Chemical Education Today
Letters Redundant Assumption
5, for instance P(V – nb) = nRT, the G. A. Hirn equation (4), where b is a constant.
The very useful article “On the Importance of Ideality” published in this Journal (1) reports, as happens frequently, that “The ideal gas is defined by the two relations
Note
PV = nRT
(1)
(∂U/∂V )T,n = (∂U/∂P)T,n = 0
(2)
and [where n is the total quantity in moles of substance in the gas phase] or U is a function of T and n only”.1 Square brackets delimit an addition of mine. This is redundant, because eq 1 implies eq 2. In fact since the differentials have the same formal properties of numbers (2), dividing the general relation dU = TdS – PdV by dV at constant T, and then using one of the Maxwell’s relations (∂U/∂V )T,n = T(∂S/∂V )T,n – P = T(∂P/∂T )V,n – P
(3)
But differentiating eq 1 at constant V we get VdP = RdT, or (∂P/∂T )V,n = P/T so that eq 2 follows. The inverse is not true, or eq 1 fulfills eq 2 but is not the only relation that does so. If eq 2 holds, from eq 3 we get dP/P = dT/T at fixed V
(4) (5)
where f (V ) is an arbitrary function of V. Thus eq 5 is fulfilled by eq 1 but also by other equations of the form of eq
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Literature Cited 1. Battino, R.; Wood, S. E.; Williamson, A. G. J. Chem. Educ. 2001, 78, 1364–1368. 2. Mellor, J. W. Higher Mathematics; Dover: New York, 1955; p 10. 3. Bronwell, A. Advanced Mathematics in Physics and Engineering; McGraw-Hill: New York 1953, p 233. 4. Lunelli, B. Principi di termodinamica chimica (Principles of Chemical Thermodynamics, in Italian); Pitagora: Bologna, 2000; Chapter 3.03.2. 5. Lunelli, B. Principi di termodinamica chimica (Principles of Chemical Thermodynamics, in Italian); Pitagora: Bologna, 2000; Chapter 3.03.9. 6. Denbigh, K. The Principles of Chemical Equilibrium, 4th ed.; Cambridge University Press: Cambridge, 1981, p 128. Bruno Lunelli
a partial differential equation, from which (3) ln(P/T ) = ln f (V )
1. “Perfect” instead of “ideal” avoids confusion with the “ideal gaseous solutions” (5), the gas mixtures following the Lewis and Randall fugacity rule (6), giving chemical potentials dependent of the mole fraction analogously to ideal (condensed) solutions.
Dipartimento di Chimica “G. Ciamician” Università di Bologna Bologna, Italy
[email protected] Vol. 81 No. 9 September 2004
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Journal of Chemical Education
1267
