Page 213 - Advanced thermodynamics for engineers

9.5 ENTROPY OF MIXTURES 199

9.5 ENTROPY OF MIXTURES
Consider a mixture of two ideal gases, a and b. The entropy of the mixture is equal to the sum of the
entropies which each component of the mixture would have if it alone occupied the whole volume of
the mixture at the same temperature. This concept can present some difficulty because it means that
when gases are separate but all at the same temperature and pressure they have a lower entropy than
when they are mixed together in a volume equal to the sums of their previous volumes: the situation
can be envisaged from Fig. 9.2. This can be analysed by considering the mixing process in the
following way:
1. Gases a and b, at the same pressure, are contained in a control volume but prevented from mixing
by an impermeable membrane.
2. The membrane is then removed, and the components mix due to diffusion i.e. due to the
concentration gradient there will be a net migration of molecules from their original volumes.
This means that the probability of finding a particular molecule at any particular point in the
volume is decreased, and the system is in a less ordered state. This change in probability can be
related to an increase in entropy by statistical thermodynamics. Hence, qualitatively it can be
expected that mixing will give rise to an increase in entropy.
p a ¼ p b ¼ p.
Considering the mixing process from a macroscopic viewpoint, when the membrane is
broken the pressure is unaffected but the partial pressures of the individual components are
decreased.
The expression for entropy of a gas is (Eqn (9.20))
p
s m ¼ s m ðTÞ < ln þ s 0;m
p 0
where s m (T) ¼ function of T alone. Hence, considering the pressure term, which is actually the partial
pressure of a component, a decrease in the partial pressure will cause an increase in the entropy of the
gas. Equation (9.20) can be simplified by writing the pressure ratio as p r ¼ p=p 0 , giving
(9.20a)
s m ¼ s m ðTÞ < ln p r þ s 0;m
Now consider the change from an analytical viewpoint. Consider the two gases a and b.

Gas 'a ' Gas 'b '
p p

T T

FIGURE 9.2
Two gases at the same pressure, p, contained in an insulated container, and separated by a membrane.