Chapter 13 | 697Discussion This result is about 6 percent greater than the result obtained in
part (b) by using Kay’s rule. But it is more than twice the result obtained by
assuming the mixture to be an ideal gas.When two gases or two miscible liquids are brought into contact, they mix
and form a homogeneous mixture or solution without requiring any work
input. That is, the natural tendency of miscible substances brought into con-
tact is to mix with each other. As such, these are irreversible processes, and
thus it is impossible for the reverse process of separation to occur sponta-
neously. For example, pure nitrogen and oxygen gases readily mix when
brought into contact, but a mixture of nitrogen and oxygen (such as air)
never separates into pure nitrogen and oxygen when left unattended.
Mixing and separation processes are commonly used in practice. Separa-
tion processes require a work (or, more generally, exergy) input, and mini-
mizing this required work input is an important part of the design process of
separation plants. The presence of dissimilar molecules in a mixture affect
each other, and therefore the influence of composition on the properties must
be taken into consideration in any thermodynamic analysis. In this section
we analyze the general mixing processes, with particular emphasis on ideal
solutions, and determine the entropy generation and exergy destruction. We
then consider the reverse process of separation, and determine the minimum
(or reversible) work input needed for separation.
The specific Gibbs function (or Gibbs free energy) gis defined as the
combination property ghTs. Using the relation dhvdPTds, the
differential change of the Gibbs function of a pure substance is obtained by
differentiation to be
(13–27)For a mixture, the total Gibbs function is a function of two independent
intensive properties as well as the composition, and thus it can be expressed
as GG(P,T,N 1 ,N 2 ,... ,Ni). Its differential is
(13–28)where the subscript Njindicates that the mole numbers of all components in
the mixture other than component i are to be held constant during
differentiation. For a pure substance, the last term drops out since the com-
position is fixed, and the equation above must reduce to the one for a pure
substance. Comparing Eqs. 13–27 and 13–28 gives
dGV dPS dTa (13–29)
i
mi dNi¬or¬dgv dPs^ dTa
imi dyidGa
0 G
0 Pb
T,NdPa0 G
0 Tb
P,NdTa
ia0 G
0 Nib
P,T,NjdNi¬¬ 1 mixture 2
dgv dPs dT¬or¬dGV dPS dT¬¬ 1 pure substance 2
TOPIC OF SPECIAL INTEREST* Chemical Potential and the Separation Work of Mixtures
*This section can be skipped without a loss in continuity.