CHAP. 4: APPLICATION OF THERMODYNAMICS [CONTENTS] 126
Example
Using the relation following from the van der Waals equation, estimate the inversion temperature
at zero pressure for several common gases given that you know the critical temperatures of the
gases. Which gases cool and which heat up during isenthalpic expansion at room temperatures
and low pressures?
Data: helium,Tc= 5.3 K; hydrogen,Tc= 33.2 K; nitrogen,Tc= 126.3 K; oxygen,Tc= 154.6 K;
carbon dioxide,Tc= 304.2 K.
Solution
The constantsa, bof the van der Waals equation can be estimated using the critical temperature
and pressure from relations (2.24). By substituting these relations into the equation for the
inversion temperature from the preceding example, we get
Tinv=
2 a
Rb
=
27
4
Tc, [p= 0].
The estimated inversion temperatures are as follows (experimental values are given in brackets):
helium, 35.8 K (40 K); hydrogen, 176 K (202 K); nitrogen, 852 K (621 K); oxygen, 1043 K (764
K); carbon dioxide, 2053 K (1500 K).
It follows from Figure 4.4 that at low pressures and temperatures lower than the inversion tem-
perature,μJT>0 and gases cool on expansion. At room temperatures,T ∼300 K, nitrogen,
oxygen and carbon dioxide cool on expansion at constant enthalpy while hydrogen and helium
heat up.
