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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
and
(∂U/∂V)T,n = (∂U/∂P)T,n = 0
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(2)
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[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
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(3)
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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
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(4)
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a partial differential equation, from which
(3)
where ƒ(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 5, for
instance P(V – nb) = nRT,
the G. A. Hirn equation (4), where b is a constant.
Note
- “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.
Literature
Cited
- Battino, R.; Wood, S. E.; Williamson, A. G. J. Chem. Educ. 2001,
78, 1364–1368.
- Mellor, J. W. Higher Mathematics; Dover: New York, 1955; p
10.
- Bronwell, A. Advanced Mathematics in Physics and Engineering;
McGraw-Hill: New York 1953, p 233.
- Lunelli, B. Principi di
termodinamica chimica (Principles of
Chemical Thermodynamics, in Italian); Pitagora:
Bologna,
2000; Chapter 3.03.2.
- Lunelli, B. Principi di termodinamica
chimica (Principles of Chemical Thermodynamics,
in Italian); Pitagora: Bologna,
2000; Chapter 3.03.9.
- Denbigh, K. The Principles of Chemical
Equilibrium, 4th ed.; Cambridge University
Press: Cambridge, 1981,
p 128.
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Citation
