4 Zeroth Law of Thermo- CHAPTER dynamics: Preliminary Concepts

4.1 INTRODUCTION is a branch of physics that deals with the inter-relations between and other forms of . It is based on principles, which has been formulated by generalizing experimental observations made on a variety of physical systems. It makes no attempt about what is going on at microscopic (atomic or molecular) level. It is a macroscopic (large-scale) theory of natural phenomena. Its aim is to establish relationship between directly observed macroscopic quantities viz. pressure, volume, , concentration, electric field, magnetic field, polarization, magnetization, etc. which define the state of the system under investigation. As the name itself suggests, it employs both thermal and mechanical concepts. Its principles are widely used in design of heat engines, nuclear power stations, rocket propulsion, biological phenomena etc. One of the shortcomings of thermodynamics is that it gives no picture of the internal mechanism of phenomena. The kinetic theory, on the other hand, attempts to establish relationships between macroscopic quantities and microscopic quantities. In this method, the laws of mechanics are applied to the motion of individual molecule and macroscopic properties (such as heat capacity, expansion coefficient, viscosity, diffusion, etc.) are deduced. In statistical mechanics, no attention is paid to individual molecule, instead the assembly of microscopic particles is treated statistically. This approach of describing behavior of system will be discussed in a separate chapter.

4.2 AND SURROUNDINGS To study the behavior of a substance thermodynamically, we select a finite portion of that substance. The portion that is set aside (in imagination) on which we pay our attention or on which experimental observations are made is called system. Everything outside this portion which has direct bearing on the behavior of the system is called surroundings. The system and the surroundings comprise what we call universe. The boundary which separates the system from surroundings is called wall. The wall of the system may or may not allow it to interact with the surroundings. A wall, which prevents any exchange of matter and energy between the system and surroundings is called isolating wall and the system is said to be isolated. Such a system will not be affected by its surrounding and is not of much importance from thermodynamic point of view. 70 Thermal and Statistical Physics

If the boundary of the system permits exchange of matter and energy, the system is called open system. On the other hand, if the exchange of energy and not matter occurs between the system and surroundings, the system is called closed system. A wall which prevents thermal interaction is known as and a system enclosed within an adiabatic wall is called thermally isolated. If exchange of heat takes place between the system and surroundings through the boundary wall, the boundary is called diathermic wall.

4.3 THERMODYNAMIC VARIABLES There are, in general, two points of view to study the behavior of the system or its interactions with the surroundings or both: macroscopic point of view and microscopic point of view. The macroscopic description of system involves the specification of a few fundamental measurable properties of the system such as volume, pressure, temperature, amount of substance etc., without considering the internal structure of the system. The macroscopic quantities refer to gross characteristics or large- scale properties of the system and are called macroscopic thermodynamic coordinates (variables or parameters). Of the many thermodynamic variables, only a few variables are needed for the description of the system. In microscopic description of a system assumptions are made about the structure of matter under investigation. It postulates the existence of molecules, their motion, their energy states, their interactions etc. Thus a large number of quantities, which refer to small-scale properties of the system, needed to specify the state of the system. These are called microscopic coordinates (variables or parameters). These variables cannot be measured in laboratory. A thermodynamic system is one whose state can be described in terms of thermodynamic variables. The variables defining the state of a thermodynamic system differ from system to system. For instance, the state of a gas enclosed in a vessel is described by its pressure, volume, temperature and mass. The state of a metal bar is described by its length, cross-section, tension, temperature, the state of a liquid film by its area, surface tension etc. A dielectric and a magnetic system need additional variables such as polarization and magnetization respectively. Although it might seem that the macroscopic and microscopic points of view are entirely different and incompatible, there is nevertheless, a relation between them and when both points of view are applied to the same system, they lead to the same conclusion. The relation between the two points of view lies in the fact that the macroscopic properties are really averages over a long period of time of a large number of microscopic characteristics. For example the macroscopic quantity pressure is the average rate of change of momentum due to all the molecular collisions made on a unit area of the container of the gas.

4.4 EXTENSIVE AND INTENSIVE VARIABLES The state variables which are proportional to the mass of the system are called extensive variables. For instance, volume, internal energy, enthalpy, entropy, etc. are extensive variables. On the other hand, intensive variables are those which are independent of mass of the system. Temperature, pressure and density are intensive variables. Intensive variables are invariably denoted by capital letters such as T and P. Zeroth Law of Thermodynamics: Preliminary Concepts 71

The value of extensive variable per unit mass is called specific variable. Usually an extensive variable is denoted by capital letter and its specific value by small letter. The specific value of an extensive variable is independent of mass and hence is an intensive variable. If a system with mass m occupies a volume V, then its specific volume is denoted by v and given by V Specific volume, v = m The ratio of the value of an extensive variable to the number of moles of the system is called molar specific value. For example, molar specific volume is given by

V Molar specific volume, v = , n is number of moles. n Writing thermodynamic equations in terms of specific values of extensive variables is convenient because they become independent of mass of the system.

Table 4.1 Extensive and Intensive Variables

Systems Extensive variables Specific value Intensive variables (1) Hydrostatic system Volume (V) v Pressure (P), Energy (U or E) u or e Temperature (T) Entropy (S) s Enthalpy (H) h (2) Stretched wire Length (L) l Tension (F) (3) Liquid film Area (A) a Surface tension (s) (4) Electric cell Charge (Q) q E.M.F. (e) (5) Dielectric slab Polarization (P) p Electric field (E) (6) Paramagnetic salt Magnetization (M) m Magnetic field (B or H)

4.5 : ZEROTH LAW OF THERMODYNAMICS A system is said to be in thermal equilibrium if its thermodynamic coordinates remain unchanged. Consider two gaseous systems which are separated by a wall. If the change in thermodynamic coordinates of one system has no effect on those of the other system, then the wall separating the two systems is said to be adiabatic wall. On the other hand, if the changes in thermodynamic coordinates of one system causes changes in those of the other and the composite system attains a new equilibrium state, then the wall separating them is called diathermic or . Two systems separated by a diathermic wall are in equilibrium with each other. Now consider two systems A and B separated by an adiabatic wall and a third system C separated from A and B both by a diathermic wall as shown in Fig. 4.1. In this arrangement, system A and C will be in thermal equilibrium and so will be the systems B and C. Now if the adiabatic wall between A and B is replaced by a diathermic wall, it is found that A and B are also in thermal equilibrium. From this fact we come to conclusion that if system A is in thermal equilibrium with B and B is in thermal equilibrium with C, then A and C will also be in thermal equilibrium. This result is known as Zeroth law of thermodynamics and is stated as follows. 72 Thermal and Statistical Physics

Adiabatic wall

AB

Diathermic wall

C

Fig. 4.1. When any two bodies or systems are each separately in thermal equilibrium with a third body, they are also in thermal equilibrium with each other and there exists a common useful scalar quantity called temperature which is a property of all thermodynamic systems in equilibrium states such that the temperature equality is a necessary and sufficient condition for thermal equilibrium. The above statement can also be stated in a mathematical form. Consider three systems 1, 2 and 3. Let X and Y be independent variables to describe their states. For the system 3, these variables are X3 and Y3. The condition for system 1 to be in thermal equilibrium with system 3 can be expressed in functional form as

f1,3 (X1, Y1, X3, Y3) = 0

or X3 = F13 (X1, Y1, Y3) ...(4.1) That is, out of the four variables, only three are independent. Similarly, the condition of equilibrium of system 2 and 3 can be expressed as

X3 = F23 (X2, Y2, Y3) ...(4.2) Eliminating X3 from Eqs. (4.1) and (4.2) we have

F13 (X1, Y1, Y3) – F23 (X2, Y2, Y3) = 0 ...(4.3) But according to the zeroth law, system 1 and 2 will also be in thermal equilibrium with one another. This requires that

f12 (X1, Y1, X2, Y2) = 0 ...(4.4) To satisfy the Eqs. (4.3) and (4.4) simultaneously, the function F13 and F23 must be of the form

F13 = f1 (X1,Y1) f (Y3) + h (Y3) F23 = f2 (X2, Y2) f (Y3) + h (Y3) These imply that Eq. (4.3) should be of the form

f1 (X1, Y1) = f2 (X2, Y2) or in general we can write

f1 (X1, Y1) = f2 (X2, Y2) = f3 (X3, Y3) This equation shows that when two or more systems are in thermal equilibrium, for each system there exists a single valued function of state variables which has a common numerical value for all of them. The common value is known as empirical temperature. Thus we may write, in general, for thermal equilibrium f (X, Y) = q Zeroth Law of Thermodynamics: Preliminary Concepts 73

This equation is called the equation of state and q is the empirical temperature. Thus temperature is a physical quantity which describes the thermal state of the system. It gives the feeling of hotness and coldness of a body. The temperature defined in this way has no objective use and hence cannot be used in scientific methods. If it is to be used in scientific analysis, it must have objective definition and should be assigned numerical value. The instruments employed for the measurement of tempera- ture are called thermometers. The construction of a thermometer involves following methods : (i) A suitable material and its temperature dependent property is chosen. (ii) Two fixed states of the material, which can be easily reproduced are selected. In ordinary glass thermometer, the material used is mercury and its temperature dependent property which is used is the length of column in glass tube. The fixed states are the ice point and the steam point. These two states are arbitrarily assigned zero and 100 on Celsius scale and the interval between 0 and 100 is divided into 100 parts. Each part represents 1°C. Many other properties of materials have been used to devise thermometers. Unfortunately, all the thermometers do not agree in the whole range of their operation. To remove indefiniteness or inconsistency in their readings, the International Temperature scale was revised in 1968 and triple point of water (temperature at which the three phases ice, water and vapor coexist) was assigned a temperature 273.16 K (= 0.01°C). This scale of temperature is known as Giaque Scale.

If Ptr represents the pressure of gas thermometer at triple point of water, P the pressure at unknown temperature T, then

F P I T = lim27316 . kelvin P G J tr Æ 0 H Ptr K The Celsius temperature scale employs a degree of the same magnitude as that of the ideal gas scale, but its zero point is shifted so that the Celsius temperature of triple point of water is 0.01 degree Celsius (0.01°C). Thus the zero on the Celsius temperature scale is 273.15 on Kelvin or absolute scale of temperature.

4.6 THERMODYNAMIC EQUILIBRIUM If the temperature at all points in a system under investigation has the same value, which is equal to that of the surroundings, the system is said to be in a state of thermal equilibrium. If there exists any temperature gradient inside an isolated system, after a sufficiently long time, the temperature will level out and the system will come in thermal equilibrium. When there is no unbalanced force in the interior of a system and none between a system and its surroundings, the system is said to be in a state of mechanical equilibrium. It should be borne in mind that for a system to be in mechanical equilibrium it is not necessary that the pressure at all points be the same. For instance, a liquid in a vessel has maximum hydrostatic pressure at the bottom and minimum at the top level, still it may be in mechanical equilibrium. When a system does not undergo any change in density or chemical composition, it is said to be in chemical equilibrium. A system in thermal, mechanical and chemical equilibrium is said to be in the state of thermodynamic equilibrium. In thermodynamic equilibrium, all the parameters of the system have definite values and remain unchanged as long as the external conditions of the system remain unchanged. If we choose any two thermodynamic variables and plot one along x-axis and the other along y-axis, the system in 74 Thermal and Statistical Physics equilibrium will be represented by a point on the graph. If the state variables of a system undergo erratic change, it is said to be in non-equilibrium state and this state cannot be represented by a point on such a graph.

4.7 THERMODYNAMIC PROCESS If a system initially in non-equilibrium state, left to itself, will pass over to equilibrium state. The transition of a system from one equilibrium state to another is accompanied by changes in state variables. The term process is used to describe this transition. A process, in which a system passes through a continuous sequence of equilibrium states, is called quasi-static process. An infinitely slow process is a quasi-static process.

4.8 REVERSIBLE AND IRREVERSIBLE PROCESSES A reversible process is one which when retraced in opposite direction, the system and the surroundings pass through the same intermediate states as encountered in the forward process, the thermal and mechanical effects in the reverse process should be equal in magnitude but opposite in sign to those in the forward process. A reversible process should not produce a permanent change in the system and its surroundings. The changes produced in the forward process should be completely erased in the reverse process. A reversible process can be carried out in either direction. “A reversible process is one in which the system and the surroundings both can be restored to their initial states by reversing the direction of the process.” If a system absorbs heat Q and performs W in the direct process, then by extracting heat Q from the system and doing work W on the system, the system should pass through the same intermediate states in the reversible process. The conditions for a process to be reversible are (i) There should be no dissipating forces acting within the system. (ii) The process should be carried out infinitesimally slowly so that the system is at each step in equilibrium state i.e., the process should be quasi-static. These conditions are never realized in practice. Hence perfectly reversible processes are only abstractions. By minimizing the dissipating forces and allowing the system under investigation to go over from one state to another quasi-statically, a real process may be approximated to a reversible one. Consider a process in which a body exchanges heat with another body, which we shall call heat reservoir. Let the heat capacity of the reservoir be infinitely great. This means that addition to or extraction of finite amount of heat from the reservoir will not cause any change in its temperature. The process of heat exchange between the body and the reservoir can be reversible only if upon receiving heat and returning it in the reverse process to the reservoir, the body has the same temperature as that of the reservoir. Strictly speaking, when receiving heat the temperature of the body must be lower than that of the reservoir by an infinitely small value (otherwise no heat will flow from the reservoir to the body) and when giving up heat, the temperature of the body must be higher than that of the reservoir by an infinitely small value. Consequently, the only reversible process, attended by heat exchange with reservoir whose temperature remains constant, is an isothermal process at the temperature of the reservoir. A system in equilibrium state is specified by thermodynamic coordinates. Since in a reversible process the system is always very near to the equilibrium states, and therefore it can be represented by a continuous curve on P-V diagram and the analysis of a reversible process is easier. Zeroth Law of Thermodynamics: Preliminary Concepts 75

A process which does not satisfy the conditions of reversible process is called irreversible process. Now consider a viscous liquid in contact with a heat bath. By shaking the liquid, work may be done against the viscous forces. This work appears as heat, which is transferred to the heat bath. If W is the work done on the liquid and Q is the heat transferred to the heat bath in the direct process, then by extracting heat Q from the heat bath, we never get the same amount of work W in the reverse process. (According to second law of thermodynamics, the extraction of heat and its complete conversion into work is not possible.) In general, the processes, in which dissipative forces like friction, viscosity, electrical resistance, magnetic hysteresis are involved, are irreversible. All natural processes i.e., those which occur spontaneously are irreversible. A process of between two bodies with finite temperature difference is irreversible.

4.9 EQUATION OF STATE The parameters describing the state of a system are not independent. Instead they are interrelated through an equation called the equation of state. For a system described by P, V, T, a relation of the type f (P, V, T) = 0 ...(4.5) always exists and is called the equation of state. This type of equation cannot be derived from thermodynamic methods. It is obtained either from experimental observations or from microscopic considerations. Ideal and real gases are the two important systems for our study. By ideal gas, we mean a gas in which the interaction between the molecules is negligibly small and the volume occupied by the molecules themselves is insignificant in comparison to the volume of the gas. For n moles of ideal gas the relation between P, V, T and n is PV = nRT ...(4.6) In terms of specific volume v = V/n, this equation is written as Pv = RT ...(4.7) In real gases, the intermolecular forces and the volume of the molecules are not negligible. However, at sufficiently high temperature and low pressure, real gases behave like ideal gases. Deviations from ideal behavior are observed at low temperature and high pressure. In order to explain the behavior of real gases, J. D. van der Waals modified the ideal gas equation taking intermolecular forces and finite size of molecules into consideration. His equation of state is

ʈna2 PVnb+ = nRT (for n moles) ...(4.8) Á˜2 ()- ˯V In terms of specific volume v = V/n, this equation is written as ʈa Á˜Pb+ 2 ()v - = RT ...(4.9) ˯v where a and b are constants called van der Waals constants. 76 Thermal and Statistical Physics

4.10 COEFFICIENT OF EXPANSION (EXPANSIVITY) The coefficient of volume expansion b is defined as the fractional change in volume per unit change in temperature at constant pressure. If a P, V, T system undergoes expansion from volume V to V + DV due to rise in temperature from T to T + DT at constant pressure P then b is defined as follows.

1 ʈDV = lim b DÆT 0 Á˜ VT˯D P 1 ʈ∂V or b = Á˜ ...(4.10) VT˯∂ P In terms of molar specific volume v,

1 ʈ∂v b = Á˜ v ˯∂T P For an ideal gas, Pv = RT ʈ∂v and P Á˜ = R ˯∂T P 1 ʈR 1 Therefore b = Á˜ = v ˯P T

Fig. 4.2. Expansitivity of water Thus the expansivity of an ideal gas depends only on its temperature. In general, expansivity is a function of both temperature and pressure. The change in volume of water when it is heated from 0°C is shown in the Fig. 4.2. The specific volume of water decreases on heating in the temperature range from 0 to 4°C and increases beyond 4°C. Thus expansitivity of water (β) at 4°C is negative.

4.11 COMPRESSIBILITY The isothermal compressibility of a substance is defined as the fractional change in volume per unit change in pressure at constant temperature.

1 ʈ∂v kT = - Á˜ ...(4.11) v ˯∂P T The negative sign indicates that an increase in pressure is accompanied by a decrease in volume. Zeroth Law of Thermodynamics: Preliminary Concepts 77

If the change in pressure is carried out under adiabatic condition, then the compressibility defined as above is called adiabatic compressibility.

1 ʈ∂v k S = - Á˜ ...(4.12) v ˯∂P S where the subscript S indicates that process is adiabatic in which entropy S, remains constant. For an ideal gas

Pv = RT ʈ∂v ∵ P Á˜+ v =0 ˯∂P T 1 Whence k = T P For a real substance the compressibility is a function of both temperature and pressure. The reciprocal of compressibility is called bulk modulus E, of the substance.

1 ʈ∂P 1 ʈ∂P ET = = - v Á˜and ES = = - v Á˜ kT ˯∂v T kS ˯∂v S

4.12 RELATION BETWEEN PARTIAL DERIVATIVES Let x, y, z be the thermodynamic variables of a system and each one be expressible in terms of the remaining two. If z = z (x, y) F ∂z I F ∂z I then dz = G J dx + G J dy ...(4.13) H ∂x K y H ∂yK x Similarly, if x = x (y, z) F ∂xI F ∂x I then dx = G J dy + G J dz ...(4.14) H ∂yK z H ∂z K y Substituting the value of dx from (4.14) in (4.13) we obtain

F ∂z I LF ∂x I F ∂x I O F ∂zI dz = G J M dy + G J dzP + dy H ∂x K G ∂yJ H ∂z K G ∂yJ y NMH K z y QP H K x LF ∂z I F ∂x I F ∂z I O F ∂z I F ∂x I = MG J + P dy + G J G J dz H ∂x K G ∂yJ G ∂yJ H ∂x K H ∂z K NM y H K zxH K QP yy L F ∂z I F ∂x I O LF ∂z I F ∂x I F ∂z I O M1 - G J G J P dz = MG J + P dy ...(4.15) H ∂xK H ∂z K H ∂x K G ∂yJ G ∂yJ NM yyQP NM y H K zxH K QP 78 Thermal and Statistical Physics

Let us apply Eq. (4.15) to two neighboring equilibrium states. If the two states have the same y, then dy = 0. Equation (4.15) reduces to

L F ∂z I F ∂x I O M1 - G J G J P =0 x z NM H ∂ K yyH ∂ K QP F ∂z I 1 or G J = ...(4.16) H ∂xK ∂x y GF JI H ∂z K y Similar relations hold for other variables i.e.,

F ∂xI 1 ʈ∂y 1 G J = and Á˜ = ...(4.17) H ∂yK ʈ∂y ˯∂z x F ∂zI z Á˜ ˯∂x G J z H ∂yK x Let us apply the Eq. (4.15) to two equilibrium states having the same value of z, i.e., dz = 0 then

ʈ∂∂zxʈ F ∂z I . = - Á˜Ë¯Á˜ G J ∂xyy Ë ∂ ¯ z H ∂yK x

ʈ∂∂∂xyzʈ ʈ or .. = –1 ...(4.18) Á˜Ë¯Á˜ ˯Á˜ Ë∂yzx ¯z ∂xy ∂ If the variables x, y, z denote pressure P, volume V, and temperature T, respectively then equation (4.18) takes the form

ʈʈʈ∂∂∂PVT Á˜Á˜Á˜.. = –1 ...(4.19) ˯˯˯∂∂∂VTPTPV Notice that P, V, T occur cyclically in Eq. (4.19).

Solved Examples

Example 1. Show that the coefficient of expansion of a van der Waals gas is Rb2 af- = vv b 3 2 RTvv-- 2a af b 1P∂ −  1 ∂v v ∂T Solution. By definition, b =  = v ...(1) v ∂T P ∂P  ∂v T van der Waals equation of state is RT a P = - 2 v - b v Zeroth Law of Thermodynamics: Preliminary Concepts 79

ʈ∂P RPʈ∂ RT2 a ∵ = and = -+23 ˯Á˜T b ˯Á˜ - b ∂ v vv-∂T ()v v Substituting these values in (1) we get

È˘ÊˆR Í˙Á˜ 2 1 ˯- b Rbvv()- b = - Í˙v = Í˙RT2 a 3 2 v RTvv--2 a() b Í˙-+23 Î˚()v - b v 1 For ideal gas a = 0, b = 0, then b = T Example 2. Show that the compressibility of van der Waals gas is 2 2 af- b k = vv . 3 2 RTvv-- 2abaf 1 ʈ∂v Solution. By definition k =– Á˜ ...(1) v ˯∂P T RT a For van der Waals gas P = - 2 v - b v 3 ʈ∂P RT2 a -+-RTvv2 a() b Therefore, = -+ = ˯Á˜ 23 3 2 ∂v T ()v - b v vv()- b

2 2 vv()- b k \ = 3 2 RTv--2ab() v 1 For ideal gas a = 0, b = 0, k = . P Example 3. Find the equation of state of a hypothetical substance whose isothermal compressibility is a 2bT given by k = and expansivity is given by b = , where a and b are constants. v v 1 ∂v 21bT ʈ∂ a Solution. Given that b = = and - v =  Á˜ v ∂T P vv˯∂P T v ʈ∂v ʈ∂v That is, Á˜ = 2andbT Á˜ = – a ˯∂T P ˯∂P T Integration of these equations yield 2 v = bT + f (P) and v = – aP + f (T) Combining these equations we have 2 v – bT + aP = const. 80 Thermal and Statistical Physics

Questions and Problems 1. State Zeroth law of thermodynamics. Explain the following terms: (i) Thermodynamic variables (parameters) (ii) Extensive and intensive variables (iii) Thermodynamic equilibrium (iv) Thermodynamic process (v) Reversible and irreversible processes (vi) Equation of state (vii) Expansivity and compressibility