These diagrams combine information about the composition of the liquid and vapour phases at a given temperature, and how the compositions of the phases change with pressure. They have the general form (where we maintain our convention of A being more volatile than B):
| The point I indicates the vapour pressure of a liquid phase of composition xA. The point II indicates the composition of the vapour, yA, that is in equilibrium with the liquid at this pressure. i.e. if the two phases are at equilibrium at this pressure, then the liquid phase has a composition with mole fraction xA of A, while the vapour phase has a mole fraction yA of A. Note it is not necessary for the two phases to be in equilibrium at this pressure. If the mole fraction of A in the entire system is sufficiently low (to the left of xA on the diagram) only a liquid phase exists. Conversely, if the total mole fraction of A is high enough (to the right of yA on the diagram) there will be only a vapour phase. |
It is important to note the labeling of the horizontal axis at this point. It gives the overall composition, in terms of the mole fraction of A in the entire system, zA. At all points above the diagonal line, the pressure is greater than the vapour pressure of the system and only a liquid phase exists. In this instance, zA is equivalent to xA, the mole fraction of A in the liquid. At all points below the lower curved line, only a vapour phase exists and zA is equivalent to yA, the vapour phase mole fraction of A.
At points between the two lines, two phases, one liquid and one vapour, coexist. The amount of information that can be derived from this region is large, and worthy of slightly more detailed study:
Consider that the system starts off at point Φ. Here it is composed of a single liquid phase, with composition ΦA. If the pressure on this liquid mixture is then reduced, no change in the overall composition occurs, so the state of the system moves down the vertical line in the diagram.
The system remains as a single liquid phase until the point Φ1 (corresponding to a pressure of p1) is reached. At this point, the liquid may exist in equilibrium with its vapour.
The composition of the vapour phase at this pressure is given by the point Φ1′ (the point on the lower curve at which the pressure is p1.) The horizontal line that joins the points Φ1 and Φ1′ is called a tie line. At the point Φ1 the composition of the liquid is negligibly different from that at Φ, as there is virtually no vapour present, but the small amount that there is will have the composition Φ1′.Consider that the system starts off at point Φ. Here it is composed of a single liquid phase, with composition ΦA. If the pressure on this liquid mixture is then reduced, no change in the overall composition occurs, so the state of the system moves down the vertical line in the diagram.
The system remains as a single liquid phase until the point Φ1 (corresponding to a pressure of p1) is reached. At this point, the liquid may exist in equilibrium with its vapour.
Lowering the pressure to p2 takes the system further down the vertical line to the point Φ2″. The overall composition of the system is still given by ΦA, but there are now two separate phases that make up the system, each with a different composition. The liquid phase has a composition given by the point Φ2, and the vapour it is in equilibrium with has a composition Φ2′. i.e. the mole fraction of A in the liquid phase at p2 is given by the x coordinate of point Φ2, while the mole fraction of A in the vapour phase is given by the x coordinate of the point Φ2′.
If the pressure is lowered further to p3, then the composition of the liquid and vapour readjust to those given by the points Φ3 and Φ3′. Since the composition of the vapour at Φ3′ is the same as ΦA, the overall composition of the system, it follows that there must be a negligibly small amount of liquid present at this pressure. What liquid there is, however, will have a composition given by Φ3.
Further reduction of the pressure takes the system into the region where only vapour is present, and thus the vapour must have the same composition as the overall system, which is in turn the same as the composition of the original liquid.
Given that the overall composition of the system remains the same throughout, it is possible to see that the position of a point in the two-phase region of such a diagram indicates the relative amounts of each phase present. This observation is set out as the lever rule. If there are two phases, α and β, in equilibrium, then we measure the distances lα and lβ along the tie line (these are the distances from the point under consideration to the edge of the appropriate single phase region) and apply the lever rule:
where nα and nβ are the amounts of the two phases in moles.
The rule is the same in form as the rule for two weights at the ends of a seesaw being in equilibrium. (The turning force of each weight is given by its mass multiplied by its distance from the pivot, and the two turning forces must be equal for the seesaw to balance. This is the classical mechanical theory of levers, which is where the rule gets its name from.) It may be helpful to bear this analogy in mind when applying the rule.where nα and nβ are the amounts of the two phases in moles.
We can immediately see, for instance, that whichever phase X for which the distance lX is shorter will be present in greater amount than the other phase. (the weight nearer the pivot on a seesaw must be heavier than the other weight for the seesaw to balance.)
Note that although this page has focused exclusively upon a liquid and vapour phase in equilibrium, analogous diagrams may be derived for other such systems with two phases of different composition in equilibrium, for example between a solid and a vapour phase.