David W. McClure Portland State University Portland, Oregon 97207
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A Comment on Showina dQ and dW to be Inexact
Practically any undergraduate physical chemistry student can recite the usually well inculcated fact that, unlike the thermodynamic state functions dU, dG, etc., the differentials of heat and work are inexact. Justification of this statement, if given at all, is commonly done by the student alluding vaguely to the assumed existence of experiments which confirm it or, in some cases, to having seen a line integration of d W (as well as a state function such as dV for comparison purposes) over two different paths with the same terminal ~ 0 i n t s . l .On ~ finding that W (path 1) Z W (path 2) one concludes correctly that indeed d W is inexact. Of course no conclusion of any kind is possible for the volume where AV (path 1) = AV (path 21, since all paths in the domain have not been tested. Apart from the inefficiency of testing for exactness by line integration, there is the more basic problem that many students in their first year of . physical chemistry find the . mathematics at best abstruse. In any case, once having introduced inexactness for d W one usually resorts to either simply stating that dQ is inexact without justification, or by arguing that since AU is a function of the terminal points only, and independent of how variations in Q and Ware achieved, then inexactness of dQ (and dW) is implied. The trouble with this argument is that the relatively sophisticated notion of what exactly constitutes a pathway in thermodynamics is seldom clear to the beginning student; so the argument is again simply accepted, hut not understood. On the other hand, the use of the easily derived recipmcity conditions are pedagogically more sound because their use is simple and efficient, since equality is both a necessary and sufficient condition for exactness. With this view in mind, we have outlined the use of the reciprocity conditions for showing dQ and d W to he inexact. In addition, employing the appropriate integrating factor effects exactness which, in the case of heat, leads naturally into the second law. We consider a closed simple system undergoing a reversible change in state. Since only hydrostatic work is involved and because the process is reversible, we can write
dW = -PdV = - P
The reciprocity condition for exactness of the Pfaffian requires
[$ P
(%),,I,
=
[$ (%>,I,,
which reduces to
($),,
a av +
a av
[aP(n),,]. =
[n(aP),l,,
The Schwarz theorem guarantees that the order of differentiation is immaterial provided certain existence and continuity conditions, which are physically reasonable, are met; so exactness thus implies that, in general
(g) ,,,
=
0
,"L
which is not true. It follows then that d W is an inexact differential, a fact frequently denoted by writing aW. Multiplication of eqn. (1) by the reciprocal pressure, P-', leads to exactness however, establishing P-l as an integrating factor for dW. In the case of non-isometric heat transfer we may write, since the system is closed and the process revenihle, dU = dQ + dW = dQ - PdV Choosing U = U IV, T)we have
'Margenau, H., and Murphy, G. M., "The Mathematics of Physics and Chemistry," 2nd Ed., D. Van Nastrand Co., Inc., New Jersey, 1956, p. 8. 2
3
Blinder, S. M., J. CHEM. EDUC., 43,85 (1966). Kleindienst, H., J. CHEM. EDUC., 50,835 (1913)
Volume 51. Number
11. November 1974
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707
Exactness now requires Exactness then requires
and, again, since the order of differentiation is irrelevant, exactness implies
which in general, as the zeroth law asserts, is not true. It thus follows that dQ is inexact. Incidentally, if the process is isometric, eqn. (2) reduces to dQ = d U as expected. Use of the reciprocal temperature, T-1,as an integrating factor in eqn. (2). is interesting. We have
708
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Journal of Chemical Education
This reduces, since the order of differentiation may be reversed, to showing that
which is a well known true identity. Of course this argument, if used as a prelude to introducing the second law is circular since eqn. (3) can only be derived by invoking the second law. Logistically however, no difficulty arises since students will have been exposed to this relation well in advance of the second law, as it is required for non-ideal gas calculations.
