Page 215 - Advanced thermodynamics for engineers

9.6 CONCLUDING REMARKS 201

Hence,

1
X X X
Þ
ðs m T mix ¼ x i s m i ðTÞþ x i s 0;m i þ< x i ln ln p r : (9.83)
x i
Considering the terms on the right hand side of Eqn (9.83).
Þ
1st and 2nd terms: summation of entropies before mixing, ðS m T sep .
3rd term: change of entropy due to mixing. This is due to change of partial pressures when mixed.
4th term: pressure term.
The change of entropy due to mixing is given by

Þ
Þ
DS ¼ðS m T mix ðS m T sep (9.84)
which is the difference between Eqns (9.83) and (9.77). This gives
1
X
DS ¼< x i ln (9.85)
x i
The numerical value of DS must be positive because x i is always less than unity.
Equation (9.85) shows that there is an entropy increase due to mixing, and this is caused by the
reduction in the order of the molecules. Before mixing it is possible to go into one side of the
container and guarantee taking a particular molecule because only molecules of a are in the left-
hand container, and only molecules of b are in the right-hand container. After mixing it is not
possible to know whether the molecule obtained will be of a or b: the order of the system has been
reduced.
What happens if both a and b in Fig. 9.2 are the same gas?
Superficially it might be imagined that there will still be an increase of entropy on mixing.
However, since a and b are now the same there is no change in partial pressure due to mixing.This
means that the molar fraction is unaltered and

DS ¼ 0: (9.86)

9.6 CONCLUDING REMARKS

A detailed study of ideal gases and ideal gas mixtures has been undertaken in preparation for later
chapters. Equations have been developed for all properties, and simplified enthalpy coefficients have
been introduced for nine commonly encountered gases. Tables of gas properties have been presented
for most gases occurring in combustion calculations.
Equations for gas mixtures have been developed and the effects of mixing on entropy and Gibbs
energy have been shown.