INTRODUCTION 11.2 Temperature and heat 11.3 Measurement of We all have common sense notions of heat and temperature. Temperature is a measure of ‘hotness’ of a body. A kettle temperature with boiling water is hotter than a box containing ice. In physics, we need to define the notion of heat, temperature, 11.4 Ideal-gas equation and etc., more carefully. In this chapter, you will learn what heat is and how it is measured, and study the various proceses by absolute temperature which heat flows from one body to another. Along the way, you will find out why blacksmiths heat the iron ring before 11.5 Thermal expansion fitting on the rim of a wooden wheel of a horse cart and why 11.6 Specific heat capacity the wind at the beach often reverses direction after the sun 11.7 Calorimetry goes down. You will also learn what happens when water boils 11.8 Change of state or freezes, and its temperature does not change during these 11.9 Heat transfer processes even though a great deal of heat is flowing into or out of it. 11.10 Newton’s law of cooling 11.2 TEMPERATURE AND HEAT Summary Points to ponder We can begin studying thermal properties of matter with Exercises definitions of temperature and heat. Temperature is a relative Additional Exercises measure, or indication of hotness or coldness. A hot utensil is said to have a high temperature, and ice cube to have a low temperature. An object that has a higher temperature than another object is said to be hotter. Note that hot and cold are relative terms, like tall and short. We can perceive temperature by touch. However, this temperature sense is somewhat unreliable and its range is too limited to be useful for scientific purposes. We know from experience that a glass of ice-cold water left on a table on a hot summer day eventually warms up whereas a cup of hot tea on the same table cools down. It means that when the temperature of body, ice-cold water or hot tea in this case, and its surrounding medium are different, heat transfer takes place between the system and the surrounding medium, until the body and the surrounding medium are at the same temperature. We also know that in the case of glass tumbler of ice-cold water, heat flows from the environment to 2020-21

THERMAL PROPERTIES OF MATTER 279 the glass tumbler, whereas in the case of hot Fig. 11.1 A plot of Fahrenheit temperature (tF) versus tea, it flows from the cup of hot tea to the Celsius temperature (tc). environment. So, we can say that heat is the A relationship for converting between the two form of energy transferred between two (or scales may be obtained from a graph of more) systems or a system and its wtFeahmhopsreeerneahqtuueiratetito(etnCm)ispineraatsutrreaig(thFt) versus celsius surroundings by virtue of temperature line (Fig. 11.1), difference. The SI unit of heat energy transferred is expressed in joule (J) while SI unit t F – 32 = tC (11.1) of temperature is Kelvin (K), and degree Celsius 180 100 (oC) is a commonly used unit of temperature. When an object is heated, many changes may 11.4 IDEAL-GAS EQUATION AND take place. Its temperature may rise, it may expand or change state. We will study the effect ABSOLUTE TEMPERATURE of heat on different bodies in later sections. Liquid-in-glass thermometers show different 11.3 MEASUREMENT OF TEMPERATURE readings for temperatures other than the fixed points because of differing expansion properties. A measure of temperature is obtained using a A thermometer that uses a gas, however, gives thermometer. Many physical properties of the same readings regardless of which gas is materials change sufficiently with temperature. used. Experiments show that all gases at low Some such properties are used as the basis for densities exhibit same expansion behaviour. The constructing thermometers. The commonly used variables that describe the behaviour of a given property is variation of the volume of a liquid quantity (mass) of gas are pressure, volume, and with temperature. For example, in common temperature (P, V, and T )(where T = t + 273.15; liquid–in–glass thermometers, mercury, alcohol t is the temperature in °C). When temperature etc., are used whose volume varies linearly with is held constant, the pressure and volume of a temperature over a wide range. quantity of gas are related as PV = constant. This relationship is known as Boyle’s law, after Thermometers are calibrated so that a Robert Boyle (1627–1691), the English Chemist numerical value may be assigned to a given who discovered it. When the pressure is held temperature in an appropriate scale. For the constant, the volume of a quantity of the gas is definition of any standard scale, two fixed related to the temperature as V/T = constant. reference points are needed. Since all This relationship is known as Charles’ law, substances change dimensions with after French scientist Jacques Charles (1747– temperature, an absolute reference for 1823). Low-density gases obey these expansion is not available. However, the laws, which may be combined into a single necessary fixed points may be correlated to the physical phenomena that always occur at the same temperature. The ice point and the steam point of water are two convenient fixed points and are known as the freezing and boiling points, respectively. These two points are the temperatures at which pure water freezes and boils under standard pressure. The two familiar temperature scales are the Fahrenheit temperature scale and the Celsius temperature scale. The ice and steam point have values 32 °F and 212 °F, respectively, on the Fahrenheit scale and 0 °C and 100 °C on the Celsius scale. On the Fahrenheit scale, there are 180 equal intervals between two reference points, and on the Celsius scale, there are 100. 2020-21

280 PHYSICS Fig. 11.2 Pressure versus temperature of a low Fig. 11.3 A plot of pressure versus temperature and density gas kept at constant volume. extrapolation of lines for low density gases indicates the same absolute zero relationship. Notice that since PV = constant temperature. and V/T = constant for a given quantity of gas, named after the British scientist Lord Kelvin. On then PV/T should also be a constant. This this scale, – 273.15 °C is taken as the zero point, relationship is known as ideal gas law. It can be that is 0 K (Fig. 11.4). written in a more general form that applies not just to a given quantity of a single gas but to any quantity of any low-density gas and is known as ideal-gas equation: PV = µR T or PV = µRT (11.2) where, µ is the number of moles in the sample of gas and R is called universal gas constant: R = 8.31 J mol–1 K–1 In Eq. 11.2, we have learnt that the pressure and volume are directly proportional to temperature : PV ∝ T. This relationship allows a gas to be used to measure temperature in a constant volume gas thermometer. Holding the volume of a gas constant, it gives P ∝T. Thus, Fig. 11.4 Comparision of the Kelvin, Celsius and with a constant-volume gas thermometer, Fahrenheit temperature scales. temperature is read in terms of pressure. A plot The size of unit in Kelvin and Celsius of pressure versus temperature gives a straight temperature scales is the same. So, temperature on these scales are related by line in this case, as shown in Fig. 11.2. However, measurements on real gases deviate T = tC + 273.15 (11.3) from the values predicted by the ideal gas law at low temperature. But the relationship is linear 11.5 THERMAL EXPANSION over a large temperature range, and it looks as You may have observed that sometimes sealed bottles with metallic lids are so tightly screwed though the pressure might reach zero with that one has to put the lid in hot water for some time to open it. This would allow the metallic lid decreasing temperature if the gas continued to to expand, thereby loosening it to unscrew easily. In case of liquids, you may have observed be a gas. The absolute minimum temperature that mercury in a thermometer rises, when the thermometer is put in slightly warm water. If for an ideal gas, therefore, inferred by we take out the thermometer from the warm extrapolating the straight line to the axis, as in Fig. 11.3. This temperature is found to be – 273.15 °C and is designated as absolute zero. Absolute zero is the foundation of the Kelvin temperature scale or absolute scale temperature 2020-21

THERMAL PROPERTIES OF MATTER 281 water the level of mercury falls again. Similarly, Table 11.1 Values of coef ficient of linear in case of gases, a balloon partially inflated in a expansion for some material cool room may expand to full size when placed in warm water. On the other hand, a fully Material α (10–5 K–1) inflated balloon when immersed in cold water l would start shrinking due to contraction of the Aluminium air inside. Brass 2.5 Iron It is our common experience that most Copper 1.8 substances expand on heating and contract on Silver cooling. A change in the temperature of a body Gold 1.2 causes change in its dimensions. The increase Glass (pyrex) in the dimensions of a body due to the increase Lead 1.7 in its temperature is called thermal expansion. The expansion in length is called linear 1.9 expansion. The expansion in area is called area expansion. The expansion in volume is called 1.4 volume expansion (Fig. 11.5). 0.32 0.29 Similarly, we consider the fractional change in volume, ∆V , of a substance for temperature V change ∆T and define the coefficient of volume expansion (or volume expansivity), α V as αV =  ∆V  1 (11.5) V ∆T ∆A ∆V Here αV is also a characteristic of the A V substance but is not strictly a constant. It ∆l = al ∆T = 2a l ∆T = 3a l ∆T depends in general on temperature (Fig 11.6). It l is seen that αV becomes constant only at a high temperature. (a) Linear expansion (b) Area expansion (c) Volume expansion Fig. 11.5 Thermal Expansion. If the substance is in the form of a long rod, then for small change in temperature, ∆T, the fractional change in length, ∆l/l, is directly proportional to ∆T. ∆l = α1 ∆T (11.4) l where α1 is known as the coefficient of linear Fig. 11.6 Coefficient of volume expansion of copper expansion (or linear expansivity) and is as a function of temperature. characteristic of the material of the rod. In Table Table 11.2 gives the values of coefficient of 11.1, typical average values of the coefficient of volume expansion of some common substances linear expansion for some material in the in the temperature range 0–100 °C. You can see temperature range 0 °C to 100°C are given. From that thermal expansion of these substances this Table, compare the value of αl for glass and (solids and liquids) is rather small, with material, copper. We find that copper expands about five times more than glass for the same rise in temperature. Normally, metals expand more and have relatively high values of αl. 2020-21

282 PHYSICS like pyrex glass and invar (a special iron-nickel lakes and ponds, freeze at the top first. As a lake atahlllciosoyh)Tohalabv(elienthgwapneaorflti)inciduslatmhrloayrtleotwthhevaanvluameluseeorfcoαufVr.yαFvraofnomdr cools toward 4 °C, water near the surface loses expands more than mercury for the same rise energy to the atmosphere, becomes denser, and in temperature. sinks; the warmer, less dense water near the bottom rises. However, once the colder water on Table 11.2 Values of coefficient of volume expansion for some substances top reaches temperature below 4 °C, it becomes Material α ( K–1) less dense and remains at the surface, where it v freezes. If water did not have this property, lakes Aluminium and ponds would freeze from the bottom up, Brass 7 × 10–5 which would destroy much of their animal and Iron 6 × 10–5 plant life. Paraffin 3.55 × 10–5 Glass (ordinary) 58.8 × 10–5 Gases, at ordinary temperature, expand more Glass (pyrex) 2.5 × 10–5 Hard rubber 1 × 10–5 than solids and liquids. For liquids, the Invar 2.4 × 10–4 coefficient of volume expansion is relatively Mercury 2 × 10–6 Water 18.2 × 10–5 independent of the temperature. However, for Alcohol (ethanol) 20.7 × 10–5 110 × 10–5 gases it is dependent on temperature. For an ideal gas, the coefficient of volume expansion at constant pressure can be found from the ideal gas equation: PV = µRT At constant pressure P∆V = µR ∆T Water exhibits an anomalous behaviour; it ∆V = ∆T VT contracts on heating between 0 °C and 4 °C. The volume of a given amount of water decreases i.e., αv =1 for ideal gas (11.6) as it is cooled from room temperature, until its T temperature reaches 4 °C, [Fig. 11.7(a)]. Below At 0 °C, αv = 3.7 × 10–3 K–1, which is much 4 °C, the volume increases, and therefore, the larger than that for solids and liquids. density decreases [Fig. 11.7(b)]. Equation (11.6) shows the temperature This means that water has the maximum density at 4 °C. This property has an important tdeempepnedraentucreeo. fFoαrv;aigtadseactreroaosmes with increasing environmental effect: bodies of water, such as temperature and constant pressure, αv is about 3300 × 10–6 K–1, as Temperature (°C) Temperature (°C) (a) (b) Fig. 11.7 Thermal expansion of water. 2020-21

THERMAL PROPERTIES OF MATTER 283 much as order(s) of magnitude larger than the Answer ∆A3 = (∆a) (∆b) coefficient of volume expansion of typical liquids. ∆b b There is a simple relation between the ∆Al = a (∆b) ∆a a coefficien tofolfinveoalruemxpe aenxspioann(sαilo).nIm(αavg)inaen d coefficient a cube of length, l, that expands equally in all directions, when its temperature increases by ∆T. We have s∆ol ,=∆αVl l ∆T = (l+∆l)3 – l3 3l2 ∆l (11.7) In Equation (11.7), terms in (∆l)2 and (∆l)3 have been neglected since ∆l is small compared to l. So ∆A2 = b (∆a) ∆V = 3V ∆l = 3V αl ∆T Fig. 11.8 l (11.8) Consider a rectangular sheet of the solid which gives material of length a and breadth b (Fig. 11.8 ). αv = 3αl (11.9) When the temperature increases by ∆T, a i=nαcrlb∆∆eAAa∆sTe==.sFa∆brAyo∆1mb∆+a+∆FAi=bg2.α∆+1laa1∆∆+.A8T(3,∆aatnh)de(b∆inbinc) rceraeaseseisn by ∆b What happens by preventing the thermal area expansion of a rod by fixing its ends rigidly? Clearly, the rod acquires a compressive strain due to the external forces provided by the rigid = a αlb ∆T + b ααll a ∆T +αl(Aαl)∆2 Ta(b2(+∆Tα)l2∆T) support at the ends. The corresponding stress = αl ab ∆T (2 + ∆T) = set up in the rod is called thermal stress. For example, consider a steel rail of length 5 m and Since αl 10–5 K–1, from Table 11.1, the area of cross-section 40 cm2 that is prevented pinrocdoumcpt aαrl i∆sTiofnorwfritahct2ioannadl tmemapyebreatnuergeleisctsemda. ll from expanding while the temperature rises by Hence, 10 °C. The coefficient of linear expansion of steel is α =l(steel) 1.2 × 10–5 K–1. Thus, the compressive strain is ∆l = αl(steel) ∆T = 1.2 × 10–5 × 10=1.2 × 10–4. l Example 11.2 A blacksmith fixes iron ring Youngs modulus of steel is Yst(srteeesl) s= 2 × 1011 N m–2. on the rim of the wooden wheel of a horse Therefore, the thermal developed is cart. The diameter of the rim and the iron ring are 5.243 m and 5.231 m, respectively ∆F = Ysteel  ∆l  = 2.4 × 107 N m–2, which at 27 °C. To what temperature should the A l ring be heated so as to fit the rim of the wheel? corresponds to an external force of ∆F = AYsteel  ∆l  = 2.4 × 107 × 40 × 10–4 j 105N. Answer l If two such steel rails, fixed at their outer ends, Given, T1 = 27 °C are in contact at their inner ends, a force of this magnitude can easily bend the rails. LT1 = 5.231 m Example 11.1 Show that the coefficient LT2 = 5.243 m So, of area expansion, (∆A/A)/∆T, of a rectangular sheet of the solid is twice its LT2 =LT1 [1+αl (T2–T1)] linear expansivity, αl. 5.243 m = 5.231 m [1 + 1.20×10–5 K–1 (T2–27 °C)] or T2 = 218 °C. 2020-21

284 PHYSICS 11.6 SPECIFIC HEAT CAPACITY heat absorbed or given off to change the Take some water in a vessel and start heating it temperature of unit mass of it by one unit. This on a burner. Soon you will notice that bubbles begin to move upward. As the temperature is quantity is referred to as the specific heat raised the motion of water particles increases capacity of the substance. till it becomes turbulent as water starts boiling. What are the factors on which the quantity of If ∆Q stands for the amount of heat absorbed heat required to raise the temperature of a or given off by a substance of mass m when it substance depend? In order to answer this undergoes a temperature change ∆T, then the question in the first step, heat a given quantity specific heat capacity, of that substance is given of water to raise its temperature by, say 20 °C and note the time taken. Again take the same by amount of water and raise its temperature by 40 °C using the same source of heat. Note the s= S =1 ∆Q (11.11) time taken by using a stopwatch. You will find m m ∆T it takes about twice the time and therefore, double the quantity of heat required raising twice The specific heat capacity is the property of the temperature of same amount of water. the substance which determines the change in In the second step, now suppose you take double the amount of water and heat it, using the temperature of the substance (undergoing the same heating arrangement, to raise the temperature by 20 °C, you will find the time no phase change) when a given quantity of heat taken is again twice that required in the first step. is absorbed (or given off) by it. It is defined as the In the third step, in place of water, now heat amount of heat per unit mass absorbed or given the same quantity of some oil, say mustard oil, and raise the temperature again by 20 °C. Now off by the substance to change its temperature note the time by the same stopwatch. You will find the time taken will be shorter and therefore, by one unit. It depends on the nature of the the quantity of heat required would be less than that required by the same amount of water for substance and its temperature. The SI unit of the same rise in temperature. specific heat capacity is J kg–1 K–1. The above observations show that the quantity of heat required to warm a given substance If the amount of substance is specified in depends on its mass, m, the change in temperature, ∆T and the nature of substance. terms of moles µ, instead of mass m in kg, we The change in temperature of a substance, when can define heat capacity per mole of the a given quantity of heat is absorbed or rejected by it, is characterised by a quantity called the substance by heat capacity of that substance. We define heat capacity, S of a substance as C=S =1 ∆Q (11.12) µµ ∆T where C is known as molar specific heat capacity of the substance. Like S, C also depends on the nature of the substance and its temperature. The SI unit of molar specific heat capacity is J mol–1 K–1. However, in connection with specific heat capacity of gases, additional conditions may be needed to define C. In this case, heat transfer can be achieved by keeping either pressure or volume constant. If the gas is held under constant pressure during the heat transfer, then S = ∆Q (11.10) it is called the molar specific heat capacity at ∆T constant pressure and is denoted by Cp. On the other hand, if the volume of the gas is maintained during the heat transfer, then the where ∆Q is the amount of heat supplied to corresponding molar specific heat capacity is the substance to change its temperature from T called molar specific heat capacity at constant volume and is denoted by Cv. For details see to T + ∆T. Chapter 12. Table 11.3 lists measured specific You have observed that if equal amount of heat capacity of some substances at atmospheric heat is added to equal masses of different pressure and ordinary temperature while Table substances, the resulting temperature changes 11.4 lists molar specific heat capacities of some will not be the same. It implies that every gases. From Table 11.3 you can note that water substance has a unique value for the amount of 2020-21

THERMAL PROPERTIES OF MATTER 285 Table 11.3 Specific heat capacity of some substances at room temperature and atmospheric pressure Substance Specific heat capacity Substance Specific heat capacity (J kg–1 K–1) (J kg–1 K–1) Aluminium 900.0 Ice 2060 Carbon 506.5 Glass 840 Copper 386.4 Iron 450 Lead 127.7 Kerosene Silver 236.1 Edible oil 2118 Tungesten 134.4 Mercury 1965 Water 4186.0 140 has the highest specific heat capacity compared equal to the heat gained by the colder body, to other substances. For this reason water is also provided no heat is allowed to escape to the used as a coolant in automobile radiators, as surroundings. A device in which heat well as, a heater in hot water bags. Owing to its measurement can be done is called a high specific heat capacity, water warms up calorimeter. It consists of a metallic vessel and more slowly than land during summer, and stirrer of the same material, like copper or consequently wind from the sea has a cooling aluminium. The vessel is kept inside a wooden effect. Now, you can tell why in desert areas, jacket, which contains heat insulating material, the earth surface warms up quickly during the like glass wool etc. The outer jacket acts as a day and cools quickly at night. heat shield and reduces the heat loss from the inner vessel. There is an opening in the outer Table 11.4 Molar specific heat capacities of jacket through which a mercury thermometer can be inserted into the calorimeter (Fig. 11.20). some gases The following example provides a method by which the specific heat capacity of a given solid Gas Cp (J mol–1K–1) Cv(J mol–1K–1) can be determinated by using the principle, heat gained is equal to the heat lost. He 20.8 12.5 Example 11.3 A sphere of 0.047 kg H2 28.8 20.4 aluminium is placed for sufficient time in a N2 29.1 20.8 vessel containing boiling water, so that the O2 29.4 21.1 sphere is at 100 °C. It is then immediately CO2 37.0 28.5 transfered to 0.14 kg copper calorimeter containing 0.25 kg water at 20 °C. The 11.7 CALORIMETRY temperature of water rises and attains a steady state at 23 °C. Calculate the specific A system is said to be isolated if no exchange or heat capacity of aluminium. transfer of heat occurs between the system and its surroundings. When different parts of an Answer In solving this example, we shall use isolated system are at different temperature, a the fact that at a steady state, heat given by an quantity of heat transfers from the part at higher aluminium sphere will be equal to the heat temperature to the part at lower temperature. absorbed by the water and calorimeter. The heat lost by the part at higher temperature MInaitsiaslotfeamlupemraintuiurme osf aplhuemrein(miu1m) =sp0h.0e4re7=k10g0 °C is equal to the heat gained by the part at lower Final temperature = 23 °C temperature. Change in temperature (∆T)=(100 °C -23 °C) = 77 °C Let specific heat capacity of aluminium be sAl. Calorimetry means measurement of heat. When a body at higher temperature is brought in contact with another body at lower temperature, the heat lost by the hot body is 2020-21

286 PHYSICS The amount of heat lost by the aluminium Fig. 11.9 A plot of temperature versus time showing sphere = m1sAl ∆T = 0.047kg × sAl × 77 °C the changes in the state of ice on heating Mass of water (m2) = 0.25 kg Mass of calorimeter (m3) = 0.14 kg (not to scale). Initial temperature of water and calorimeter=20 °C The change of state from solid to liquid is Final temperature of the mixture = 23 °C called melting and from liquid to solid is called fusion. It is observed that the temperature Change in temperature (∆T2) = 23 °C – 20 °C = 3 °C remains constant until the entire amount of the Specific heat capacity of water (sw) solid substance melts. That is, both the solid = 4.18 × 103 J kg–1 K–1 and the liquid states of the substance coexist Specific heat capacity of copper calorimeter in thermal equilibrium during the change of states from solid to liquid. The temperature = 0.386 × 103 J kg–1 K–1 at which the solid and the liquid states of the substance is in thermal equilibrium with each The amount of heat gained by water and other is called its melting point. It is calorimeter = m2 sw ∆T2 + m3scu∆T2 characteristic of the substance. It also depends on pressure. The melting point of a substance = (m2sw + m3scu) (∆T2) at standard atomspheric pressure is called its = (0.25 kg × 4.18 × 103 J kg–1 K–1 + 0.14 kg × normal melting point. Let us do the following activity to understand the process of melting 0.386 × 103 J kg–1 K–1) (23 °C – 20 °C) of ice. In the steady state heat lost by the aluminium Take a slab of ice. Take a metallic wire and sphere = heat gained by water + heat gained by fix two blocks, say 5 kg each, at its ends. Put calorimeter. the wire over the slab as shown in Fig. 11.10. You will observe that the wire passes through So, 0.047 kg × sAl × 77 °C the ice slab. This happens due to the fact that = (0.25 kg × 4.18 × 103 J kg–1 K–1+ 0.14 kg × just below the wire, ice melts at lower temperature due to increase in pressure. When 0.386 × 103 J kg–1 K–1)(3 °C) the wire has passed, water above the wire freezes again. Thus, the wire passes through the slab sAl = 0.911 kJ kg –1 K–1 and the slab does not split. This phenomenon of refreezing is called regelation. Skating is 11.8 CHANGE OF STATE possible on snow due to the formation of water under the skates. Water is formed due to the Matter normally exists in three states: solid, increase of pressure and it acts as a liquid and gas. A transition from one of these lubricant. states to another is called a change of state. Two common changes of states are solid to liquid and liquid to gas (and, vice versa). These changes can occur when the exchange of heat takes place between the substance and its surroundings. To study the change of state on heating or cooling, let us perform the following activity. Take some cubes of ice in a beaker. Note the temperature of ice. Start heating it slowly on a constant heat source. Note the temperature after every minute. Continuously stir the mixture of water and ice. Draw a graph between temperature and time (Fig. 11.9). You will observe no change in the temperature as long as there is ice in the beaker. In the above process, the temperature of the system does not change even though heat is being continuously supplied. The heat supplied is being utilised in changing the state from solid (ice) to liquid (water). 2020-21

THERMAL PROPERTIES OF MATTER 287 Fig. 11.10 100 °C when it again becomes steady. The heat supplied is now being utilised to change water After the whole of ice gets converted into water from liquid state to vapour or gaseous state. and as we continue further heating, we shall see that temperature begins to rise (Fig.11.9). The The change of state from liquid to vapour (or temperature keeps on rising till it reaches nearly gas) is called vaporisation. It is observed that the temperature remains constant until the entire amount of the liquid is converted into vapour. That is, both the liquid and vapour states of the substance coexist in thermal equilibrium, during the change of state from liquid to vapour. The temperature at which the liquid and the vapour states of the substance coexist is called its boiling point. Let us do the following activity to understand the process of boiling of water. Take a round-bottom flask, more than half filled with water. Keep it over a burner and fix a Triple Point The temperature of a substance remains constant during its change of state (phase change). A graph between the temperature T and the Pressure P of the substance is called a phase diagram or P – T diagram. The following figure shows the phase diagram of water and CO2. Such a phase diagram divides the P – T plane into a solid-region, the vapour-region and the liquid-region. The regions are separated by the curves such as sublimation curve (BO), fusion curve (AO) and vaporisation curve (CO). The points on sublimation curve represent states in which solid and vapour phases coexist. The point on the sublimation curve BO represent states in which the solid and vapour phases co-exist. Points on the fusion curve AO represent states in which solid and liquid phase coexist. Points on the vapourisation curve CO represent states in which the liquid and vapour phases coexist. The temperature and pressure at which the fusion curve, the vaporisation curve and the sublimation curve meet and all the three phases of a substance coexist is called the triple point of the substance. For example the triple point of water is represented by the temperature 273.16 K and pressure 6.11×10–3 Pa. (a) (b) Fig. 11.11: Pressure-temperature phase diagrams for (a) water and (b) CO2 (not to the scale). 2020-21

288 PHYSICS thermometer and steam outlet through the cork cork. Keep the f lask turned upside down on the stand. Pour ice-cold water on the flask. Water of the flask (Fig. 11.11). As water gets heated in vapours in the flask condense reducing the the flask, note first that the air, which was pressure on the water surface inside the flask. dissolved in the water, will come out as small Water begins to boil again, now at a lower bubbles. Later, bubbles of steam will form at temperature. Thus boiling point decreases with the bottom but as they rise to the cooler water decrease in pressure. near the top, they condense and disappear. Finally, as the temperature of the entire mass This explains why cooking is difficult on hills. of the water reaches 100 °C, bubbles of steam At high altitudes, atmospheric pressure is lower, reach the surface and boiling is said to occur. reducing the boiling point of water as compared The steam in the flask may not be visible but as to that at sea level. On the other hand, boiling it comes out of the flask, it condenses as tiny point is increased inside a pressure cooker by droplets of water, giving a foggy appearance. increasing the pressure. Hence cooking is faster. The boiling point of a substance at standard atmospheric pressure is called its normal boiling point. However, all substances do not pass through the three states: solid-liquid-gas. There are certain substances which normally pass from the solid to the vapour state directly and vice versa. The change from solid state to vapour state without passing through the liquid state is called sublimation, and the substance is said to sublime. Dry ice (solid CO2) sublimes, so also iodine. During the sublimation process both the solid and vapour states of a substance coexist in thermal equilibrium. Fig. 11.11 Boiling process. 11.8.1 Latent Heat If now the steam outlet is closed for a few In Section 11.8, we have learnt that certain seconds to increase the pressure in the flask, amount of heat energy is transferred between a you will notice that boiling stops. More heat substance and its surroundings when it would be required to raise the temperature undergoes a change of state. The amount of heat (depending on the increase in pressure) before per unit mass transferred during change of state boiling begins again. Thus boiling point increases of the substance is called latent heat of the with increase in pressure. substance for the process. For example, if heat is added to a given quantity of ice at –10 °C, the Let us now remove the burner. Allow water to temperature of ice increases until it reaches its cool to about 80 °C. Remove the thermometer and melting point (0 °C). At this temperature, the steam outlet. Close the flask with the airtight addition of more heat does not increase the temperature but causes the ice to melt, or changes its state. Once the entire ice melts, adding more heat will cause the temperature of the water to rise. A similar situation occurs during liquid gas change of state at the boiling point. Adding more heat to boiling water causes vaporisation, without increase in temperature. 2020-21

THERMAL PROPERTIES OF MATTER 289 Table 11.5 Temperatures of the change of state and latent heats for various substances at 1 atm pressure Substance Melting (105JLfkg–1) Boiling (105JLvkg–1) Point (°C) Point (°C) Ethanol 1.0 8.5 Gold –114 0.645 78 15.8 Lead 1063 0.25 2660 Mercury 0.12 1744 8.67 Nitrogen 328 0.26 2.7 Oxygen –39 0.14 357 2.0 Water –210 3.33 –196 2.1 –219 –183 22.6 0 100 The heat required during a change of state Note that when heat is added (or removed) during a change of state, the temperature depends upon the heat of transformation and remains constant. Note in Fig. 11.12 that the slopes of the phase lines are not all the same, the mass of the substance undergoing a change which indicate that specific heats of the various states are not equal. For water, the latent heat of of state. Thus, if mass m of a substance fusion and vaporisation are Lf = 3.33 × 105 J kg–1 undergoes a change from one state to the other, and Lv = 22.6 × 105 J kg–1, respectively. That is, then the quantity of heat required is given by 3.33 × 105 J of heat is needed to melt 1 kg ice at 0 °C, and 22.6 × 105 J of heat is needed to convert Q=mL 1 kg water into steam at 100 °C. So, steam at 100 °C carries 22.6 × 105 J kg–1 more heat than or L = Q/m (11.13) water at 100 °C. This is why burns from steam are usually more serious than those from where L is known as latent heat and is a boiling water. characteristic of the substance. Its SI unit is Example 11.4 When 0.15 kg of ice at 0 °C is mixed with 0.30 kg of water at 50 °C in a J kg–1. The value of L also depends on the container, the resulting temperature is 6.7 °C. Calculate the heat of fusion of ice. pressure. Its value is usually quoted at standard (swater = 4186 J kg–1 K–1) atmospheric pressure. The latent heat for a solid- Answer Heat lost by water = msw (θf–θi)w liquid state change is called the latent heat of = (0.30 kg ) (4186 J kg–1 K–1) (50.0 °C – 6.7 °C) fusion (Lf), and that for a liquid-gas state change = 54376.14 J is called the latent heat of vaporisation (Lv). Heat required to melt ice = m2Lf = (0.15 kg) Lf These are often referred to as the heat of fusion Heat required to raise temperature of ice and the heat of vaporisation. A plot of water to final temperature = mIsw (θf–θi)I = (0.15 kg) (4186 J kg–1 K –1) (6.7 °C – 0 °C) temperature versus heat for a quantity of water = 4206.93 J Heat lost = heat gained is shown in Fig. 11.12. The latent heats of some 54376.14 J = (0.15 kg ) Lf + 4206.93 J Lf = 3.34×105 J kg–1. substances, their freezing and boiling points, are given in Table 11.5. Fig. 11.12 Temperature versus heat for water at 1 atm pressure (not to scale). 2020-21

290 PHYSICS Example 11.5 Calculate the heat required temperature difference. What are the different to convert 3 kg of ice at –12 °C kept in a ways by which this energy transfer takes calorimeter to steam at 100 °C at place? There are three distinct modes of heat atmospheric pressure. Given specific heat transfer: conduction, convection and radiation capacity of ice = 2100 J kg–1 K–1, specific heat (Fig. 11.13). capacity of water = 4186 J kg– 1 K–1, latent heat of fusion of ice = 3.35 × 105 J kg–1 Fig. 11.13 Heating by conduction, convection and and latent heat of steam = 2.256 ×106 J kg–1. radiation. Answer We have Mass of the ice, m = 3 kg specific heat capacity of ice, sice = 2100 J kg–1 K–1 specific heat capacity of water, swater = 4186 J kg–1 K–1 latent heat of fusion of ice, Lf ice = 3.35 × 105 J kg–1 latent heat of steam, Lsteam = 2.256 × 106 J kg–1 Now, Q = heat required to convert 3 kg of 11.9.1 Conduction So, ice at –12 °C to steam at 100 °C, Conduction is the mechanism of transfer of heat Q1 = heat required to convert ice at between two adjacent parts of a body because –12 °C to ice at 0 °C. of their temperature difference. Suppose, one end of a metallic rod is put in a flame, the other end = m sice ∆T1 = (3 kg) (2100 J kg–1. of the rod will soon be so hot that you cannot K–1) [0–(–12)]°C = 75600 J hold it by your bare hands. Here, heat transfer takes place by conduction from the hot end of Q2 = heat required to melt ice at the rod through its different parts to the other 0 °C to water at 0 °C end. Gases are poor thermal conductors, while liquids have conductivities intermediate between = m Lf ice = (3 kg) (3.35 × 105 J kg–1) solids and gases. = 1005000 J Q3 = heat required to convert water Heat conduction may be described quantitatively as the time rate of heat flow in a at 0 °C to water at 100 °C. material for a given temperature difference. = msw ∆T2 = (3kg) (4186J kg–1 K–1) Consider a metallic bar of length L and uniform cross-section A with its two ends maintained at (100 °C) different temperatures. This can be done, for = 1255800 J example, by putting the ends in thermal contact Q4 = heat required to convert water wTidDiet,ahrlelcasorpgneedcrtitievisoeenlryvtho(Fiaritgs.tah1te1te.s1mid4pe).esrLoaefttuthureessb,aassrasyua,mrTeeCfautnhlldye insulated so that no heat is exchanged between at 100 °C to steam at 100 °C. the sides and the surroundings. = m Lsteam = (3 kg) (2.256×106 After sometime, a steady state is reached; the J kg–1) temperature of the bar decreases uniformly with = 6768000 J dCisstaunpcpelifersomheTaCttoaTt Da; (TcCo>nTsDt)a. nTht eraretsee, rvwohiricaht Q = Q1 + Q2 + Q3 + Q4 transfers through the bar and is given out at = 75600J + 1005000 J the same rate to the reservoir at D. It is found + 1255800 J + 6768000 J = 9.1×106 J 11.9 HEAT TRANSFER We have seen that heat is energy transfer from one system to another or from one part of a system to another part, arising due to 2020-21

THERMAL PROPERTIES OF MATTER 291 prohibited and keeps the room cooler. In some situations, heat transfer is critical. In a nuclear reactor, for example, elaborate heat transfer systems need to be installed so that the enormous energy produced by nuclear fission in the core transits out sufficiently fast, thus preventing the core from overheating. Table 11.6 Thermal conductivities of some material Fig. 11.14 Steady state heat flow by conduction in Material Thermal conductivity (J s–1 m–1 K–1 ) a bar with its two ends maintained at Metals temperatures TC and TD; (TC > TD). 406 Silver 385 experimentally that in this steady state, the rate Copper 205 of flow of heat (or heat current) H is proportional Aluminium 109 to the temperature difference (TC – TD) and the Brass 50.2 area of cross-section A and is inversely Steel 34.7 proportional to the length L : Lead 8.3 Mercury H = KA TC – TD (11.14) L The constant of proportionality K is called the thermal conductivity of the material. The Non-metals greater the value of K for a material, the more Insulating brick Concrete rapidly will it conduct heat. The SI unit of K is Body fat 0.15 Felt 0.8 J s–1 m–1 K–1 or W m–1 K–1. The thermal Glass Ice 0.20 conductivities of various substances are listed Glass wool 0.04 Wood in Table 11.6. These values vary slightly with Water 0.8 1.6 temperature, but can be considered to be 0.04 0.12 constant over a normal temperature range. 0.8 Compare the relatively large thermal conductivities of good thermal conductors and, metals, with the relatively small thermal conductivities of some good thermal insulators, such as wood and glass wool. You may have Gases noticed that some cooking pots have copper coating on the bottom. Being a good conductor Air 0.024 Argon 0.016 of heat, copper promotes the distribution of heat Hydrogen 0.14 over the bottom of a pot for uniform cooking. Plastic foams, on the other hand, are good Example 11.6 What is the temperature of the steel-copper junction in the steady insulators, mainly because they contain pockets state of the system shown in Fig. 11.15. Length of the steel rod = 15.0 cm, length of air. Recall that gases are poor conductors, of the copper rod = 10.0 cm, temperature of the furnace = 300 °C, temperature of and note the low thermal conductivity of air in the other end = 0 °C. The area of cross section of the steel rod is twice that of the the Table 11.5. Heat retention and transfer are copper rod. (Thermal conductivity of steel = 50.2 J s–1 m–1K–1; and of copper important in many other applications. Houses = 385 J s–1m–1K–1). made of concrete roofs get very hot during summer days because thermal conductivity of concrete (though much smaller than that of a metal) is still not small enough. Therefore, people, usually, prefer to give a layer of earth or foam insulation on the ceiling so that heat transfer is 2020-21

292 PHYSICS Answer Given, L1 = L2= L = 0.1 m, A1 = A2= A= 0.02 m2 K1 = 79 W m–1 K–1, K2 = 109 W m–1 K–1, T1 = 373 K, and yT2 = 273 K. Under s tead st ate con di t i o n, the heat Fig. 11.15 current (H1) through iron bar is equal to the heat current (=HH2)2through brass bar. Answer The insulating material around the So, H = H1 rods reduces heat loss from the sides of the rods. Therefore, heat flows only along the length of ( )= the rods. Consider any cross section of the rod. K1A1 T1 – T0 = K 2 A2 (T0 – T2 ) In the steady state, heat flowing into the element L1 L2 must equal the heat flowing out of it; otherwise there would be a net gain or loss of heat by the lFeoardAs 1to= A2 = A and L1 = L2 = L, this equation element and its temperature would not be steady. Thus in the steady state, rate of heat TKh1 u(Ts1,–thT0e) =juKn2ct(Tio0n– tTe2m) perature T0 of the two flowing across a cross section of the rod is the bars is same at every point along the length of the combined steel-copper rod. Let T be the T0 = (K1T1 + K2T2 ) temperature of the steel-copper junction in the (K1 + K2 ) steady state. Then, Using this equation, the heat current H through either bar is K1 A1 (300 − T ) = K2 A2 (T – 0) ( )H = K1A T1 – T0 = K2 A(T0 – T2 ) L1 L 2 LL where 1 and 2 refer to the steel and copper rod sLre–21s=mp1e–01ct.K0iv–ce1,mlyw,.eKh1Fa=ovr5e0A.12 = 2 Am2,–1 LK1 = 15.0 cm, J s–1 –1, K2 = 385 J 50.2 × 2 (300 − T ) 385T Using these equations, the heat current H′ through tehqeucivoamlepnotutnhderbmaraol fcloenndguthctLiv1 i+tyL2K=′, 2L 15 = 10 and the of which gives T = 44.4 °C the compound bar are given by b=Erxa0as.m0s2pbleamr12,1(L.K72 1A==n0i7.r1o9nmWb,armA(2L–11=K=–001)..01a2mndm, A2a1, H′= K′ A (T1 – T2 ) = H 2L aKs2 = 109 W m–1K–1) are soldered end to end K ′ = 2 K1 K2 shown in Fig. 11.16. The free ends of K1 + K2 the iron bar and brass bar are maintained = (K1T1 + K2T2 ) (K1 + K2 ) at 373 K and 273 K respectively. Obtain (i) T0 expressions for and hence compute (i) the ( ) ( )( ) ( )79 W m–1K –1 373 K + 109 W m–1K –1 273 K temperature of the junction of the two bars, = 79 W m –1K –1 + 109 W m –1K –1 (ii) the equivalent thermal conductivity of the compound bar, and (iii) the heat current through the compound bar. = 315 K (ii) K′ = 2K1 K2 K1 + K2 2 × (79 W m–1 K –1 ) × (109 W m –1 K –1 ) = 79 W m–1 K –1 +109 W m–1 K –1 Fig 11.16 = 91.6 W m–1 K–1 2020-21

THERMAL PROPERTIES OF MATTER 293 (iii) H′=H = K′ A (T1 – T2 ) of water do. This occurs both because water has a greater specific heat capacity and because 2L mixing currents disperse the absorbed heat ( ) ( )= × (373 K–273 K ) throughout the great volume of water. The air 91.6 W m–1 K –1 × 0.02 m2 in contact with the warm ground is heated by conduction. It expands, becoming less dense 2× (0.1 m) than the surrounding cooler air. As a result, the = 916.1 W warm air rises (air currents) and the other air moves (winds) to fill the space-creating a sea 11.9.2 Convection breeze near a large body of water. Cooler air descends, and a thermal convection cycle is set Convection is a mode of heat transfer by actual motion of matter. It is possible only in fluids. up, which transfers heat away from the land. Convection can be natural or forced. In natural At night, the ground loses its heat more quickly, convection, gravity plays an important part. and the water surface is warmer than the land. When a fluid is heated from below, the hot part As a result, the cycle is reveresed (Fig. 11.17). expands and, therefore, becomes less dense. Because of buoyancy, it rises and the upper The other example of natural convection is colder part replaces it. This again gets heated, the steady surface wind on the earth blowing rises up and is replaced by the relatively colder in from north-east towards the equator, the part of the fluid. The process goes on. This mode so-called trade wind. A resonable explanation of heat transfer is evidently different from conduction. Convection involves bulk transport is as follows: the equatorial and polar regions of of different parts of the fluid. the earth receive unequal solar heat. Air at the earth’s surface near the equator is hot, while In forced convection, material is forced to move the air in the upper atmosphere of the poles is by a pump or by some other physical means. The common examples of forced convection systems cool. In the absence of any other factor, a are forced-air heating systems in home, the convection current would be set up, with the human circulatory system, and the cooling air at the equatorial surface rising and moving system of an automobile engine. In the human out towards the poles, descending and body, the heart acts as the pump that circulates blood through different parts of the body, streaming in towards the equator. The rotation transferring heat by forced convection and of the earth, however, modifies this convection maintaining it at a uniform temperature. current. Because of this, air close to the equator has an eastward speed of 1600 km/h, while it Natural convection is responsible for many familiar phenomena. During the day, the is zero close to the poles. As a result, the air ground heats up more quickly than large bodies descends not at the poles but at 30° N (North) latitude and returns to the equator. This is called trade wind. Fig. 11.17 Convection cycles. 2020-21

294 PHYSICS 11.9.3 Radiation contents of the bottle. The outer wall similarly reflects back any incoming radiation. The space Conduction and convection require some between the walls is evacuted to reduce material as a transport medium. These modes conduction and convection losses and the flask of heat transfer cannot operate between bodies is supported on an insulator, like cork. The separated by a distance in vacuum. But the device is, therefore, useful for preventing hot earth does receive heat from the Sun across a contents (like, milk) from getting cold, or huge distance. Similarly, we quickly feel the alternatively, to store cold contents (like, ice). warmth of the fire nearby even though air conducts poorly and before convection takes 11.9.4 Blackbody Radiation some time to set in. The third mechanism for heat transfer needs no medium; it is called We have so far not mentioned the wavelength radiation and the energy so transferred by content of thermal radiation. The important electromagnetic waves is called radiant energy. thing about thermal radiation at any In an electromagnetic wave, electric and temperature is that it is not of one (or a few) magnetic fields oscillate in space and time. Like wavelength(s) but has a continuous spectrum any wave, electromagnetic waves can have from the small to the long wavelengths. The different wavelengths and can travel in vacuum energy content of radiation, however, varies for with the same speed, namely the speed of light different wavelengths. Figure 11.18 gives the i.e., 3 × 108 m s–1 . You will learn these matters experimental curves for radiation energy per unit in more detail later, but you now know why heat area per unit wavelength emitted by a blackbody transfer by radiation does not need any medium versus wavelength for different temperatures. and why it is so fast. This is how heat is transferred to the earth from the Sun through Fig. 11.18: Energy emitted versus wavelength empty space. All bodies emit radiant energy, for a blackbody at different whether they are solid, liquid or gas. The temperatures electromagnetic radiation emitted by a body by virtue of its temperature, like radiation by a red Notice that the wavelength λ for which energy hot iron or light from a filament lamp is called is the maximum decreasesm with increasing thermal radiation. temperature. iTshkenorewlantaiosnWbieetnw’seeDnisλpmlaacnedmTenist When this thermal radiation falls on other given by what bodies, it is partly reflected and partly absorbed. The amount of heat that a body can absorb by Law: radiation depends on the colour of the body. λ T = constant (11.15) We find that black bodies absorb and emit m radiant energy better than bodies of lighter colours. This fact finds many applications in our The value of the constant (Wien’s constant) daily life. We wear white or light coloured clothes is 2.9 × 10–3 m K. This law explains why the in summer, so that they absorb the least heat colour of a piece of iron heated in a hot flame from the Sun. However, during winter, we use first becomes dull red, then reddish yellow, and dark coloured clothes, which absorb heat from the sun and keep our body warm. The bottoms of finally white hot. Wien’s law is useful for utensils for cooking food are blackened so that estimating the surface temperatures of celestial they absorb maximum heat from fire and transfer it to the vegetables to be cooked. Similarly, a Dewar flask or thermos bottle is a device to minimise heat transfer between the contents of the bottle and outside. It consists of a double-walled glass vessel with the inner and outer walls coated with silver. Radiation from the inner wall is reflected back to the 2020-21

THERMAL PROPERTIES OF MATTER 295 bodies like, the moon, Sun and other stars. Light For a body with emissivity e, the relation modifies to from the moon is found to have a maximum intensity near the wavelength 14 µm. By Wien’s H = eσ A (T4 – Ts4) (11.18) law, the surface of the moon is estimated to have As an example, let us estimate the heat a temperature of 200 K. Solar radiation has a maximum at λm = 4753 Å. This corresponds to radiated by our bodies. Suppose the surface area T = 6060 K. Remember, this is the temperature of a person’s body is about 1.9 m2 and the room of the surface of the sun, not its interior. The most significant feature of the temperature is 22°C. The internal body blackbody radiation curves in Fig. 11.18 is that temperature, as we know, is about 37°C. The they are universal. They depend only on the temperature and not on the size, shape or skin temperature may be 28°C (say). The material of the blackbody. Attempts to explain blackbody radiation theoretically, at the emissivity of the skin is about 0.97 for the beginning of the twentieth century, spurred the quantum revolution in physics, as you will relevant region of electromagnetic radiation. The learn in later courses. rate of heat loss is: Energy can be transferred by radiation over H = 5.67 × 10–8 × 1.9 × 0.97 × {(301)4 – (295)4} large distances, without a medium (i.e., in = 66.4 W vacuum). The total electromagnetic energy which is more than half the rate of energy radiated by a body at absolute temperature T production by the body at rest (120 W). To is proportional to its size, its ability to radiate prevent this heat loss effectively (better than ordinary clothing), modern arctic clothing has (called emissivity) and most importantly to its an additional thin shiny metallic layer next to the skin, which reflects the body’s radiation. temperature. For a body, which is a perfect 11.9.5 Greenhouse Effect radiator, the energy emitted per unit time (H) The earth’s surface is a source of thermal is given by radiation as it absorbs energy received from the Sun. The wavelength of this radiation lies in the H = AσT 4 (11.16) long wavelength (infrared) region. But a large portion of this radiation is absorbed by where A is the area and T is the absolute greenhouse gases, namely, carbon dioxide temperature of the body. This relation obtained (cChOlo2r)o;flmuoertohcaanrbeo(nCH(C4F);xCnlix)t;roaunsd otrxoipdoesp(Nh2eOri)c; experimentally by Stefan and later proved ionzotnuern(,Og3i)v. eTshmisohreeaetsnuerpgtyhteoaetamrtohs,prheesruelwtinhigchin, theoretically by Boltzmann is known as Stefan- warmer surface. This increases the intensity of Boltzmann law and the constant σ is called radiation from the surface. The cycle of Stefan-Boltzmann constant. Its value in SI units processes described above is repeated until no is 5.67 × 10–8 W m–2 K–4. Most bodies emit only a radiation is available for absorption. The net fraction of the rate given by Eq. 11.16. A substance result is heating up of earth’s surface and like lamp black comes close to the limit. One, atmosphere. This is known as Greenhouse therefore, defines a dimensionless fraction e Effect. Without the Greenhouse Effect, the called emissivity and writes, temperature of the earth would have been –18°C. H = AeσT 4 (11.17) Concentration of greenhouse gases has enhanced due to human activities, making the Here, e = 1 for a perfect radiator. For a tungsten earth warmer. According to an estimate, average temperature of earth has increased by 0.3 to lamp, for example, e is about 0.4. Thus, a tungsten 0.6°C, since the beginning of this century because of this enhancement. By the middle of lamp at a temperature of 3000 K and a surface the next century, the earth’s global temperature may be 1 to 3°C higher than today. This global area of 0.3 cm2 radiates at the rate H = 0.3 × 10–4 × 0.4 × 5.67 × 10–8 × (3000)4 = 60 W. A body at temperature T, with surroundings at temperatures Ts, emits, as well as, receives energy. For a perfect radiator, the net rate of loss of radiant energy is H = σA (T 4 – Ts4) 2020-21

296 PHYSICS warming may cause problem for human life, From the graph you can infer how the cooling plants and animals. Because of global warming, of hot water depends on the difference of its ice caps are melting faster, sea level is rising, temperature from that of the surroundings. You and weather pattern is changing. Many coastal will also notice that initially the rate of cooling cities are at the risk of getting submerged. The is higher and decreases as the temperature of enhanced Greenhouse Effect may also result in the body falls. expansion of deserts. All over the world, efforts are being made to minimise the effect of global The above activity shows that a hot body loses warming. heat to its surroundings in the form of heat radiation. The rate of loss of heat depends on 11.10 NEWTON’S LAW OF COOLING the difference in temperature between the body and its surroundings. Newton was the first to We all know that hot water or milk when left on study, in a systematic manner, the relation a table begins to cool, gradually. Ultimately it between the heat lost by a body in a given attains the temperature of the surroundings. To enclosure and its temperature. study how slow or fast a given body can cool on exchanging heat with its surroundings, let us According to Newton’s law of cooling, the rate perform the following activity. of loss of heat, – dQ/dt of the body is directly proportional to the difference of temperature Take some water, say 300 mL, in a ∆T = (T2–T1) of the body and the surroundings. calorimeter with a stirrer and cover it with a The law holds good only for small difference of two-holed lid. Fix the stirrer through one hole temperature. Also, the loss of heat by radiation and fix a thermometer through another hole depends upon the nature of the surface of the in the lid and make sure that the bulb of body and the area of the exposed surface. We thermometer is immersed in the water. Note can write the reading of the thermometer. This reading T1 is the temperature of the surroundings. – (11.19) Heat the water kept in the calorimeter till it attains a temperature, say 40 °C above room where k is a positive constant depending upon temperature (i.e., temperature of the the area and nature of the surface of the body. surroundings). Then, stop heating the water by removing the heat source. Start the Suppose a body of mass m and specific heat stop-watch and note the reading of the thermometer after a fixed interval of time, say capacity s is at temperature uTn2.dLi netg sT.1 be the after every one minute of stirring gently with tempera ture o f the surro If the the stirrer. Continue to note the temperature (T2) of water till it attains a temperature about temperature falls by a small amount dT2 in time 5 °C above that of the surroundings. Then, plot dt, then the amount of heat lost is a graph by taking each value of temperature ∆T = T2 – T1 along y-axis and the coresponding dQ = ms dT2 value of t along x-axis (Fig. 11.19). ∴ Rate of loss of heat is given by ∆ dQ =ms dT2 (11.20) dt dt Fig. 11.19 Curve showing cooling of hot water with time. From Eqs. (11.15) and (11.16) we have –m s dT2 =k (T2 – T1 ) dt dT2 = – k dt = – K dt (11.21) T2 – T1 ms where K = k/m s On integrating, loge (T2 – T1) = – K t + c (11.22) or T2 = T1 + C′ e–Kt; where C′ = ec (11.23) Equation 11.23 enables you to calculate the time of cooling of a body through a particular range of temperature. 2020-21

THERMAL PROPERTIES OF MATTER 297 For small temperature differences, the rate time. A graph is plotted )b. eTthweeennatluogree (oTf2–tTh1e) of cooling, due to conduction, convection, and [or ln(T2–T1)] and time (t radiation combined, is proportional to the difference in temperature. It is a valid graph is observed to be a straight line having approximation in the transfer of heat from a radiator to a room, the loss of heat through the a negative slope as shown in Fig. 11.20(b). This wall of a room, or the cooling of a cup of tea on the table. is in support of Eq. 11.22. Example 11.8 A pan filled with hot food cools from 94 °C to 86 °C in 2 minutes when the room temperature is at 20 °C. How long will it take to cool from 71 °C to 69 °C? Answer The average temperature of 94 °C and 86 °C is 90 °C, which is 70 °C above the room temperature. Under these conditions the pan cools 8 °C in 2 minutes. Using Eq. (11.21), we have Change in temperature = K ∆T Time 8 °C = K (70 °C) 2 min Fig. 11.20 Verification of Newton’s Law of cooling. The average of 69 °C and 71 °C is 70 °C, which is 50 °C above room temperature. K is the same Newton’s law of cooling can be verified with for this situation as for the original. the help of the experimental set-up shown in Fig. 11.20(a). The set-up consists of a double- 2 °C = K (50 °C) walled vessel (V) containing water between Time the two walls. A copper calorimeter (C) containing hot water is placed inside the When we divide above two equations, we double-walled vessel. Two thermometers through the corks are used to note the have Ttree1msoppfeehcrtoaitvteuwlryae.tseTrTem2inopfberweatawttueerereniontfhcheaoltdoowriuambteeletreiwrnaatlnhlsde, calorimeter is noted after equal intervals of 8 °C/2 min = K (70 °C) 2 °C/time K (50 °C) Time = 0.7 min = 42 s SUMMARY 1. Heat is a form of energy that flows between a body and its surrounding medium by virtue of temperature difference between them. The degree of hotness of the body is quantitatively represented by temperature. 2. A temperature-measuring device (thermometer) makes use of some measurable property (called thermometric property) that changes with temperature. Different thermometers lead to different temperature scales. To construct a temperature scale, two fixed points are chosen and assigned some arbitrary values of temperature. The two numbers fix the origin of the scale and the size of its unit. 3. The Celsius temperature (tC) and the Farenheit temperare (tF)are related by tF = (9/5) tC + 32 4. The ideal gas equation connecting pressure (P), volume (V) and absolute temperature (T) is : PV = µRT where µ is the number of moles and R is the universal gas constant. 2020-21

298 PHYSICS 5. In the absolute temperature scale, the zero of the scale corresponds to the temperature where every substance in nature has the least possible molecular activity. The Kelvin absolute temperature scale (T ) has the same unit size as the Celsius scale (Tc ), but differs in the origin : TC = T – 273.15 6. The coefficient of linear expansion (αl ) and volume expansion (αv ) are defined by the relations : ∆l = αl ∆T l ∆V = αV ∆T V where ∆l and ∆V denote the change in length l and volume V for a change of temperature ∆T. The relation between them is : αv = 3 αl 7. The specific heat capacity of a substance is defined by s = 1 ∆Q m ∆T where m is the mass of the substance and ∆Q is the heat required to change its temperature by ∆T. The molar specific heat capacity of a substance is defined by C = 1 ∆Q µ ∆T where µ is the number of moles of the substance. 8. The latent heat olfiqfuusidionat(Ltf)hies the heat per unit mass required to change a substance from solid into same temperature and pressure. The latent heat of tvoapthoreisvaatpioonur(Lsvt)aitsetwheithhoeuatt per unit mass required to change a substance from liquid change in the temperature and pressure. 9. The three modes of heat transfer are conduction, convection and radiation. 10. In conduction, heat is transferred between neighbouring parts of a body through molecular collisions, without any flow of matter. For a bar of length L and uniform cross section A with its ends maintained at temperatures TC and TD, the rate of flow of heat H is : T −T H=K A C D L where K is the thermal conductivity of the material of the bar. 11. Newton’s Law of Cooling says that the rate of cooling of a body is proportional to the excess temperature of the body over the surroundings : dQ = – k (T2 – T1 ) dt Where T1 is the temperature of the surrounding medium and T2 is the temperature of the body. 2020-21

THERMAL PROPERTIES OF MATTER 299 POINTS TO PONDER 1. The relation connecting Kelvin temperature (T ) and the Celsius temperature tc T = tc + 273.15 and the assignment T = 273.16 K for the triple point of water are exact relations (by choice). With this choice, the Celsius temperature of the melting point of water and boiling point of water (both at 1 atm pressure) are very close to, but not exactly equal to 0 °C and 100 °C respectively. In the original Celsius scale, these latter fixed points were exactly at 0 °C and 100 °C (by choice), but now the triple point of water is the preferred choice for the fixed point, because it has a unique temperature. 2. A liquid in equilibrium with vapour has the same pressure and temperature throughout the system; the two phases in equilibrium differ in their molar volume (i.e. density). This is true for a system with any number of phases in equilibrium. 3. Heat transfer always involves temperature difference between two systems or two parts of the same system. Any energy transfer that does not involve temperature difference in some way is not heat. 4. Convection involves flow of matter within a fluid due to unequal temperatures of its parts. A hot bar placed under a running tap loses heat by conduction between the surface of the bar and water and not by convection within water. EXERCISES 11.1 The triple points of neon and carbon dioxide are 24.57 K and 216.55 K respectively. 11.2 Express these temperatures on the Celsius and Fahrenheit scales. 11.3 Two absolute scales A and B have triple points of water defined to be 200 A and 350 11.4 B. What is the relation between TA and TB ? The electrical resistance in ohms of a certain thermometer varies with temperature according to the approximate law : R = Ro [1 + α (T – To )] The resistance is 101.6 Ω at the triple-point of water 273.16 K, and 165.5 Ω at the normal melting point of lead (600.5 K). What is the temperature when the resistance is 123.4 Ω ? Answer the following : (a) The triple-point of water is a standard fixed point in modern thermometry. 2020-21

300 PHYSICS Why ? What is wrong in taking the melting point of ice and the boiling point of water as standard fixed points (as was originally done in the Celsius scale) ? (b) There were two fixed points in the original Celsius scale as mentioned above which were assigned the number 0 °C and 100 °C respectively. On the absolute scale, one of the fixed points is the triple-point of water, which on the Kelvin absolute scale is assigned the number 273.16 K. What is the other fixed point on this (Kelvin) scale ? (c) The absolute temperature (Kelvin scale) T is related to the temperature tc on the Celsius scale by tc = T – 273.15 Why do we have 273.15 in this relation, and not 273.16 ? (d) What is the temperature of the triple-point of water on an absolute scale whose unit interval size is equal to that of the Fahrenheit scale ? 11.5 Two ideal gas thermometers A and B use oxygen and hydrogen respectively. The following observations are made : Temperature Pressure Pressure thermometer A thermometer B Triple-point of water 1.250 × 105 Pa 0.200 × 105 Pa Normal melting point 1.797 × 105 Pa 0.287 × 105 Pa of sulphur (a) What is the absolute temperature of normal melting point of sulphur as read by thermometers A and B ? (b) What do you think is the reason behind the slight difference in answers of thermometers A and B ? (The thermometers are not faulty). What further procedure is needed in the experiment to reduce the discrepancy between the two readings ? 11.6 A steel tape 1m long is correctly calibrated for a temperature of 27.0 °C. The length of a steel rod measured by this tape is found to be 63.0 cm on a hot day when the temperature is 45.0 °C. What is the actual length of the steel rod on that day ? What is the length of the same steel rod on a day when the temperature is 27.0 °C ? Coefficient of linear expansion of steel = 1.20 × 10–5 K–1 . 11.7 A large steel wheel is to be fitted on to a shaft of the same material. At 27 °C, the 11.8 outer diameter of the shaft is 8.70 cm and the diameter of the central hole in the wheel is 8.69 cm. The shaft is cooled using ‘dry ice’. At what temperature of the shaft does the wheel slip on the shaft? Assume coefficient of linear expansion of the steel to be constant over the required temperature range : αsteel = 1.20 × 10–5 K–1. A hole is drilled in a copper sheet. The diameter of the hole is 4.24 cm at 27.0 °C. What is the change in the diameter of the hole when the sheet is heated to 227 °C? Coefficient of linear expansion of copper = 1.70 × 10–5 K–1. 11.9 A brass wire 1.8 m long at 27 °C is held taut with little tension between two rigid supports. If the wire is cooled to a temperature of –39 °C, what is the tension developed in the wire, if its diameter is 2.0 mm ? Co-efficient of linear expansion of brass = 2.0 × 10–5 K–1; Young’s modulus of brass = 0.91 × 1011 Pa. 11.10 A brass rod of length 50 cm and diameter 3.0 mm is joined to a steel rod of the same length and diameter. What is the change in length of the combined rod at 250 °C, if the original lengths are at 40.0 °C? Is there a ‘thermal stress’ developed at the junction ? The ends of the rod are free to expand (Co-efficient of linear expansion of brass = 2.0 × 10–5 K–1, steel = 1.2 × 10–5 K–1 ). 2020-21

THERMAL PROPERTIES OF MATTER 301 11.11 The coefficient of volume expansion of glycerine is 49 × 10–5 K–1. What is the 11.12 fractional change in its density for a 30 °C rise in temperature ? 11.13 11.14 A 10 kW drilling machine is used to drill a bore in a small aluminium block of mass 8.0 kg. How much is the rise in temperature of the block in 2.5 minutes, 11.15 assuming 50% of power is used up in heating the machine itself or lost to the surroundings. Specific heat of aluminium = 0.91 J g–1 K–1. A copper block of mass 2.5 kg is heated in a furnace to a temperature of 500 °C and then placed on a large ice block. What is the maximum amount of ice that can melt? (Specific heat of copper = 0.39 J g–1 K–1; heat of fusion of water = 335 J g–1 ). In an experiment on the specific heat of a metal, a 0.20 kg block of the metal at 150 °C is dropped in a copper calorimeter (of water equivalent 0.025 kg) containing 150 cm3 of water at 27 °C. The final temperature is 40 °C. Compute the specific heat of the metal. If heat losses to the surroundings are not negligible, is your answer greater or smaller than the actual value for specific heat of the metal ? Given below are observations on molar specific heats at room temperature of some common gases. Gas Molar specific heat (Cv ) (cal mo1–1 K–1) Hydrogen Nitrogen 4.87 Oxygen Nitric oxide 4.97 Carbon monoxide Chlorine 5.02 4.99 5.01 6.17 The measured molar specific heats of these gases are markedly different from those for monatomic gases. Typically, molar specific heat of a monatomic gas is 2.92 cal/mol K. Explain this difference. What can you infer from the somewhat larger (than the rest) value for chlorine ? 11.16 A child running a temperature of 101°F is given an antipyrin (i.e. a medicine that lowers fever) which causes an increase in the rate of evaporation of sweat from his body. If the fever is brought down to 98 °F in 20 minutes, what is the average rate of extra evaporation caused, by the drug. Assume the evaporation mechanism to be the only way by which heat is lost. The mass of the child is 30 kg. The specific heat of human body is approximately the same as that of water, and latent heat of evaporation of water at that temperature is about 580 cal g–1. 11.17 A ‘thermacole’ icebox is a cheap and an efficient method for storing small quantities of cooked food in summer in particular. A cubical icebox of side 30 cm has a thickness of 5.0 cm. If 4.0 kg of ice is put in the box, estimate the amount of ice remaining after 6 h. The outside temperature is 45 °C, and co-efficient of thermal conductivity of thermacole is 0.01 J s–1 m–1 K–1. [Heat of fusion of water = 335 × 103 J kg–1] 11.18 A brass boiler has a base area of 0.15 m2 and thickness 1.0 cm. It boils water at the rate of 6.0 kg/min when placed on a gas stove. Estimate the temperature of the part of the flame in contact with the boiler. Thermal conductivity of brass = 109 J s–1 m–1 K–1 ; Heat of vaporisation of water = 2256 × 103 J kg–1. 11.19 Explain why : (a) a body with large reflectivity is a poor emitter (b) a brass tumbler feels much colder than a wooden tray on a chilly day (c) an optical pyrometer (for measuring high temperatures) calibrated for an ideal black body radiation gives too low a value for the temperature of a red hot iron piece in the open, but gives a correct value for the temperature when the same piece is in the furnace 2020-21

302 PHYSICS 11.20 (d) the earth without its atmosphere would be inhospitably cold (e) heating systems based on circulation of steam are more efficient in warming a building than those based on circulation of hot water A body cools from 80 °C to 50 °C in 5 minutes. Calculate the time it takes to cool from 60 °C to 30 °C. The temperature of the surroundings is 20 °C. ADDITIONAL EXERCISES 11.21 Answer the following questions based on the P-T phase diagram of carbon dioxide: 11.22 (a) At what temperature and pressure can the solid, liquid and vapour phases of CO2 co-exist in equilibrium ? (b) What is the effect of decrease of pressure on the fusion and boiling point of CO2 ? (c) What are the critical temperature and pressure for CO2 ? What is their significance ? (d) Is CO2 solid, liquid or gas at (a) –70 °C under 1 atm, (b) –60 °C under 10 atm, (c) 15 °C under 56 atm ? Answer the following questions based on the P – T phase diagram of CO2: (a) CO2 at 1 atm pressure and temperature – 60 °C is compressed isothermally. Does it go through a liquid phase ? (b) What happens when CO2 at 4 atm pressure is cooled from room temperature at constant pressure ? (c) Describe qualitatively the changes in a given mass of solid CO2 at 10 atm pressure and temperature –65 °C as it is heated up to room temperature at constant pressure. (d) CO2 is heated to a temperature 70 °C and compressed isothermally. What changes in its properties do you expect to observe ? 2020-21

CHAPTER TWELVE THERMODYNAMICS 12.1 Introduction 12.1 INTRODUCTION 12.2 12.3 Thermal equilibrium In previous chapter we have studied thermal properties of matter. In this chapter we shall study laws that govern 12.4 Zeroth law of thermal energy. We shall study the processes where work is Thermodynamics converted into heat and vice versa. In winter, when we rub 12.5 Heat, internal energy and our palms together, we feel warmer; here work done in rubbing work produces the ‘heat’. Conversely, in a steam engine, the ‘heat’ 12.6 First law of of the steam is used to do useful work in moving the pistons, 12.7 thermodynamics which in turn rotate the wheels of the train. Specific heat capacity 12.8 In physics, we need to define the notions of heat, 12.9 Thermodynamic state temperature, work, etc. more carefully. Historically, it took a 12.10 variables and equation of long time to arrive at the proper concept of ‘heat’. Before the state modern picture, heat was regarded as a fine invisible fluid 12.11 Thermodynamic processes filling in the pores of a substance. On contact between a hot body and a cold body, the fluid (called caloric) flowed from 12.12 Heat engines the colder to the hotter body ! This is similar to what happens when a horizontal pipe connects two tanks containing water 12.13 Refrigerators and heat up to different heights. The flow continues until the levels of pumps water in the two tanks are the same. Likewise, in the ‘caloric’ picture of heat, heat flows until the ‘caloric levels’ (i.e., the Second law of temperatures) equalise. thermodynamics Reversible and irreversible In time, the picture of heat as a fluid was discarded in processes favour of the modern concept of heat as a form of energy. An Carnot engine important experiment in this connection was due to Benjamin Thomson (also known as Count Rumford) in 1798. He Summary observed that boring of a brass cannon generated a lot of Points to ponder heat, indeed enough to boil water. More significantly, the Exercises amount of heat produced depended on the work done (by the horses employed for turning the drill) but not on the sharpness of the drill. In the caloric picture, a sharper drill would scoop out more heat fluid from the pores; but this was not observed. A most natural explanation of the observations was that heat was a form of energy and the experiment demonstrated conversion of energy from one form to another–from work to heat. 2020-21

304 PHYSICS Thermodynamics is the branch of physics that in a different context : we say the state of a system deals with the concepts of heat and temperature is an equilibrium state if the macroscopic and the inter-conversion of heat and other forms variables that characterise the system do not of energy. Thermodynamics is a macroscopic change in time. For example, a gas inside a closed science. It deals with bulk systems and does not rigid container, completely insulated from its go into the molecular constitution of matter. In surroundings, with fixed values of pressure, fact, its concepts and laws were formulated in the volume, temperature, mass and composition that nineteenth century before the molecular picture do not change with time, is in a state of of matter was firmly established. Thermodynamic thermodynamic equilibrium. description involves relatively few macroscopic variables of the system, which are suggested by (a) common sense and can be usually measured directly. A microscopic description of a gas, for (b) example, would involve specifying the co-ordinates and velocities of the huge number of molecules Fig. 12.1 (a) Systems A and B (two gases) separated constituting the gas. The description in kinetic by an adiabatic wall – an insulating wall theory of gases is not so detailed but it does involve that does not allow flow of heat. (b) The molecular distribution of velocities. same systems A and B separated by a Thermodynamic description of a gas, on the other diathermic wall – a conducting wall that hand, avoids the molecular description altogether. allows heat to flow from one to another. In Instead, the state of a gas in thermodynamics is this case, thermal equilibrium is attained specified by macroscopic variables such as pressure, volume, temperature, mass and in due course. composition that are felt by our sense perceptions In general, whether or not a system is in a state and are measurable*. of equilibrium depends on the surroundings and the nature of the wall that separates the system The distinction between mechanics and from the surroundings. Consider two gases A and thermodynamics is worth bearing in mind. In B occupying two different containers. We know mechanics, our interest is in the motion of particles experimentally that pressure and volume of a or bodies under the action of forces and torques. given mass of gas can be chosen to be its two Thermodynamics is not concerned with the independent variables. Let the pressure and motion of the system as a whole. It is concerned volume of the gases be (PA, VA) and (PB, VB) with the internal macroscopic state of the body. respectively. Suppose first that the two systems When a bullet is fired from a gun, what changes are put in proximity but are separated by an is the mechanical state of the bullet (its kinetic energy, in particular), not its temperature. When the bullet pierces a wood and stops, the kinetic energy of the bullet gets converted into heat, changing the temperature of the bullet and the surrounding layers of wood. Temperature is related to the energy of the internal (disordered) motion of the bullet, not to the motion of the bullet as a whole. 12.2 THERMAL EQUILIBRIUM Equilibrium in mechanics means that the net external force and torque on a system are zero. The term ‘equilibrium’ in thermodynamics appears * Thermodynamics may also involve other variables that are not so obvious to our senses e.g. entropy, enthalpy, etc., and they are all macroscopic variables. However, a thermodynamic state is specified by five state variables viz., pressure, volume, temperature, internal energy and entropy. Entropy is a measure of disorderness in the system. Enthalpy is a measure of total heat content of the system. 2020-21

THERMODYNAMICS 305 adiabatic wall – an insulating wall (can be law in 1931 long after the first and second Laws movable) that does not allow flow of energy (heat) of thermodynamics were stated and so numbered. The Zeroth Law clearly suggests that when two from one to another. The systems are insulated systems A and B, are in thermal equilibrium, from the rest of the surroundings also by similar there must be a physical quantity that has the same value for both. This thermodynamic adiabatic walls. The situation is shown variable whose value is equal for two systems in thermal equilibrium is called temperature (T ). schematically in Fig. 12.1 (a). In this case, it is Thus, if A and B are separately in equilibrium with C, TA = TC and TB = TC. This implies that found that any possible pair of values (PA, VA) will TA = TB i.e. the systems A and B are also in be in equilibrium with any possible pair of values thermal equilibrium. (PB, VB ). Next, suppose that the adiabatic wall is We have arrived at the concept of temperature replaced by a diathermic wall – a conducting wall formally via the Zeroth Law. The next question is : how to assign numerical values to that allows energy flow (heat) from one to another. temperatures of different bodies ? In other words, how do we construct a scale of temperature ? It is then found that the macroscopic variables of Thermometry deals with this basic question to which we turn in the next section. the systems A and B change spontaneously until both the systems attain equilibrium states. After that there is no change in their states. The situation is shown in Fig. 12.1(b). The pressure and volume variables of the two gases change to (PB ′, VB ′) Banadre(PiAn′,eVquA i′)lisburicuhmthwaitththeeancehwosthtaetre*s. of A and There is no more energy flow from one to another. We then say that the system A is in thermal equilibrium with the system B. What characterises the situation of thermal equilibrium between two systems ? You can guess the answer from your experience. In thermal equilibrium, the temperatures of the two systems are equal. We shall see how does one arrive at the concept of temperature in thermodynamics? The Zeroth law of thermodynamics provides the clue. 12.3 ZEROTH LAW OF THERMODYNAMICS (a) Imagine two systems A and B, separated by an (b) adiabatic wall, while each is in contact with a third system C, via a conducting wall [Fig. 12.2(a)]. The Fig. 12.2 (a) Systems A and B are separated by an states of the systems (i.e., their macroscopic adiabatic wall, while each is in contact variables) will change until both A and B come to with a third system C via a conducting thermal equilibrium with C. After this is achieved, wall. (b) The adiabatic wall between A suppose that the adiabatic wall between A and B and B is replaced by a conducting wall, is replaced by a conducting wall and C is insulated while C is insulated from A and B by an from A and B by an adiabatic wall [Fig.12.2(b)]. It is found that the states of A and B change no adiabatic wall. further i.e. they are found to be in thermal equilibrium with each other. This observation forms the basis of the Zeroth Law of Thermodynamics, which states that ‘two systems in thermal equilibrium with a third system separately are in thermal equilibrium with each other’. R.H. Fowler formulated this * Both the variables need not change. It depends on the constraints. For instance, if the gases are in containers of fixed volume, only the pressures of the gases would change to achieve thermal equilibrium. 2020-21

306 PHYSICS 12.4 HEAT, INTERNAL ENERGY AND WORK associated with various random motions of its molecules. We will see in the next chapter that The Zeroth Law of Thermodynamics led us to in a gas this motion is not only translational the concept of temperature that agrees with our (i.e. motion from one point to another in the commonsense notion. Temperature is a marker volume of the container); it also includes of the ‘hotness’ of a body. It determines the rotational and vibrational motion of the direction of flow of heat when two bodies are molecules (Fig. 12.3). placed in thermal contact. Heat flows from the body at a higher temperature to the one at lower Fig. 12.3 (a) Internal energy U of a gas is the sum temperature. The flow stops when the of the kinetic and potential energies of its temperatures equalise; the two bodies are then molecules when the box is at rest. Kinetic in thermal equilibrium. We saw in some detail energy due to various types of motion how to construct temperature scales to assign (translational, rotational, vibrational) is to temperatures to different bodies. We now be included in U. (b) If the same box is describe the concepts of heat and other relevant moving as a whole with some velocity, quantities like internal energy and work. the kinetic energy of the box is not to be included in U. The concept of internal energy of a system is not difficult to understand. We know that every Fig. 12.4 Heat and work are two distinct modes of bulk system consists of a large number of energy transfer to a system that results in molecules. Internal energy is simply the sum of change in its internal energy. (a) Heat is the kinetic energies and potential energies of energy transfer due to temperature these molecules. We remarked earlier that in difference between the system and the thermodynamics, the kinetic energy of the surroundings. (b) Work is energy transfer system, as a whole, is not relevant. Internal brought about by means (e.g. moving the energy is thus, the sum of molecular kinetic and piston by raising or lowering some weight potential energies in the frame of reference connected to it) that do not involve such a relative to which the centre of mass of the system temperature difference. is at rest. Thus, it includes only the (disordered) energy associated with the random motion of molecules of the system. We denote the internal energy of a system by U. Though we have invoked the molecular picture to understand the meaning of internal energy, as far as thermodynamics is concerned, U is simply a macroscopic variable of the system. The important thing about internal energy is that it depends only on the state of the system, not on how that state was achieved. Internal energy U of a system is an example of a thermodynamic ‘state variable’ – its value depends only on the given state of the system, not on history i.e. not on the ‘path’ taken to arrive at that state. Thus, the internal energy of a given mass of gas depends on its state described by specific values of pressure, volume and temperature. It does not depend on how this state of the gas came about. Pressure, volume, temperature, and internal energy are thermodynamic state variables of the system (gas) (see section 12.7). If we neglect the small intermolecular forces in a gas, the internal energy of a gas is just the sum of kinetic energies 2020-21

THERMODYNAMICS 307 What are the ways of changing internal 12.5 FIRST LAW OF THERMODYNAMICS energy of a system ? Consider again, for simplicity, the system to be a certain mass of We have seen that the internal energy U of a gas contained in a cylinder with a movable piston as shown in Fig. 12.4. Experience shows system can change through two modes of energy there are two ways of changing the state of the gas (and hence its internal energy). One way is transfer : heat and work. Let to put the cylinder in contact with a body at a higher temperature than that of the gas. The ∆Q = Heat supplied to the system by the temperature difference will cause a flow of energy (heat) from the hotter body to the gas, surroundings thus increasing the internal energy of the gas. The other way is to push the piston down i.e. to ∆W = Work done by the system on the do work on the system, which again results in increasing the internal energy of the gas. Of surroundings course, both these things could happen in the reverse direction. With surroundings at a lower ∆U = Change in internal energy of the system temperature, heat would flow from the gas to the surroundings. Likewise, the gas could push The general principle of conservation of the piston up and do work on the surroundings. In short, heat and work are two different modes energy then implies that of altering the state of a thermodynamic system and changing its internal energy. ∆Q = ∆U + ∆W (12.1) The notion of heat should be carefully i.e. the energy (∆Q) supplied to the system goes distinguished from the notion of internal energy. Heat is certainly energy, but it is the energy in in partly to increase the internal energy of the transit. This is not just a play of words. The system (∆U) and the rest in work on the distinction is of basic significance. The state of environment (∆W). Equation (12.1) is known as a thermodynamic system is characterised by its internal energy, not heat. A statement like ‘a the First Law of Thermodynamics. It is simply gas in a given state has a certain amount of heat’ is as meaningless as the statement that the general law of conservation of energy applied ‘a gas in a given state has a certain amount of work’. In contrast, ‘a gas in a given state to any system in which the energy transfer from has a certain amount of internal energy’ is a perfectly meaningful statement. Similarly, the or to the surroundings is taken into account. statements ‘a certain amount of heat is supplied to the system’ or ‘a certain amount Let us put Eq. (12.1) in the alternative form of work was done by the system’ are perfectly meaningful. ∆Q – ∆W = ∆U (12.2) To summarise, heat and work in Now, the system may go from an initial state thermodynamics are not state variables. They to the final state in a number of ways. For are modes of energy transfer to a system example, to change the state of a gas from resulting in change in its internal energy, (P1, V1) to (P2, V2), we can first change the which, as already mentioned, is a state variable. volume of the gas from V1 to V2, keeping its pressure constant i.e. we can first go the state In ordinary language, we often confuse heat (P1, V2) and then change the pressure of the with internal energy. The distinction between gas from P1 to P2, keeping volume constant, to them is sometimes ignored in elementary take the gas to (P2, V2). Alternatively, we can physics books. For proper understanding of first keep the volume constant and then keep thermodynamics, however, the distinction is the pressure constant. Since U is a state crucial. variable, ∆U depends only on the initial and final states and not on the path taken by the gas to go from one to the other. However, ∆Q and ∆W will, in general, depend on the path taken to go from the initial to final states. From the First Law of Thermodynamics, Eq. (12.2), it is clear that the combination ∆Q – ∆W, is however, path independent. This shows that if a system is taken through a process in which ∆U = 0 (for example, isothermal expansion of an ideal gas, see section 12.8), ∆Q = ∆W i.e., heat supplied to the system is used up entirely by the system in doing work on the environment. 2020-21

308 PHYSICS If the system is a gas in a cylinder with a If the amount of substance is specified in movable piston, the gas in moving the piston does terms of moles µ (instead of mass m in kg ), we work. Since force is pressure times area, and can define heat capacity per mole of the area times displacement is volume, work done by the system against a constant pressure P is substance by ∆W = P ∆V C = S = 1 ∆Q (12.6) µ µ ∆T where ∆V is the change in volume of the gas. C is known as molar specific heat capacity of Thus, for this case, Eq. (12.1) gives the substance. Like s, C is independent of the amount of substance. C depends on the nature ∆Q = ∆U + P ∆V (12.3) of the substance, its temperature and the conditions under which heat is supplied. The As an application of Eq. (12.3), consider the unit of C is J mo1–1 K–1. As we shall see later (in change in internal energy for 1 g of water when connection with specific heat capacity of gases), we go from its liquid to vapour phase. The additional conditions may be needed to define measured latent heat of water is 2256 J/g. i.e., C or s. The idea in defining C is that simple for 1 g of water ∆Q = 2256 J. At atmospheric predictions can be made in regard to molar pressure, 1 g of water has a volume 1 cm3 in specific heat capacities. liquid phase and 1671 cm3 in vapour phase. Table 12.1 lists measured specific and molar Therefore, heat capacities of solids at atmospheric pressure and ordinary room temperature. ∆W =P (Vg –Vl ) = 1.013 ×105 ×(1671×10–6) =169.2 J We will see in Chapter 13 that predictions of Equation (12.3) then gives specific heats of gases generally agree with experiment. We can use the same law of ∆U = 2256 – 169.2 = 2086.8 J equipartition of energy that we use there to predict molar specific heat capacities of solids We see that most of the heat goes to increase (See Section 13.5 and 13.6). Consider a solid of the internal energy of water in transition from N atoms, each vibrating about its mean the liquid to the vapour phase. position. An oscillator in one dimension has average energy of 2 × ½ kBT = kBT. In three 12.6 SPECIFIC HEAT CAPACITY dimensions, the average energy is 3 kBT. For a mole of a solid, the total energy is Suppose an amount of heat ∆Q supplied to a U = 3 kBT × NA = 3 RT (∵kBT × NA = R ) substance changes its temperature from T to T + ∆T. We define heat capacity of a substance Now, at constant pressure, ∆Q = ∆U + P ∆V ≅ (see Chapter 11) to be S = ∆Q (12.4) ∆U, since for a solid ∆V is negligible. Therefore, ∆T C= ∆Q = ∆U = 3R We expect ∆Q and, therefore, heat capacity S ∆T ∆T (12.7) to be proportional to the mass of the substance. Table 12.1 Specific and molar heat capacities of some solids at room Further, it could also depend on the temperature and atmospheric temperature, i.e., a different amount of heat may pressure be needed for a unit rise in temperature at different temperatures. To define a constant Speci c–v heat Molar speci c (J kg–1 K–1) heat (J mol–1 K–1) characteristic of the substance and Substance independent of its amount, we divide S by the mass of the substance m in kg : s= S =  1  ∆Q (12.5) m m ∆T s is known as the specific heat capacity of the As Table 12.1 shows, the experimentally measured values which generally agrees with substance. It depends on the nature of the substance and its temperature. The unit of specific heat capacity is J kg–1 K–1. 2020-21

THERMODYNAMICS 309 predicted value 3R at ordinary temperatures. ideal gas, we have a simple relation. (Carbon is an exception.) The agreement is known to break down at low temperatures. Cp – Cv = R (12.8) Specific heat capacity of water where Cp and Cv are molar specific heat capacities of an ideal gas at constant pressure The old unit of heat was calorie. One calorie was earlier defined to be the amount of heat and volume respectively and R is the universal required to raise the temperature of 1g of water by 1°C. With more precise measurements, it was gas constant. To prove the relation, we begin found that the specific heat of water varies slightly with temperature. Figure 12.5 shows with Eq. (12.3) for 1 mole of the gas : this variation in the temperature range 0 to 100 °C. ∆Q = ∆U + P ∆V If ∆Q is absorbed at constant volume, ∆V = 0 Cv =  ∆Q  =  ∆U  v =  ∆U  (12.9) ∆T ∆T ∆T v where the subscript v is dropped in the last step, since U of an ideal gas depends only on temperature. (The subscript denotes the quantity kept fixed.) If, on the other hand, ∆Q is absorbed at constant pressure, Cp =  ∆Q  =  ∆U  + P  ∆V  (12.10) ∆T ∆T ∆T p p p Fig. 12.5 Variation of specific heat capacity of water The subscript p can be dropped from the with temperature. first term since U of an ideal gas depends only on T. Now, for a mole of an ideal gas For a precise definition of calorie, it was, therefore, necessary to specify the unit PV = RT temperature interval. One calorie is defined which gives to be the amount of heat required to raise the temperature of 1g of water from 14.5 °C to P  ∆V  p = R (12.11) 15.5 °C. Since heat is just a form of energy, ∆T it is preferable to use the unit joule, J. In SI units, the specific heat capacity of water Equations (12.9) to (12.11) give the desired is 4186 J kg–1 K–1 i.e. 4.186 J g–1 K–1. The so relation, Eq. (12.8). called mechanical equivalent of heat defined as the amount of work needed to produce 12.7 THERMODYNAMIC STATE VARIABLES 1 cal of heat is in fact just a conversion factor AND EQUATION OF STATE between two different units of energy : calorie to joule. Since in SI units, we use the unit joule Every equilibrium state of a thermodynamic for heat, work or any other form of energy, the system is completely described by specific term mechanical equivalent is now values of some macroscopic variables, also superfluous and need not be used. called state variables. For example, an equilibrium state of a gas is completely As already remarked, the specific heat specified by the values of pressure, volume, capacity depends on the process or the temperature, and mass (and composition if conditions under which heat capacity transfer there is a mixture of gases). A thermodynamic takes place. For gases, for example, we can system is not always in equilibrium. For example, define two specific heats : specific heat a gas allowed to expand freely against vacuum capacity at constant volume and specific is not an equilibrium state [Fig. 12.6(a)]. During heat capacity at constant pressure. For an the rapid expansion, pressure of the gas may 2020-21

310 PHYSICS not be uniform throughout. Similarly, a mixture temperature do not. To decide which variable is of gases undergoing an explosive chemical extensive and which intensive, think of a reaction (e.g. a mixture of petrol vapour and relevant system in equilibrium, and imagine that air when ignited by a spark) is not an it is divided into two equal parts. The variables equilibrium state; again its temperature and that remain unchanged for each part are pressure are not uniform [Fig. 12.6(b)]. intensive. The variables whose values get halved Eventually, the gas attains a uniform in each part are extensive. It is easily seen, for temperature and pressure and comes to example, that internal energy U, volume V, total thermal and mechanical equilibrium with its mass M are extensive variables. Pressure P, surroundings. temperature T, and density ρ are intensive variables. It is a good practice to check the consistency of thermodynamic equations using this classification of variables. For example, in the equation ∆Q = ∆U + P ∆V quantities on both sides are extensive*. (The product of an intensive variable like P and an extensive quantity ∆V is extensive.) Fig. 12.6 (a) The partition in the box is suddenly 12.8 THERMODYNAMIC PROCESSES removed leading to free expansion of the gas. (b) A mixture of gases undergoing an 12.8.1 Quasi-static process explosive chemical reaction. In both situations, the gas is not in equilibrium and Consider a gas in thermal and mechanical equilibrium with its surroundings. The pressure cannot be described by state variables. of the gas in that case equals the external pressure and its temperature is the same as In short, thermodynamic state variables that of its surroundings. Suppose that the describe equilibrium states of systems. The external pressure is suddenly reduced (say by various state variables are not necessarily lifting the weight on the movable piston in the independent. The connection between the state container). The piston will accelerate outward. variables is called the equation of state. For During the process, the gas passes through example, for an ideal gas, the equation of state states that are not equilibrium states. The non- is the ideal gas relation equilibrium states do not have well-defined pressure and temperature. In the same way, if PV=µRT a finite temperature difference exists between the gas and its surroundings, there will be a For a fixed amount of the gas i.e. given µ, there rapid exchange of heat during which the gas are thus, only two independent variables, say P will pass through non-equilibrium states. In and V or T and V. The pressure-volume curve due course, the gas will settle to an equilibrium for a fixed temperature is called an isotherm. state with well-defined temperature and Real gases may have more complicated pressure equal to those of the surroundings. The equations of state. free expansion of a gas in vacuum and a mixture of gases undergoing an explosive chemical The thermodynamic state variables are of two reaction, mentioned in section 12.7 are also kinds: extensive and intensive. Extensive examples where the system goes through non- variables indicate the ‘size’ of the system. equilibrium states. Intensive variables such as pressure and Non-equilibrium states of a system are difficult to deal with. It is, therefore, convenient to imagine an idealised process in which at every stage the system is an equilibrium state. Such a * As emphasised earlier, Q is not a state variable. However, ∆Q is clearly proportional to the total mass of system and hence is extensive. 2020-21

THERMODYNAMICS 311 process is, in principle, infinitely slow, hence the A process in which the temperature of the name quasi-static (meaning nearly static). The system is kept fixed throughout is called an system changes its variables (P, T, V ) so slowly isothermal process. The expansion of a gas in a metallic cylinder placed in a large reservoir of that it remains in thermal and mechanical fixed temperature is an example of an isothermal equilibrium with its surroundings throughout. process. (Heat transferred from the reservoir to the system does not materially affect the In a quasi-static process, at every stage, the temperature of the reservoir, because of its very difference in the pressure of the system and the large heat capacity.) In isobaric processes the external pressure is infinitesimally small. The pressure is constant while in isochoric processes the volume is constant. Finally, if same is true of the temperature difference the system is insulated from the surroundings between the system and its surroundings and no heat flows between the system and the surroundings, the process is adiabatic. The (Fig.12.7). To take a gas from the state (P, T ) to definitions of these special processes are another state (P ′, T ′ ) via a quasi-static process, summarised in Table. 12.2 we change the external pressure by a very small amount, allow the system to equalise its pressure Table 12.2 Some special thermodynamic with that of the surroundings and continue the processes process infinitely slowly until the system achieves the pressure P ′. Similarly, to change the temperature, we introduce an infinitesimal temperature difference between the system and the surrounding reservoirs and by choosing reservoirs of progressively different temperatures T to T ′, the system achieves the temperature T ′. Fig. 12.7 In a quasi-static process, the temperature We now consider these processes in some detail : of the surrounding reservoir and the external pressure differ only infinitesimally 12.8.2 Isothermal process from the temperature and pressure of the For an isothermal process (T fixed), the ideal gas system. equation gives A quasi-static process is obviously a PV = constant hypothetical construct. In practice, processes i.e., pressure of a given mass of gas varies inversely that are sufficiently slow and do not involve as its volume. This is nothing but Boyle’s Law. accelerated motion of the piston, large temperature gradient, etc., are reasonably Suppose an ideal gas goes isothermally (at approximation to an ideal quasi-static process. temperature T ) from its initial state (P1, V1) to We shall from now on deal with quasi-static the final state (P2, V 2). At any intermediate stage processes only, except when stated otherwise. with pressure P and volume change from V to V + ∆V (∆V small) ∆W = P ∆ V Taking (∆V → 0) and summing the quantity ∆W over the entire process, V2 W = ∫ P dV V1 = µ V2 dV = µRT In V2 V1 RT ∫ (12.12) V1 V 2020-21

312 PHYSICS where in the second step we have made use of We can calculate, as before, the work done in the ideal gas equation PV = µ RT and taken the an adiabatic change of an ideal gas from the constants out of the integral. For an ideal gas, state (P1, V1, T1) to the state (P2, V2, T2). internal energy depends only on temperature. Thus, there is no change in the internal energy V2 of an ideal gas in an isothermal process. The First Law of Thermodynamics then implies that W = ∫ P dV heat supplied to the gas equals the work done by the gas : Q = W. Note from Eq. (12.12) that V1 for V2 > V1, W > 0; and for V2 < V1, W < 0. That is, in an isothermal expansion, the gas absorbs (12.15) heat and does work while in an isothermal compression, work is done on the gas by the From Eq. (12.14), the constant is P1V1γ or P2V2γ environment and heat is released. 12.8.3 Adiabatic process In an adiabatic process, the system is insulated W 1  P2V2γ − P1V1γ  = 1− γ  V2γ −1 V1γ −1  from the surroundings and heat absorbed or released is zero. From Eq. (12.1), we see that = µR(T1 − T2 ) γ −1 work done by the gas results in decrease in its [ ]1 =1−γ internal energy (and hence its temperature for P2V2 − P1V1 (12.16) an ideal gas). We quote without proof (the result As expected, if work is done by the gas in an that you will learn in higher courses) that for adiabatic process (W > 0), from Eq. (12.16), an adiabatic process of an ideal gas. T2 < T1. On the other hand, if work is done on P V γ = const (12.13) the gas (W < 0), we get T2 > T1 i.e., the temperature of the gas rises. where γ is the ratio of specific heats (ordinary or molar) at constant pressure and at constant volume. 12.8.4 Isochoric process γ = Cp In an isochoric process, V is constant. No work Cv is done on or by the gas. From Eq. (12.1), the heat absorbed by the gas goes entirely to change Thus if an ideal gas undergoes a change in its internal energy and its temperature. The change in temperature for a given amount of its state adiabatically from (P1, V1) to (P2, V2) : heat is determined by the specific heat of the gas at constant volume. P1 V1γ = P2 V2γ (12.14) Figure12.8 shows the P-V curves of an ideal gas for two adiabatic processes connecting two 12.8.5 Isobaric process isotherms. In an isobaric process, P is fixed. Work done by the gas is W = P (V2 – V1) = µ R (T2 – T1) (12.17) Since temperature changes, so does internal energy. The heat absorbed goes partly to increase internal energy and partly to do work. The change in temperature for a given amount of heat is determined by the specific heat of the gas at constant pressure. Fig. 12.8 P-V curves for isothermal and adiabatic 12.8.6 Cyclic process processes of an ideal gas. In a cyclic process, the system returns to its initial state. Since internal energy is a state variable, ∆U = 0 for a cyclic process. From 2020-21

THERMODYNAMICS 313 Eq. (12.1), the total heat absorbed equals the and W is the work done on the environment in work done by the system. a cycle. In a cycle, a certain amount of heat (Q2) may also be rejected to the environment. Then, 12.9 HEAT ENGINES according to the First Law of Thermodynamics, over one complete cycle, Heat engine is a device by which a system is made to undergo a cyclic process that results W = Q1 – Q2 (12.19) in conversion of heat to work. (1) It consists of a working substance–the i.e., system. For example, a mixture of fuel η =1 − Q2 (12.20) vapour and air in a gasoline or diesel engine Q1 or steam in a steam engine are the working substances. For Q2 = 0, η = 1, i.e., the engine will have (2) The working substance goes through a cycle 100% efficiency in converting heat into work. consisting of several processes. In some of Note that the First Law of Thermodynamics i.e., these processes, it absorbs a total amount the energy conservation law does not rule out of heat Q1 from an external reservoir at some such an engine. But experience shows that high temperature T1. such an ideal engine with η = 1 is never possible, (3) In some other processes of the cycle, the even if we can eliminate various kinds of losses working substance releases a total amount associated with actual heat engines. It turns of heat Q2 to an external reservoir at some out that there is a fundamental limit on the lower temperature T2. efficiency of a heat engine set by an independent (4) The work done (W ) by the system in a cycle principle of nature, called the Second Law of is transferred to the environment via some Thermodynamics (section 12.11). arrangement (e.g. the working substance may be in a cylinder with a moving piston The mechanism of conversion of heat into that transfers mechanical energy to the work varies for different heat engines. Basically, wheels of a vehicle via a shaft). there are two ways : the system (say a gas or a The basic features of a heat engine are mixture of gases) is heated by an external schematically represented in Fig. 12.9. furnace, as in a steam engine; or it is heated internally by an exothermic chemical reaction as in an internal combustion engine. The various steps involved in a cycle also differ from one engine to another. Fig. 12.9 Schematic representation of a heat engine. 12.10 REFRIGERATORS AND HEAT PUMPS The engine takes heat Q1 from a hot A refrigerator is the reverse of a heat engine. reservoir at temperature T1, releases heat Here the working substance extracts heat Q2 Q2 to a cold reservoir at temperature T2 from the cold reservoir at temperature T2, some and delivers work W to the surroundings. external work W is done on it and heat Q1 is released to the hot reservoir at temperature T1 The cycle is repeated again and again to get (Fig. 12.10). useful work for some purpose. The discipline of Fig. 12.10 Schematic representation of a refrigerator or a heat pump, the reverse of a heat thermodynamics has its roots in the study of heat engine. engines. A basic question relates to the efficiency of a heat engine. The efficiency (η) of a heat engine is defined by η= W (12.18) Q1 where Q1 is the heat input i.e., the heat absorbed by the system in one complete cycle 2020-21

314 PHYSICS Pioneers of Thermodynamics Lord Kelvin (William Thomson) (1824-1907), born in Belfast, Ireland, is among the foremost British scientists of the nineteenth century. Thomson played a key role in the development of the law of conservation of energy suggested by the work of James Joule (1818-1889), Julius Mayer (1814- 1878) and Hermann Helmholtz (1821-1894). He collaborated with Joule on the so-called Joule-Thomson effect : cooling of a gas when it expands into vacuum. He introduced the notion of the absolute zero of temperature and proposed the absolute temperature scale, now called the Kelvin scale in his honour. From the work of Sadi Carnot (1796-1832), Thomson arrived at a form of the Second Law of Thermodynamics. Thomson was a versatile physicist, with notable contributions to electromagnetic theory and hydrodynamics. Rudolf Clausius (1822-1888), born in Poland, is generally regarded as the discoverer of the Second Law of Thermodynamics. Based on the work of Carnot and Thomson, Clausius arrived at the important notion of entropy that led him to a fundamental version of the Second Law of Thermodynamics that states that the entropy of an isolated system can never decrease. Clausius also worked on the kinetic theory of gases and obtained the first reliable estimates of molecular size, speed, mean free path, etc. A heat pump is the same as a refrigerator. where Q2 is the heat extracted from the cold What term we use depends on the purpose of reservoir and W is the work done on the the device. If the purpose is to cool a portion of space, like the inside of a chamber, and higher system–the refrigerant. (α for heat pump is temperature reservoir is surrounding, we call Qex1/cWee)dN1o,teαthcaatnwbheilegrηeabtyedr etfhinaintio1n. the device a refrigerator; if the idea is to pump defined as heat into a portion of space (the room in a can never building when the outside environment is cold), the device is called a heat pump. By energy conservation, the heat released to the In a refrigerator the working substance hot reservoir is (usually, in gaseous form) goes through the following steps : (a) sudden expansion of the gas Q1 = W + Q2 from high to low pressure which cools it and converts it into a vapour-liquid mixture, (b) i.e., α = Q2 (12.22) absorption by the cold fluid of heat from the Q1 – Q2 region to be cooled converting it into vapour, (c) heating up of the vapour due to external work In a heat engine, heat cannot be fully done on the system, and (d) release of heat by the vapour to the surroundings, bringing it to converted to work; likewise a refrigerator cannot the initial state and completing the cycle. work without some external work done on the The coefficient of performance (α) of a refrigerator is given by system, i.e., the coefficient of performance in Eq. (12.21) cannot be infinite. 12.11 SECOND LAW OF THERMODYNAMICS α = Q2 (12.21) The First Law of Thermodynamics is the principle W of conservation of energy. Common experience shows that there are many conceivable processes that are perfectly allowed by the First Law and yet are never observed. For example, nobody has ever seen a book lying on a table jumping to a height by itself. But such a thing 2020-21

THERMODYNAMICS 315 would be possible if the principle of conservation Experience suggests that for most processes in of energy were the only restriction. The table nature this is not possible. The spontaneous could cool spontaneously, converting some of its processes of nature are irreversible. Several internal energy into an equal amount of examples can be cited. The base of a vessel on mechanical energy of the book, which would an oven is hotter than its other parts. When then hop to a height with potential energy equal the vessel is removed, heat is transferred from to the mechanical energy it acquired. But this the base to the other parts, bringing the vessel never happens. Clearly, some additional basic to a uniform temperature (which in due course principle of nature forbids the above, even cools to the temperature of the surroundings). though it satisfies the energy conservation The process cannot be reversed; a part of the principle. This principle, which disallows many vessel will not get cooler spontaneously and phenomena consistent with the First Law of warm up the base. It will violate the Second Law Thermodynamics is known as the Second Law of Thermodynamics, if it did. The free expansion of Thermodynamics. of a gas is irreversible. The combustion reaction of a mixture of petrol and air ignited by a spark The Second Law of Thermodynamics gives a cannot be reversed. Cooking gas leaking from a fundamental limitation to the efficiency of a heat gas cylinder in the kitchen diffuses to the engine and the co-efficient of performance of a entire room. The diffusion process will not refrigerator. In simple terms, it says that spontaneously reverse and bring the gas back efficiency of a heat engine can never be unity. to the cylinder. The stirring of a liquid in thermal According to Eq. (12.20), this implies that heat contact with a reservoir will convert the work released to the cold reservoir can never be made done into heat, increasing the internal energy zero. For a refrigerator, the Second Law says that of the reservoir. The process cannot be reversed the co-efficient of performance can never be exactly; otherwise it would amount to conversion infinite. According to Eq. (12.21), this implies of heat entirely into work, violating the Second that external work (W ) can never be zero. The Law of Thermodynamics. Irreversibility is a rule following two statements, one due to Kelvin and rather an exception in nature. Planck denying the possibility of a perfect heat engine, and another due to Clausius denying Irreversibility arises mainly from two causes: the possibility of a perfect refrigerator or heat one, many processes (like a free expansion, or pump, are a concise summary of these an explosive chemical reaction) take the system observations. to non-equilibrium states; two, most processes involve friction, viscosity and other dissipative Kelvin-Planck statement effects (e.g., a moving body coming to a stop and losing its mechanical energy as heat to the floor No process is possible whose sole result is the and the body; a rotating blade in a liquid coming absorption of heat from a reservoir and the to a stop due to viscosity and losing its complete conversion of the heat into work. mechanical energy with corresponding gain in the internal energy of the liquid). Since Clausius statement dissipative effects are present everywhere and can be minimised but not fully eliminated, most No process is possible whose sole result is the processes that we deal with are irreversible. transfer of heat from a colder object to a hotter object. A thermodynamic process (state i → state f ) is reversible if the process can be turned back It can be proved that the two statements such that both the system and the surroundings above are completely equivalent. return to their original states, with no other change anywhere else in the universe. From the 12.12 REVERSIBLE AND IRREVERSIBLE preceding discussion, a reversible process is an PROCESSES idealised notion. A process is reversible only if it is quasi-static (system in equilibrium with the Imagine some process in which a thermodynamic surroundings at every stage) and there are no system goes from an initial state i to a final dissipative effects. For example, a quasi-static state f. During the process the system absorbs heat Q from the surroundings and performs work W on it. Can we reverse this process and bring both the system and surroundings to their initial states with no other effect anywhere ? 2020-21

316 PHYSICS isothermal expansion of an ideal gas in a from temperature T1 to T2 and then back from cylinder fitted with a frictionless movable piston temperature T2 to T1. Which processes should is a reversible process. we employ for this purpose that are reversible? A little reflection shows that we can only adopt Why is reversibility such a basic concept in reversible adiabatic processes for these thermodynamics ? As we have seen, one of the purposes, which involve no heat flow from any concerns of thermodynamics is the efficiency reservoir. If we employ any other process that is with which heat can be converted into work. not adiabatic, say an isochoric process, to take The Second Law of Thermodynamics rules out the system from one temperature to another, we the possibility of a perfect heat engine with 100% shall need a series of reservoirs in the efficiency. But what is the highest efficiency temperature range T2 to T1 to ensure that at each possible for a heat engine working between two stage the process is quasi-static. (Remember reservoirs at temperatures T1 and T2 ? It turns again that for a process to be quasi-static and out that a heat engine based on idealised reversible, there should be no finite temperature reversible processes achieves the highest difference between the system and the reservoir.) efficiency possible. All other engines involving But we are considering a reversible engine that irreversibility in any way (as would be the case operates between only two temperatures. Thus for practical engines) have lower than this adiabatic processes must bring about the limiting efficiency. temperature change in the system from T1 to T2 and T2 to T1 in this engine. 12.13 CARNOT ENGINE Fig. 12.11 Carnot cycle for a heat engine with an Suppose we have a hot reservoir at temperature ideal gas as the working substance. T1 and a cold reservoir at temperature T2. What is the maximum efficiency possible for a heat A reversible heat engine operating between engine operating between the two reservoirs and two temperatures is called a Carnot engine. We what cycle of processes should be adopted to have just argued that such an engine must have achieve the maximum efficiency ? Sadi Carnot, the following sequence of steps constituting one a French engineer, first considered this question cycle, called the Carnot cycle, shown in Fig. in 1824. Interestingly, Carnot arrived at the 12.11. We have taken the working substance of correct answer, even though the basic concepts the Carnot engine to be an ideal gas. of heat and thermodynamics had yet to be firmly (a) Step 1 → 2 Isothermal expansion of the gas established. taking its state from (P1, V1, T1) to We expect the ideal engine operating between (P2, V2, T1). two temperatures to be a reversible engine. The heat absorbed by the gas (Q1) from the Irreversibility is associated with dissipative reservoir at temperature T1 is given by effects, as remarked in the preceding section, and lowers efficiency. A process is reversible if it is quasi-static and non-dissipative. We have seen that a process is not quasi-static if it involves finite temperature difference between the system and the reservoir. This implies that in a reversible heat engine operating between two temperatures, heat should be absorbed (from the hot reservoir) isothermally and released (to the cold reservoir) isothermally. We thus have identified two steps of the reversible heat engine : isothermal process at temperature T1 absorbing heat Q1 from the hot reservoir, and another isothermal process at temperature T2 releasing heat Q2 to the cold reservoir. To complete a cycle, we need to take the system 2020-21

THERMODYNAMICS 317 Eq. (12.12). This is also the work done (W1 → 2) V2  T2  1 /(γ −1) by the gas on the environment. V3  T1  i.e. = (12.29) W1 → 2 = Q1 = µ R T1 ln  V2  (12.23) Similarly, since step 4 → 1 is an adiabatic  V1  process (b) Step 2 → 3 Adiabatic expansion of the gas from (P2, V2, T1) to (P3, V3, T2) γ −1 = T1 γ −1 Work done by the gas, using T2 V4 V1 Eq. (12.16), is   1 / γ −1   µR (T1 − T2 ) i.e. V1 = T2 (12.30) V4 T1 W2→3 = γ −1 (12.24) (c) Step 3 → 4 Isothermal compression of the From Eqs. (12.29) and (12.30), gas from (P3, V3, T2) to (P4, V4, T2). V3 = V2 (12.31) V4 V1 Heat released (Q2) by the gas to the reservoir at temperature T2 is given by Eq. (12.12). This Using Eq. (12.31) in Eq. (12.28), we get is also the work done (W3 → 4) on the gas by the environment. η = 1 − T2 T1  V3  (Carnot engine) (12.32)  V4  W3 → 4 = Q2 = µRT2 ln (12.25) (d) Step 4 → 1 Adiabatic compression of the We have already seen that a Carnot engine gas from (P4, V4, T2) to (P1,V1, T1). is a reversible engine. Indeed it is the only Work done on the gas, [using Eq.(12.16), is reversible engine possible that works between W4 →1 = µR  T1 − T2  (12.26) two reservoirs at different temperatures. Each  γ - 1  step of the Carnot cycle given in Fig. 12.11 can From Eqs. (12.23) to (12.26) total work done by the gas in one complete cycle is be reversed. This will amount to taking heat Q2 from the cold reservoir at T2, doing work W on the system, and transferring heat Q1 to the hot reservoir. This will be a reversible refrigerator. W = W1 → 2 + W2 → 3 – W3 → 4 – W4 → 1 We next establish the important result = µ RT1 ln  V2  – µ RT2 ln  V3  (12.27) (sometimes called Carnot’s theorem) that  V1   V4  (a) working between two given temperatures T1 The efficiency η of the Carnot engine is and T2 of the hot and cold reservoirs respectively, no engine can have efficiency more than that of η = W = 1 − Q2 Q1 Q1 the Carnot engine and (b) the efficiency of the Carnot engine is independent of the nature of the working substance. To prove the result (a), imagine a reversible (Carnot) engine R and an irreversible engine I In  V3  working between the same source (hot reservoir)  V4  = −  T2  and sink (cold reservoir). Let us couple the T1 1  V2  (12.28) engines, I and R, in such a way so that I acts  V1  In like a heat engine and R acts as a refrigerator. Now since step 2 → 3 is an adiabatic process, Let I absorb rheeleaatsQe 1thferohmeatthQe1-sWou′ rtcoet,hdeesliinvekr. work W ′ and We arrange so that R returns the same heat Q1 to the source, taking heat Q2 from the sink and γ −1 = γ −1 requiring work W = Q1 – Q2 to be done on it. T1 V2 T2 V3 2020-21

318 PHYSICS Now suppose ηR < ηI i.e. if R were to act than that of the Carnot engine. A similar as an engine it would give less work output than that of I i.e. W < W ′ for a given Q1. With R argument can be constructed to show that a acting like a refrigerator, this would mean Q2 = Q1 – W > Q1 – W ′. Thus, on the whole, reversible engine with one particular substance the coupled I-R system extracts heat (Q1 – W) – (Q1 – W ′) = (W ′ – W ) from the cold cannot be more efficient than the one using reservoir and delivers the same amount of work another substance. The maximum efficiency of in one cycle, without any change in the source a Carnot engine given by Eq. (12.32) is or anywhere else. This is clearly against the independent of the nature of the system Kelvin-Planck statement of the Second Law of performing the Carnot cycle of operations. Thus Thermodynamics. Hence the assertion η > η I R we are justified in using an ideal gas as a system is wrong. No engine can have efficiency greater in the calculation of efficiency η of a Carnot engine. The ideal gas has a simple equation of state, which allows us to readily calculate η, but the final result for η, [Eq. (12.32)], is true for any Carnot engine. This final remark shows that in a Carnot cycle, I Q1 = T1 (12.33) Q2 T2 R is a universal relation independent of the nature W of the system. Here Q1 and Q2 are respectively, the heat absorbed and released isothermally Fig. 12.12 An irreversible engine (I) coupled to a (from the hot and to the cold reservoirs) in a reversible refrigerator (R). If W ′ > W, this Carnot engine. Equation (12.33), can, therefore, would amount to extraction of heat be used as a relation to define a truly universal W′ – W from the sink and its full thermodynamic temperature scale that is conversion to work, in contradiction with independent of any particular properties of the the Second Law of Thermodynamics. system used in the Carnot cycle. Of course, for an ideal gas as a working substance, this universal temperature is the same as the ideal gas temperature introduced in section 12.11. SUMMARY 1. The zeroth law of thermodynamics states that ‘two systems in thermal equilibrium with a third system separately are in thermal equilibrium with each other’. The Zeroth Law leads to the concept of temperature. 2. Internal energy of a system is the sum of kinetic energies and potential energies of the molecular constituents of the system. It does not include the over-all kinetic energy of the system. Heat and work are two modes of energy transfer to the system. Heat is the energy transfer arising due to temperature difference between the system and the surroundings. Work is energy transfer brought about by other means, such as moving the piston of a cylinder containing the gas, by raising or lowering some weight connected to it. 3. The first law of thermodynamics is the general law of conservation of energy applied to any system in which energy transfer from or to the surroundings (through heat and work) is taken into account. It states that ∆Q = ∆U + ∆W where ∆Q is the heat supplied to the system, ∆W is the work done by the system and ∆U is the change in internal energy of the system. 2020-21

THERMODYNAMICS 319 4. The specific heat capacity of a substance is defined by s= 1 ∆Q m ∆T where m is the mass of the substance and ∆Q is the heat required to change its temperature by ∆T. The molar specific heat capacity of a substance is defined by C = 1 ∆Q µ ∆T where µ is the number of moles of the substance. For a solid, the law of equipartition of energy gives C = 3R which generally agrees with experiment at ordinary temperatures. Calorie is the old unit of heat. 1 calorie is the amount of heat required to raise the temperature of 1 g of water from 14.5 °C to 15.5 °C. 1 cal = 4.186 J. 5. For an ideal gas, the molar specific heat capacities at constant pressure and volume satisfy the relation Cp – Cv = R where R is the universal gas constant. 6. Equilibrium states of a thermodynamic system are described by state variables. The value of a state variable depends only on the particular state, not on the path used to arrive at that state. Examples of state variables are pressure (P ), volume (V ), temperature (T ), and mass (m ). Heat and work are not state variables. An Equation of State (like the ideal gas equation PV = µ RT ) is a relation connecting different state variables. 7. A quasi-static process is an infinitely slow process such that the system remains in thermal and mechanical equilibrium with the surroundings throughout. In a quasi-static process, the pressure and temperature of the environment can differ from those of the system only infinitesimally. 8. In an isothermal expansion of an ideal gas from volume V1 to V2 at temperature T the heat absorbed (Q) equals the work done (W ) by the gas, each given by Q=W= µRT ln  V2  V1 9. In an adiabatic process of an ideal gas PV γ = constant where γ = Cp Cv Work done by an ideal gas in an adiabatic change of state from (P1, V1, T1) to (P2, V2, T2) is W = µ R (T1 − T2 ) γ –1 10. Heat engine is a device in which a system undergoes a cyclic process resulting in conversion of heat into work. If Q1 is the heat absorbed Wfr,otmhetheeffiscoieunrccye,ηQo2f is the heat released to the sink, and the work output in one cycle is the engine is: η = W = 1 − Q2 Q1 Q1 2020-21

320 PHYSICS 11. In a refrigerator or a heat pump, the system extracts heat Q2 from the cold reservoir and releases Q1 amount of heat to the hot reservoir, with work W done on the system. The co-efficient of performance of a refrigerator is given by α= Q2 = Q2 W Q1 − Q2 12. The second law of thermodynamics disallows some processes consistent with the First Law of Thermodynamics. It states Kelvin-Planck statement No process is possible whose sole result is the absorption of heat from a reservoir and complete conversion of the heat into work. Clausius statement No process is possible whose sole result is the transfer of heat from a colder object to a hotter object. Put simply, the Second Law implies that no heat engine can have efficiency η equal to 1 or no refrigerator can have co-efficient of performance α equal to infinity. 13. A process is reversible if it can be reversed such that both the system and the surroundings return to their original states, with no other change anywhere else in the universe. Spontaneous processes of nature are irreversible. The idealised reversible process is a quasi-static process with no dissipative factors such as friction, viscosity, etc. 14. Carnot engine is a reversible engine operating between two temperatures T1 (source) and T2 (sink). The Carnot cycle consists of two isothermal processes connected by two adiabatic processes. The efficiency of a Carnot engine is given by η = 1 − T2 (Carnot engine) T1 No engine operating between two temperatures can have efficiency greater than that of the Carnot engine. 15. If Q > 0, heat is added to the system If Q < 0, heat is removed to the system If W > 0, Work is done by the system If W < 0, Work is done on the system Quantity Symbol Dimensions Unit Remark Co-efficienty of volume K–1 αv = 3 α1 expansion α [K–1] Heat supplied to a system v Specific heat capacity ∆Q [ML2 T–2] J Q is not a state variable Thermal Conductivity s [L2 T–2 K–1] J kg–1 K–1 [MLT–3 K–1] dt K J s–1 K–1 H = – KA dx 2020-21