# The Lewis/Randall Rule

The Lewis/Randall Rule:
A simple equation for the fugacity of a species in an ideal solution follows from the following equations. In general, for any solution:
(6.158)
Applying to ideal solution,     (6.159)
However we know (eqn. 6.77) that:  (6.160)
Also, from eqn. 6.86:
Thus, (6.161)
On comparing eqns. 6.158 & 6.160: (6.162)
The last relation is known as the Lewis/Randall rule, and applies to each species in an ideal solution at all conditions of temperature, pressure, and composition. It shows that the fugacity of each species in an ideal solution is proportional to its mole fraction; the proportionality constant is the fugacity of pure species i in the same physical state as the solution and at the same T and P.
One may write the same equation specifically for an ideal solution, whence:

(6.165)
Thus subtracting eqn. 6.165 from 6.164:
(6.166)
Thus we may write the following further generative relations:
(6.167)
(6.168)
And further:
 The sensitivity of the excess Gibbs free energy to changes in temperature and pressure may be estimated to show the effect of pressure and temperature on liquid phase properties. For example, for an equimolar mixture of benzene and cyclohexane at 298K and 1 bar are (source: J.M. Smith, H.C. Van Ness and M.M. Abbott, Introduction to Chemical Engineering Thermodynamics, 6th ed., McGraw-Hill, 2001): The above calculations suggest that to effect the same change in excess Gibbs free energy brought about a change of 1K, one needs to change the pressure to change by about 40bar. Hence the excess Gibbs free energy exhibits a relatively weak dependence on pressure.
(6.169)