Chemical Engineering question?

Question

In a closed container, a vapor mixture of some organic compounds is gradually cooled to a temperature at which a first droplet of liquid is formed. Assuming that there are n chemical components in both the vapor and liquid phases, find out the relationship between the molar fractions of component i in liquid and in vapor phase when the first droplet is formed.

Answer 1

This is off the top of my head, but......

Let's make a list of what you know.

V= volume of container
Ni = number of moles of each chemical species

There are two phases liquid and vapor which are in equilibrium with each other.

The temperature and pressure are unknown.

The chemical potentials of each component are equal to the corresponding values in the other phase, since the system is at equilibrium.

We can assume that the volume of the liquid phase is negligible.

So we need:

a relationship between the vapor composition, temperature, and pressure and its molar volume.
For this, we could use ideal gas behavior as a first approximation. More sophisticated relationships might be necessary for concentrated (high pressure) systems.

a relationship between the liquid phase composition, temperature, and vapor phase fugacity is needed. For this, one could use Henry's or Raoult'slaw as a first approximation.

constraints:

the total number of moles of each species is known

the volume is fixed

so, for a very simple solution

yi= Pi/(sum Pi)

Ptot = sum Pi

Pi= Pvap_i (T) x_i

Ptot Vtot = Ntot R T

Guess a T or P until the computed xi and yi sum to
one in each phase. You'll need a relationship for
Pvap_i(T), either table lookup or some other empricism.

Answer 2

Apply Raoult's Law