Benzene (A) is in equilibrium with 2,3,4-trimethyl
pentane (B) at 55 degrees centigrade at a total pressure of 201.74 millimeters
of mercury. The equilibrium composition
of benzene in the liquid phase is 0.0819 and in the vapor phase is 0.1869. Pure component vapor pressures are 327.05 and
178.08 millimeters of mercury for benzene and 2,3,4-trimethyl
pentane, respectively. Calculate the
values of activity coefficients and excess Gibbs free energy.
Solution:
Property
relationships for systems of variable composition:
The
fundamental property relation for homogeneous (single-phase) systems is
d(nU) is equal to Td(nS) minus
Pd(nV) plus sigma (mui
dni)
As a
result of the definitions of enthalpy, Helmholtz free
energy and Gibbs free energy, we may write:
nH is equal
to nU plus P(nV)
nA is equal
to nU minus T(nS)
nG is equal
to nU plus P(nV) minus T(nS)
Excess
Properties - Activity Coefficients
An excess
property is defined as the difference between an actual property and the
property that would be calculated for the same conditions of temperature,
pressure, and mole fraction by the equations for an ideal solution. Thus, by definition,
ME
is equal to M minus Mid
And delta
ME is equal to delta M minus delta Mid
Where ME is called the excess solution property, and
delta ME is the excess property change of mixing.
There is
also the partial molar excess property defined as
M overbariE is equal to M overbari minus M overbariid.
It should
be noted that the most important these functions is the excess Gibbs free
energy.
Pure component vapor pressure of component A, PA
is 862 millimeters of mercury.
Pure component vapor pressure of component B, PB
is 498 millimeters of mercury.
System pressure, pT is
760 millimeters of mercury.
Mole fraction
of A in the liquid phase, xA is equal to
0.0819
Activity
coefficient of A, gammaA is equal to yApT over (xA PA) or 1.4077
Activity
coefficient of B, gammaB is equal to yBpT over (xB PB) or 1.0033
GE over RT is equal to xA
log gammaA plus xB
log gammaB is 20.3 calories per mole.