Page 66 - Separation process principles 2

2.2 Phase Equilibria 31

where pi is the chemical potential or partial molar Gibbs free equality of phase temperatures and pressures,
energy of species i. When (2-3) is applied to a closed system
T(') = ~ ( = ~ 1 = . . . = T(N)
consisting of two or more phases in equilibrium at uniform (2- 10)
temperature and pressure, where each phase is an open sys- and
tem capable of mass transfer with another phase, then p(') = p(2) = p(3) = . . . = p(N) (2-1 1)
constitutes the required conditions for phase equilibria. For
a pure component, the partial fugacity, fi, becomes the
pure-component fugacity, fi. For a pure, ideal gas, fugacity
is equal to the pressure, and for a component in an ideal-gas
where the superscript 01) refers to each of N phases in equi- mixture, the partial fugacity is equal to its partial pressure,
librium. Conservation of moles of each species, in the ab- pi = yi P. Because of the close relationship between fugac-
sence of chemical reaction, requires that ity and pressure, it is convenient to define their ratio for a
pure substance as

where +, is the pure-species fugacity coefficient, which has
which, upon substitution into (2-4), gives
a value of 1.0 for an ideal gas. For a mixture, partial fugacity
coefficients are defined by

With d~(') eliminated in (2-6), each d~!~) term can be var-
ied independently of any other d N,(~) term. But this requires
that each coefficient of d~i(~) in (2-6) be zero. Therefore,

such that as ideal-gas behavior is approached, 4.v -+ 1.0
and $iL + Pf/P, where P: = vapor (saturation) pressure.
At a given temperature, the ratio of the partial fugacity of
Thus, the chemical potential of a particular species in a mul-
a component to its fugacity in some defined standard state is
ticomponent system is identical in all phases at physical
termed the activity. If the standard state is selected as the
equilibrium.
pure species at the same pressure and phase condition as the
mixture, then
Fugacities and Activity Coefficients
Chemical potential cannot be expressed as an absolute quan-
tity, and the numerical values of chemical potential are diffi-
cult to relate to more easily understood physical quantities. Since at phase equilibrium, the value of f,O is the same for
Furthermore, the chemical potential approaches an infinite each phase, substitution of (2-15) into (2-9) gives another al-
negative value as pressure approaches zero. For these rea- ternative condition for phase equilibria,
sons, the chemical potential is not favored for phase equilib-
ria calculations. Instead, fugacity, invented by G. N. Lewis
in 1901, is employed as a surrogate.
The partial fugacity of species i in a mixture is like a For an ideal solution, aiv = yi and aiL = xi.
pseudo-pressure, defined in terms of the chemical potential by To represent departure of activities from mole fractions
when solutions are nonideal, activity coeflcients based on
concentrations in mole fractions are defined by

where Cis a temperature-dependent constant. Regardless of
the value of C, it is shown by Prausnitz, Lichtenthaler, and
Azevedo [4] that (2-7) can be replaced with
For ideal solutions, yiv = 1.0 and yiL = 1 .O.
For convenient reference, thermodynamic quantities that
Thus, at equilibrium, a given species has the same partial fu- are useful in phase equilibria calculations are summarized in
gacity in each existing phase. This equality, together with Table 2.2.