A.
H. Kalantar
of Alberta Edmonton, Canada
University
I
I
Nonideal Gases and Elementary Thermodynamics
I n thermodynamics, a variety of exact relations can be found after the introduction of partial differentiation and the second law. Practical use of such relations and of partial differentiation requires an equation of state. For gases, a number of approximate equations are available. Finding, for example, an expression for the Joule-Thomson coefficient or the isothermal energy change for some gas is often an instructive exercise--whether considered from the viewpoint of simply providing practice, or having a case for comparison with experimental data, or studying different corrections to the ideal gas equation. Such exercises are also important because they require the student to rely on fundamental definitions rather than to use a single expression, good only for some special rase. The purpose of this hrief article is to present, in
Equation of state
-
one place, a tabulation of some thermodynamic quantities for certain nonideal gases. There are two reasons for this presentation: one is to encourage greater use of the more general expressions, and the other is to provide a better basis for the intelligent selection of examples. In the table, various quantities are presented for four nonideal gases; the ideal gas is included for comparison. C,-C,, (bT/bP)" (the Joule-Thomson coefficient), and (bE/bV), (the "internal pressure") are shown. Included are the results for o, G, AH, AS, and AG (Gibbs free energy) for a reversible isothermal expansion of the gmes. The van der Waals gas and two simple variants are included to show the particular effects of the volume (b) and internal pressure (a/+) correction terms. The factors a, b, e, and of course, R, are talxw to he constants. Just one mole of gas is considered. Each
Nonideal Gases
PV
RT
=
PV
=
RT
+c
RT
=
P(V - b )
(V - b) ( P
+ %)
=
RT ( P
+ g)V = RT
Some thennodynamic quantities
J.T. Coeff.
; { T ( ~ ~ - V [
0
-e
FE*
-b Cs
-1
2a
Changes for a reversible i s o t h e m l expansion, at T , fr.otn V to V ,
V, - b RT In -V-b
0
-b(P
- P,)
WIT
V , -b -RT In - -V - b -b(P - P J ) Volume 43, Number 9 , September 1966
/
477
term in the table is written so as to be positive wherever possible; if negative, it is preceded by a negative sign. For the isothermal expansion, the subscript, , , is used for the value of the quantity in the final state. The work done is taken as positive when done by the system on the surroundings. Thus AE = q - w. Only pressure-volume work is considered, thus w,.jbl. = PdV = - AA isothormsl ( A = E - T S ) . All the expressions given are exact for the equations of state considered; that is, no preliminary expansion with subsequent dropping of (smaller) terms has taken place. Thus the expression for (bT/bP)* for a van der Waals gas is correct and reducible to the more commonly found expression, p,.,. = 1/Cv[(2a/RT) - b], which becomes correct in the case of a dilute gas. In fact, expansion to large V (or low P) and subsequent simplification of the equations is a partial
fl'
478
/
Journal o f Chemical Education
check that can be performed on the results. Two other checks involve a dimensional analysis and, for the van der Waals gas and its variants, comparison of corresponding terms in different columns. This comparison makes clearer the effect of the correction terms on the results, with respect to the ideal gas case, and helps the student to see how the "extra" terms arise. That is, a careful choice of examples can provide the student with more than an exercise in manipulation; he can see the effects of &ifferent kinds of nonideality and he becomes aware of the wealcnesses of the ideal gas model. Many make use of nonideal gas equations as a tool for instruction. The table is presented to broaden this usage and, more especially, to help this usage to become an even more effective teaching instrument.
