The Lever Rule

P-x and T-x diagrams are quite useful, in that information about the compositions and relative amounts of the two phases can be easily extracted. In fact, besides giving a qualitative picture of the phase behavior of fluid mixtures, phase diagrams can also give quantitative information pertaining to the amounts of each phase present, as well as the composition of each phase.

For the case of a binary mixture, this kind of information can be extracted from P-x or T-x diagrams. However, the difficulty of extracting such information increases with the number of components in the system.

At a given temperature or pressure in a T-x or P-x diagram (respectively), a horizontal line may be drawn through the two-phase region that will connect the composition of the liquid (x_A) and vapor (y_A) in equilibrium at such condition — that is, the bubble and dew points at the given temperature or pressure, respectively. If, at the given pressure and temperature, the overall composition of the system (z_A) is found within these values (x_A < z_A < y_A in the T-x diagram or y_A < z_A < x_A in the P-x diagram), the system will be in a two-phase condition and the vapor fraction (α_G) and liquid fraction (α_L) can be determined by the lever rule:

$${\alpha }_G=\frac{z_A-x_A}{y_A-x_A}$$ (5.1a)

$${\alpha }_L=\frac{y_A-z_A}{y_A-x_A}$$ (5.1b)

Note that α_L and α_G are not independent of each other, since α_L + α_G = 1. Figure 5.5 illustrates how equations (5.1) can be realized graphically. This figure also helps us understand why these equations are called “the lever rule.” Sometimes it is also known as the “reverse arm rule,” because for the calculation of α_L (liquid) you use the “arm” within the (y_A-x_A) segment closest to the vapor, and for the vapor calculation (α_G) you use the “arm” closest to the liquid.

Figure 5.5: The Lever Rule In a P-x Diagram

At this point you will see clearly why we needed to make a clear distinction among “x_i” and “y_i” and “z_i”. Expressions (5.1) can be derived from a simple material balance. Can you prove it? [Hint: The number of moles of a component “i” per mole of mixture in the liquid phase is given by the product “xiα_L”, while the number of moles of “i” per mole of mixture in the gas is given by “yiα_G”. Since there are “z_i” moles of component “i” per mole of mixture, the following must hold: $z_i=x_i{\alpha }_z+y_i{\alpha }_G$. Can you proceed from here?]