CH353 - Physical Chemistry I
Spring 2012, Unique 52135

Lecture Summary, 19 March 2012

Binary Systems: So far we have only discussed equilibrium in systems with one component.  The next thing we need to do on our path to a general expression for chemical equilibrium is figure out what happens when two or more components are mixed.  We still are not dealing with chemical reactions, just mixing:

   pure, unmixed 1 and 2 --> homogeneous mixture of 1 and 2

Starting from our general equation for free energy at constant temperature and pressure (i.e. dependent on chemical potential only), we derived the Gibbs-Duhem equation,

  x1d(mu1) + x2d(mu2) = 0

which says that in a binary system, if we know the chemical potential of one component, we automatically know the other.  (We could of course extend this to a system of n components, but the result is the same - investigating a system with two components is all we need to do.) 

The particular binary system that we are investigating are liquid-liquid mixtures.  To figure out what is going on in the liquid, we monitor the pressure of the vapor in the head space above the liquid.  Molecules in this vapor are in equilibrium with the liquid, and so by referencing to the vapor phase we can figure out what is going on in the liquid phase.  To do this, our solution must obey two criteria: 1) the proportion of molecules at the liquid-vapor interface must be identical to the bulk solution; and 2) the probability that a molecule can escape from the liquid to vapor phase must be identical to the pure liquid.  Solutions that obey these two criteria are called "ideal" solutions and obey the following two expressions:

   Pi = xiP* (Raoult's law)
   (mu)l = (mu)*l + RT ln xi

where xi is the mole fraction of component i in the solution and 0 <= xi <= 1.

It is then trivial to use these equations to derive the following:

  deltaG(mix) < 0 always
  deltaS(mix) > 0 always
  deltaV(mix) = 0 always
  deltaH(mix) = 0 always

The criteria for ideal solutions are very restrictive, and apply only to molecules that have similar intermolecular forces (i.e. hexane and heptane).  However, the general conclusions that we draw about the thermodynamics of mixing apply even to nonideal solutions, and so this is still a useful exercise.