We often use Pxy and/or Txy diagrams to determine the compositions in VLE problems, but where do they come from?

For simplicity, let's start with Raoult's law for a binary mixture

$x_1P_1^{sat}=y_1P_{tot}$

$x_2P_2^{sat}=y_2P_{tot}$

We would like to write the pressure, $P_{tot}$ as a function of the liquid composition. So we add these two equations together:

$x_1P_1^{sat} + x_2P_2^{sat} = (y_1+y_2)P_{tot} = P_{tot}$

which we can simplify to:

$P_{tot} = x_1P_1^{sat} + (1-x_1)P_2^{sat} = P_2^{sat} + x_1(P_1^{sat}-P_2^{sat})$

Which we can plot as a straight line showing the liquid composition (see above).

Similarly, to get the pressure as a function of the vapor phase composition, we solve our equation for $x_i$ and then add them to get:

$x_1+x_2 = 1 = y_1\frac{P_{tot}}{P_1^{sat}} + y_2\frac{P_{tot}}{P_2^{sat}}$

Solving for $P_{tot}$ gives:

$P_{tot} = \frac{1}{\frac{y_1}{P_1^{sat}} + \frac{y_2}{P_2^{sat}}}$

which can be simplified to:

$P_{tot} = \frac{1}{\frac{y_1}{P_1^{sat}} + \frac{1-y_1}{P_2^{sat}}}$

This yields the curved line denoting the vapor phase composition (above).

This procedure is not fundamentally different for the case when the fugacity and/or activity coefficients are non-ideal (i.e., not 1), except that they often depend on composition as well so that our "lines" become more curved.

**Tie lines** are the name given to lines that
bridge the "coexistence space" in a phase diagram. In the example
above the lens-shaped region between the line and the curve is a
"no mans land" that separates the vapor phase compositions from the
liquid phase compositions. The bi-colored line is an example of a
tie line.

Tie lines are useful because the are graphical examples of a material balance. For example, write two balance equations for the binary mixture depicted here:

$x_1L + y_1V = z_1M$

$M = L + V$

Here $z_1$ depicts the "total" composition considering both phases at once and $M$ is the total mass in the system (again, considering both phases at once). Combining these equations gives:

$x_1L + y_1V = z_1(V+L) = z_1V + z_1L$ or (rearranged)

$L(x_1-z_1) = V(z_1-y_1)$

Rearranging one last time shows that the ratio of the line segment lengths is equal to the ratio of the amount of material in both phases.

$\frac{L}{V} = \frac{(z_1-y_1)}{(x_1-z_1)}$

The line segments are "backwards". That is, the red line (on the right) shows the relative amount of liquid (which is found to the left of the lens region), while the blue line (on the left) corresponds to the relative amount of vapor (which is found on the left of the lens shaped region).

Draw and read Txy and Pxy diagrams for VLE