A Simple Method for Showing Entropy Is a Function of State P. Djurdjevic and I. Gutman Faculty of Science, University of Kragujevac, P. 0.Box 60, YU-34000 Kragujevac. Yugoslavia
In a recent issue of this Journal' G. K. Vemulapalli raised the question of how to demonstrate in an undergraduate course of physical chemistry that entropy is a state function, without exploiting the concept of the Carnot engine. One must fully agree with Vemulapalli that introducing entropy via the examination of the operation of a reversible heat engine whose working fluid is the ideal gas is a rather cumbersome and time-consuming task with the usual outcome that the students are unable to grasp completely the meaning and the importance of the results obtained. A further argument aaainst the Carnot cycle a ~ p r o a c his that the notion of a heit engine and the reipertjie vocabulary are used nowhere else in the course of ~hysicalchemistry and therefore sound to the students somewhat artificial. What Vemulapalli actually offered in his paper was an easy and convincing proof that the entropy of the ideal gas is a function of state. We may write this as
where i = ideal aas. However, even the undergraduate students are well aware that, if a thermodvnamical relation holds for the ideal gas, it needs not necessarily he obeyed in the case of an arbitrary substance. Hence, Vemulapalli's proof has to be completed by demonstrating that eq 1 => eq 2, where
from which the implication that eq 1 => eq 2 follows in an obvious manner. Whence, for any substance dq,,/T is an exact differential and the entropy S, defined by means of d S = dq,,/T is a function of state. We would like to point out a slight modification of Vemulapalli's proof of eq 1. From the first law of thermodynamics dU=dq-PdV
(5)
and the fact that d U is a total differential
(S)"~T
d ~ ( +g ) T d ~ +
(6)
we obtain that
Already in eq 5 the term dq refers to reversihly exchanged heat because -PdV is equal to the work only in a reversible (infinitesimal) process. Now, in order that the right-hand side of eq 7 is an exact differential, we must have (by Euler's theorem)
Direct calculation gives then
that is, where s = arbitrary substance. This can be done by a simple and elementary argument. Consider an isolated system consisting of two subsystems A and B. Part A contains a certain amount of ideal gas, whereas part B contains an arbitrary substance. The two subsystems are in equilihrium and may reversibly exchange heat. Suppose now that an infinitesimal reversible process occurs in A, followed by production (or absorption) of the heat dq,.,(i). Simultaneously another infinitesimal reversihle process takes place in B, whose heat effect is dq,(s). Since the entire system is isolated, dq,&)
that is,
+ dq,&
=0
Since by Schwarz's theorem
we conclude that dq,/T if)
is an exact differential if (and only
(3)
The verification that eq 12 is obeyed by the ideal gas may serve as an easy, but quite useful exercise.
' Vemulapalll, G. K. J. Cham. Educ. 1988, 63,846.
Volume 65 Number 5
May 1988
399
