Fall 2008

ChE 545: Mass Transfer Operations I

Lecture 3: Thermodynamics and Equilibrium

Activity coefficient ( ) of a component is related to the partial molar excess Gibbs

free energy( ) for that component:

gE, molar excess Gibbs free energy, is a function of composition (xis), T, and P.

For a regular solution (no excess volume or entropy, vE = s

E = 0), the activity

coefficient is given by:

where, is the solubility parameter, is the volume fraction, and vL the molar

volume in the liquid phase.

Other expressions are available for the activity coefficients, based on the

functional relationship for the excess Gibbs free energy. These relationships are

based on various theories of solution (Wilson, NRTL, UNIQUAC, etc.).

Gibbs Phase rule stipulates the number of independent variables that need to be

specified for fixing the equilibrium state of the system consisting of C

components and P phases.

F = C – P + 2

The phase rule refers to intensive properties (independent of the system size).

The degree of freedom for a binary vapor-liquid system is 2, meaning only 2

variables need to be specified (out of P, T, x1 and y1) to fix the equilibrium

conditions. The two independent equations (assuming ideality, i.e., Raoult’s law)

are:

T-x-y, and x-y diagrams can be constructed using these equations. The region

enclosed by T-x and T-y lines on the T-x-y diagram comprises a two phase region.

Any point within this region denotes the overall composition of the system

(abscissa) at that temperature (ordinate). Horizontal line passing through this

point connecting the T-x and T-y curves is called a tie line. The intersections of

the tie line with the two curves represent the vapor and liquid concentrations at

equilibrium with each other. The relative amounts of liquid and vapor can be

determined using inverse lever rule.

Alternately,

Where, 2,1 is the separation factor (relative volatility) of component 2 with

respect to component 1, and is given by (for Raoult’s law case):

Azeotropes (constant boiling mixtures, x1 = y1) result due to the non-ideality of

the system.

o Minimum boiling azeotropes have the azeotrope boiling at a lower

temperature than the boiling point of the more volatile component (and

also the less volatile component). These result when activity coefficients

of the components are greater than 1.

o Maximum boiling azeotropes boil at a temperature higher than the boiling

point of the less volatile component (and also the more volatile

component). Activity coefficients are less than 1 in this case.