Justification and Presentation of the Second Law in a Course on Engineering Thermodynamics

G. I. Fuks

1.

The classical justifications of the second law of thermodynamics (Clausius, Thomson, Planck, and later Schiller and Carathéodory) give rise to a number of objections from a methodological standpoint. Thus, in Clausius’s method the fundamental expression for the efficiency of the Carnot cycle is derived on the basis of the properties of a hypothetical working body—an ideal gas with constant heat capacities \(c_p\) and \(c_v\). The introduction of the concept of entropy occurs with a considerable shift away from the beginning of the discussion of the question.

It seems methodologically advisable to justify separately the principles of the existence and increase of entropy. As the initial propositions for this purpose one should choose propositions that are generalizations of experience and lead most quickly and simply to the final results.

The principle of the existence of entropy can be justified if, as the initial proposition, we accept:

Equality of the temperatures of two bodies is a necessary and sufficient condition of their thermal equilibrium, i.e., of the absence of heat transfer between them.

First a system is considered consisting of two simple bodies whose state is determined by two parameters. These bodies are in an adiabatic shell that admits no thermal action from outside. The bodies of the system are separated from one another by a well-conducting heat-transfer partition (Fig. 1). In this system only direct heat exchange between bodies No. 1 and No. 2 is possible. The law of action on the pistons may be chosen arbitrarily, thereby carrying out various processes. We shall carry out in this system exclusively reversible processes, for which the change in the positions of the pistons must be performed sufficiently slowly (infinitely slowly). The presence of a heat-conducting partition—

heat-conducting partition

adiabatic insulation

Fig. 1.

heat transfer ensures equality of the temperatures of the bodies of the system, although in absolute value the temperature may change. In any elementary process, the quantity of heat received by body No. 1 \((dQ_1)\) is equal to the quantity of heat given up by the other body \((-dQ_2)\), i.e.

\[ dQ_1=-dQ_2. \tag{1} \]

The mathematical expression of the first law for an elementary process of body No. 1 as a simple body has the form

\[ dQ_1=dU_1+p_1dV_1, \tag{2} \]

where

\[ U_1=U_1(t_1,V_1)\text{—the internal energy of body No. 1,} \tag{3} \]

\[ p_1=p_1(t_1,V_1)\text{—the pressure of body No. 1.} \tag{4} \]

Using expressions (3) and (4), we bring relation (2) to the form

\[ dQ_1=M(t_1,V_1)\,dt+N(t_1,V_1)dV_1. \tag{5} \]

Expressions (2) or (5) are not complete differentials. But with two independent variables there exists for (5) an integrating divisor (factor) in the form of some function

\[ \tau'=\tau'(t_1,V_1). \tag{6} \]

Then

\[ \frac{dQ_1}{\tau'(t_1,V_1)}=dZ_1(t_1,V_1), \tag{7} \]

where \(dZ_1\) is the complete differential of the independent variables.

\[ Z_1=Z_1(t,V_1). \tag{8} \]

Let us now take the quantities \(t\) and \(Z_1\) as the independent variables. From (8) we have

\[ V_1=V_1(t_1,Z_1). \tag{9} \]

Substitution into (6) gives

\[ \tau'=\tau'\{t_1V_1(t_1Z_1)\};\qquad \tau'_x=\tau'_x(t_1Z_1). \tag{10} \]

Then (7) will be rewritten as follows:

\[ \frac{dQ_1}{\tau'_x(t_1Z_1)}=dZ_1. \]

or

\[ dQ_1=\tau'_x(t_1Z_1)\,dZ_1, \tag{11} \]

where \(t\) and \(Z_1\) are independent variables.

An analogous derivation for \(dQ_2\) gives

\[ -dQ_2=\tau''_x(t_1Z_2)\,dZ_2. \]

Taking (1) into account, we have

\[ \tau'_x(t_1Z_1)\,dZ_1=\tau''_x(t,Z_2)\,dZ_2, \]

\[ \frac{\tau'_x(t_1Z_1)}{\tau''_x(t,Z_2)}=\frac{dZ_2}{dZ_1}. \tag{12} \]

Since \(t\), \(Z_1\), and \(Z_2\) are independent variables, the right-hand side does not depend on \(t\). Consequently, the left-hand side also does not depend on \(t\). For this there are two possibilities:

1. The integrating divisor \(\tau\) depends only on \(Z\)

\[ \tau=\varphi(Z). \]

But then, for the heat supplied to the body, we have from (11)

\[ dQ = \varphi(Z)\,dZ, \]

i.e., \(dQ\) will be an exact differential, which contradicts the first law. Therefore the assumption that \(\tau=\varphi(Z)\) must be rejected as physically unsatisfactory.

1. The integrating divisor for all bodies is composed as follows:

\[ \tau=f(t)\varphi(Z), \tag{13} \]

where \(f(t)\) is the same function of temperature for all bodies. Indeed, in this case the left-hand side of (12) will not depend on \(t\), although the numerator and denominator separately are functions of \(t\) and \(Z\).

For the heat supplied to a body, the expression obtained is

\[ dQ=f(t)\varphi(Z)\,dZ. \tag{14} \]

The universal function \(f(t)\) for all bodies is called the absolute temperature,

\[ T=f(t). \tag{15} \]

The quantity \(\varphi(Z)\,dZ\) is the differential of a certain function—entropy:

\[ dS=\varphi(Z)\,dZ. \tag{16} \]

Finally,

\[ dQ=T\,dS, \tag{17} \]

where, for a simple body,

\[ S=S(T,V). \tag{18} \]

2.

Expression (18), obtained for a simple body, may be generalized to the case of a complex body (or system of bodies), the state of which is determined not by two but by an arbitrary number of parameters \(T, x_1, x_2,\ldots x_n\). Let, in the scheme shown in Fig. 1, body No. 1 be a simple body, whose state is determined by the parameters \(T\) and \(V\), and body No. 2 be any “complex” body, whose state is determined by an arbitrary number of parameters \(T, x_1, x_2,\ldots x_n\). Both bodies are enclosed in an adiabatic shell, but are separated by a heat-conducting partition. Direct heat exchange between the bodies is possible. In reversible processes both bodies will always have the same temperature \(T\), varying during the process. For body No. 1, as was indicated,

\[ dQ_1=M(T,V)\,dT+N(T,V)\,dV. \]

For body No. 2, analogously,

\[ -dQ_2=M'\,dT+N_1\,dx_1+N_2\,dx_2+\ldots+N_n\,dx_n, \tag{19} \]

where \(M'\), \(N_1\), \(N_2\) are functions of \(T, x_1, x_2,\ldots x_n\).

Since \(dQ_1+dQ_2=0\), then

\[ dQ_1=M'\,dT+N_1\,dx_1+\ldots+N_n\,dx_n. \tag{20} \]

Divide this equation by the common temperature of the bodies \(T\). We obtain

\[ \frac{dQ_1}{T}=M'_1\,dT+N'_1\,dx_1+N'_2\,dx_2+\ldots+N'_n\,dx_n, \tag{21} \]

where \(M'_1=\dfrac{M'}{T}\), \(N'_1=\dfrac{N_1}{T}\) are functions of \(T, x_1, x_2,\ldots x_n\).

But \(\dfrac{dQ_1}{T}\), as indicated, represents the change in entropy of body No. 1—\(dS_1(T,V)\). Therefore

\[ dS_1=M_1'\,dT+N_1'\,dx_1+N_2'\,dx_2+\ldots+N_n'\,dx_n . \tag{22} \]

On the other hand, the system of bodies No. 1 and No. 2 may be regarded as a certain body in an adiabatic envelope, whose state can be acted upon from outside by changing the quantities \(V, x_1, x_2,\ldots,x_n\). All the remaining parameters of this body, including the temperature \(T\), will change in accordance with the change in \(V, x_1, x_2,\ldots,x_n\). Thus,

\[ dT=\frac{\partial T}{\partial V}\,dV+\frac{\partial T}{\partial x}+\ldots+\frac{\partial T}{\partial x_n}\,dx_n . \tag{23} \]

Since the parameters of state are complete differentials, integration gives

\[ T=\varphi\left(V,x_1,x_2,\ldots x_n\right). \tag{24} \]

This means that, under conditions of thermal homogeneity of an adiabatically isolated system in which reversible processes are carried out, the value of the temperature \(T\) is a function of the parameters characterizing the external action on the system, \(V, x_1, x_2,\ldots x_n\).

Let us now, by acting on \(V, x_1, x_2,\ldots x_n\), cause bodies No. 1 and No. 2 to undergo a number of reversible processes, with the condition that in the final state the quantities \(V, x_1,\ldots x_n\) will have the same values as in the initial state. In accordance with (24), this means that the temperature of the bodies \(T\) will also prove to be equal to the initial one. Thus, both bodies No. 1 and No. 2 will undergo cyclic processes. From (22), for these processes we obtain

\[ \oint dS_1=\oint \left[M_1'\,dT+N_1'\,dx_1+\ldots+N_n'\,dx_n\right]. \tag{25} \]

Since the entropy of the first body is a complete differential of the variables \(T\) and \(V\), \(\oint dS_1=0\). Therefore

\[ \oint \left[M_1'\,dT+N_1'\,dx_1+\ldots+N_n'\,dx_n\right]=0 . \tag{26} \]

This means that the expression under the integral sign is a complete differential with respect to the variables \(T, x_1, x_2,\ldots x_n\), i.e., the differential of the entropy of body No. 2: \(dS_2(T,x_1,x_2,\ldots x_n)\).

Consequently, for body No. 2

\[ \frac{dQ_2}{T}=dS_2 \tag{27} \]

or, in general, for any body

\[ dQ=T\,dS, \tag{28} \]

where \(S\) is the entropy of the body, which is a function of the parameters of the state of the body.

3.

The existence of the expression for heat (28) makes it possible to apply the \(sT\) diagram directly (along with the \(Vp\) diagram) to the consideration of reversible and, in particular, cyclic processes. The necessity of removing heat from the working body in a cyclic cycle is clearly established. The question is posed of cyclic processes with two heat sources at constant temperatures, which logically leads to the Carnot cycle. An expression is found for the efficiency of the Carnot cycle.

The properties of the entropy of bodies and of a system when heat is supplied to and removed from them are investigated.

It is indicated that, when only reversible processes are carried out, the transfer of heat from hot bodies to cold bodies and back cannot

be the only result of the processes of the system. The concept of the work-producing capacity of heat and the methods for calculating it are established.

Consideration of these questions is greatly facilitated by the possibility of simultaneously using \(Vp\) and \(sT\) diagrams.

4.

To justify the principle of increasing entropy in real processes, two propositions are formulated:

1. The transfer of heat from a heated body to a colder body can be the only result of the processes of a system.

2. It is impossible to carry out such a set of processes whose only result would be the transfer of heat from a cold body to a more heated body.

As is known, from the second proposition, which is a refined classical formulation of Clausius, one can obtain Planck’s formulation, the proposition on the impossibility of realizing a perpetual-motion machine of the second kind, and so on. Thus, for example, if one admits the possibility of the operation of a perpetual-motion machine of the second kind, one can imagine that the work obtained will be converted into an equivalent amount of heat at an arbitrarily high temperature. The same result is led to by the assumption of the possibility of constructing a heat engine with an efficiency exceeding the efficiency of a heat engine operating only by reversible processes.

On the other hand, the propositions given above can be interpreted as follows: in a closed system only those processes can occur which lead to an increase in the entropy of the system, and only in the limiting case of reversible processes—to constancy of the value of the entropy of the system. This makes it possible to obtain a mathematical formulation of the second law for irreversible processes, to derive an expression for calculating the work-producing capacity of heat in irreversible processes, and so on. It is also useful to emphasize that the possibility of transferring internal energy from a body at high temperature to a body at a lower temperature, without any other energy effects, is likewise a specific property of thermal motion.

From the methodological standpoint, in presenting the course it is also desirable to clarify the physical meaning of entropy. For this purpose one may consider a real body, for example, \(1\,m^3\) of gas, in a rigid adiabatic shell, in a nonequilibrium state, as a “system” of small bodies. The transition to the equilibrium state is accomplished in this “system” by direct heat exchange, by diffusion, and similar irreversible processes, in which the entropy of the body increases. At the same time, there is an increase in the probability of the state of the body. Thus, the magnitude of the entropy of a body is connected with the probability of its state and with the distribution of internal energy within the body. It is also necessary to emphasize the generality of the principle of the existence of entropy and the limited character of the principle of its increase.

The three propositions put forward are not a “new” justification of the second law of thermodynamics. The proposition on the equality of the temperatures of bodies in thermal equilibrium and on the possibility of the transfer of heat “by itself” from a heated body to a colder body are, as it seems to us, present in hidden form in the classical formulation of Clausius, since it speaks of cold and more heated bodies. A clear formulation of these propositions appears expedient, since it makes it possible at once to operate with the concept of entropy. This is important because the use of entropy as a state parameter is increasing more and more, especially in connection with the introduction of the concepts of the work-producing capacity of heat, the development of thermodynamic methods for assessing the perfection of heat engines, and the development of the thermodynamics of irreversible processes.