The SECOND LAW and ENTROPY. 11. IRREVERSIBLE PROCESSES R. C. CANTELO University of Cincinnati, Cincinnati, Ohio
The concept of entropy, developed from the second law as a fundamental fiostulote, is extended to the consideration of irreversible processes. N A previous paper' the writer stated the second law of thermodynamics in the form: heat cannot he converted into work without compensation; and with this as a fundamental postulate showed that there exists a property of a body or system known as its entropy, which is determined completely by the variables which define the state of the hody or system. Thus, if S represents the entropy of the system under consideration, and X I , XS, x3, . . .are the variables which completely define the state of the system,
I
I t was shown also that for a reversihle path of change in state, TdS = dQ
(2)
Hence if A and B represent the initial and final states, respectively, of a system undergoing a change in state, we can write for the change in entropy which accompanies this change in state by a reuersible path,
It is proposed, in the present paper, to extend the concept of entropy .. to the consideration of irreversihle processes. There are two kinds of irreversihle processes. (1) Those which become reversihle in the limit as. for example, the expansion of a gas against a differing by a finite amount from the gas pressure; or the flow of electricity through a voltaic cell. Such processes are conditionally irreversible. (2) Natural spontaneous processes. Irreversibility is inherent in such processes so that it is not within our power to render them reversible by any alterations in conditions whatsoever. Such processes are said to he intrinsically irreuersible. Some such intrinsically irreversihle processes are: the flow of gas into a vacuum; the flow of heat from a hotter to a colder hody; the diffusion of gases into each other; the production of heat by friction; chemical reactions which proceed with a finite velocity. I t will be instructive to show for one of these processes I
CANTBLO,"The Second Law and Entropy,'' J. Cmu.
EDUC., 8,2198 (Nov., 1931).
that it is impossible to reverse the process completely, so that both the system and its surroundings are in their initial states. We shall choose the flow of heat from a hotter to a colder hody. Let the initial state he A , and the final state, B. In the initial state, A, we have two bodies, K, a t a temperature TI, and Kz a t a temperature Tz; TI > Tz. A finite quantity of heat Q is allowed to flow from KI to KZto give state B. Kl and Kz are so large that the transfer of the quantity of heat, Q, does not change the temperatures of K, and Kz. By means of a reversed Carnot's cycle we may take the quantity of heat Q- from Kz and by the expenditure of the amount of work W-, upon the gas used as the working fluid, deliver the quantity of heat Q+ W+ to Kl. Finally, by means of the reversible isothermal expansion of a gas, let us take the quantity of heat W- from K1 and convert it into the equivalent amount of work W+. Then KI and KZare in their initial state, A, hut something else has happened: the gas which was used in the reversihle isothermal expansion by which the quantity of heat, W-, was converted into the equivalent quantity of work, W+,has undergone a change in state from an initial state defined by pl, V, to a final state defined by 92, VZ. Hence the flow of heat from a hotter to a colder hody is an intrinsically irreversihle process. It can be shown that of all heat engines working with the same fixed temperatures of "source" and "sink," a reversible engine is the most efficient. The proof rests upon the reversihility, in the ordinary sense, of the reversihle engine. In the previous paper, it was shown-that the efficiency 1; of a reversible engine is equal to Ii, where TI
+
-
TI
and TZare, respectively, the fixedtemperatures of source and sink. Hence, if Q1-he the quantity of heat ahsorbed a t the temperature Tl by an engine operating in a Carnot's cycle with an element of irreversibility, and Qn+ he the quantity of heat rejected at the temperature Tz, the efficiency of the cycle will be less than that of the reversihle one for the same temperatures, TI and Tz. That is,
We can now extend our treatment to any irreversible cycle. We imagine that at various stages of the cycle, the heat elements 6q,, 6qz,.. .6q, are absorbed from reservoirs at the temperatures TI, Ts, . .. . T, and the heat
elements 6p11, 6pZ1,. . .6p,,' are rejected to reservoirs at the temperatures TI', T,', . ..T.'. Some of the 6p's may be zero. We imagine in addition an auxiliary source at a temperature T , where T > T I . T 2 , . .T,, and an auxiliary sink at the temperature T', where T' < TI', T Z f ,...T.'. Then, by proceeding in the same way as we did for the general reversible cycle, we can show that
We saw, previously,' that a reversible process consists of an infinite number of equilibrium states, infinitely close together. Speaking mathematically, there is an infinitely slow change of the state variables, allowing the system to pass through all intermediate states of equilibrium. Thus, if the state variables are XI, X Z , . . .x., we have for an infinitesimal reversible change at the temperature T,
We are in a position, now, to determine the change in entropy associated with an irreversible change in state. Let us consider a cyclical process ABCDA, in which ABC represents an irreversible change in state, and CDA a reversible path. For the cyclical process, the inequality (5) holds; for the reversible path CDA, however, we can write
Hence, dQ for a reversible process is an inexact differential expression in XI, a,. ..x.. On the other hand, in an irreversible process the values of the thermodynamic variables which hold for the initial state of the system are altered to those which hold for the final state in a short time. There So that 8p, is no continuous change of XI, X Z , . . .x,. the heat element absorbed during an infinitesimal stage of an irreversible process, cannot be expressed as an inexact differential of the state variables. Since for every irreversible process TdS > 6p, and since natural spontaneous processes are intrinsically irreversible, we can draw an important conclusion for irreversible changes in an isolated system. An isolated system is one for which an exchange of heat or work with the surroundings is an impossibility. For an isolated system 6p = 0, and, therefore TdS > 0; in other words, a small irreversible change in an isolated system is attended by an increase in the magnitude of that property of the system known as its entropy. Similarly, for a reversible change in an isolated system TdS = 0. For such a small reversible change in an isolated system the entropy of the system remains constant. A statement of the second law of thermodynamics is sometimes given as follows: Every process occurring in nature is accompanied by an increase in the entropy of the isolated system in which the process takes place. In chemical thermodynamics, however, we are not dealing with isolated systems. Chemical processes occur in systems surrounded by a medium, for example, air, the containing vessel, etc. In chemical thermodynamics it is preferable, therefore, to take as our criterion of a reversible process, the relation TdS = dQ, and for an irreversible one, TdS = 6g E. That is, in an irreversible change, the heat absorbed from the surroundings is less than corresponds to the increase in entropy.
.
where SA and Sc are the respective entropies in states A and C of the system undergoing the reversible change in state C -+A. Then for the path, ABC, we have
where u
> 0.
Then
An irreversible change in state, therefore, is accompanied by an increase in entropy which is greater than that corresponding to the quantity of heat absorbed during the irreversible process. The system itself in some way generates entropy. We may write the inequality (8) as follows:
dS, represents the change in entropy of the system in the (i f 1)th and ith states of the system, these states being infinitely close together, and 6p, represents the heat element absorbed a t the temperature, T,, along the infinitesimal irreversible path connecting these states.
+
