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1 Gases
2 Microscopic    Energies
3 First Law
4 2nd & 3rd    Law
5 Phase    transitions
6 Mixtures
7 Phase    Diagrams
8 Equilibrium
9 Molecular    Interactions

Phase Rule

The variance is the number of degrees of freedom we have with regards to things we can change and yet remain within one region on a phase-diagram. 

F = C – P + 2

7.1

where F is the variance, (the number of intensive properties we can vary), C is the number of components, P is the number of phases.

One-component systems

Consider the hypothetical one-component phase diagram to the left. (Fig 6.2 from Atkins)

Within any single phase region, F  = 1 – 1 + 2 = 2.
In other words, we can change T and p independently while still remaining in this phase.

On a line of equilibrium between two phases, we have
F = 1 – 2 + 2 = 1.  we cannot vary T independent of p, we need to vary them both together in order to remain at equilibrium.

At the triple point, we have F = 0.  no variance is possible.  the triple point is a point, if you make any change to one of the intensive properties, you move off the point.

There is a quadruple point in this phase diagram but that would make F negative. This is not possible so we can never have a quadruple point in a one-component system.  the short form of the phase rule for a one-component system is F = 3 – P

Two-component Systems.

If two components are present then there are generally more degrees of freedom.  Generally, T, p, and composition.  F = 4 – P is the short form of the phase rule for a two component system.

Liquid-Vapour diagrams

We saw vapor pressure diagrams for mixtures where the vapor pressure depends on the composition of the liquid phase.  Clearly, if we change the composition (mole fraction) of one component, the other one will change as well, since, xA + xB = 1 so our three "variables" are T, p, and xA (or B).  Raoult's Law gives us

pA = xA p*A        and       pB = xB p*B

7.2

However, we can also find the composition of the vapour phase from this information

7.3

Thus, for a system at equilibrium, at a certain temperature and pressure, we need specify only one component concentration in one of the phases to be able to fully specify both phases of the system.  The resulting phase diagram for this two component system is

This phase diagram is different from a single component phase diagram in that the region where equilibrium can occur is no longer an infinitely narrow line, it is now an expanded region, bounded by two extremes.  The upper region (yellow) is a single phase region (liquid), the lower region (blue) is also a single phase region (vapour) whereas the green region denotes a region in the parameter space where two phases can exist in equilibrium.  This 2d diagram represents a slice taken through a larger  dimensional surface (3d) where, T, p, and composition are all variable.  Within the single phase regions, we can adjust T, p, and composition independently.  Within the green region, we can only adjust two of the three, the third parameter will change, depending on the other two.  In this phase diagram, we have already locked T at 30ºC. So, in the green region, where two phases exist, we calculate a pseudo variance of F' = 4 – 2 = 2 – 1 (subtract variable T since it is held constant in this experiment).  In the blue or yellow regions, F' = 2.  We can adjust the pressure, or the composition independent of each other.

Consider as an example a system with an overall composition zA represented by the red line.  At low pressure, (in the blue region) the whole system in a single phase gas mixture with vapour composition yA = zA.  At high pressure, (yellow region), the whole system is in the liquid phase with liquid composition xA = zA.  The interesting point is the green intermediate pressure region where two phases are visible.  The actual composition of the two phases can be determined by the end points of the horizontal "tie line" for any given pressure.  The relative amounts of the two phases are determined by the "lever rule", as pictured here.

The Lever Rule relates the amount of both phases a and b to the length of the line segment along the tie line from the overall component to the individual phase components.

7.4

Generally, it is easier to set up an experiment where the temperature is variable and the pressure is held constant (atmospheric pressure)  In this case, we would cut a different slice through our 3d surface along an isobar, rather than along an isotherm.

In this case, the liquid region (yellow) is the lower part of the diagram and the vapour region is the upper part.  The two phase region is in green.  Because T and p are not related linearly, there are no straight lines this time, even for an ideal solution.  The arrows show three steps in a distillation process whereby a liquid is evaporated at a certain temperature, the vapour is extracted and condensed back to a liquid.  The resulting liquid then boils at a lower temperature to produce a new vapour, which is then cooled again, etc.  Each step in the distillation is called one "theoretical plate", after the process used to distill crude oil into lighter oils, gasoline, etc.  The overall process is called fractional distillation since at each step, the fraction of component A and B in the mixture changes.

 

Sometimes, the deviation from ideality is large enough that there is a minimum or a maximum in the distillation curve.  This is called the azeotrope.  For a mixture that can exhibit a minimum-boiling point, the vapour pressure will always tend towards the composition of the azeotrope, rather than to pure A or B.  For a system with a maximum-boiling temperature,the vapour will tend towards one or the other pure component A or B, away from the azeotrope.  If the overall composition is exactly the azeotrope (called an azeotropic mixture) then no amount of distillation will change the compositions.

In terms of the Excess function GE, we find a negative value in the maximum-boiling azeotrope. The solution intermolecular forces are stronger than the individual component forces, this excess Gibbs energy requires an extra heat (higher T) to make the system boil.

Immiscible liquids

In the case of distillation of immiscible liquids, the boiling will only occur when the sum of the two individual vapour pressures equals the external pressure. 



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Last updated: 
07-Apr-2010