If we need to know any combination of two of the following: concentration of phase "1" (say, liquid), concentration of phase "2" (say, vapor), $T$ of system, $P$ of system, we can start from the phase equilibrium condition

$\hat f_i^{\alpha} = \hat f_i^{\beta}$ for any two (or more!) generic phases, or

$\hat f_i^L = \hat f_i^v$ specifically for vapor-liquid equilibrium, which gives

$x_i\gamma_i f_i^{o^L} = y_i\hat \phi_i P_{tot}$

In order to use this equation, we need to know expressions for the fugacity and activity coefficients, as well as models/values for the reference fugacity of the liquid phase.

While the following procedure can be easily modified to relax these simplifications, it is instructive to examine how to use this equation for a system that satisfies Raoult's Law (i.e., one that has an ideal gas vapor phase and liquid mixture that acts as a (L-R) ideal solution), so:

$x_iP_i^{sat} = y_iP_{tot}$

Recall that we can use the Antoine (or Clausius-Clapeyron) Equation in order to get $P_i^{sat}$ in terms of $T$. Therefore, if we know $T$ and the liquid phase composition we have three unknowns in a binary mixture: $y_1, y_2,$ and $P_{tot}$. Luckily, we can also write three equations:

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

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

$y_1 + y_2 = 1$

Recall that $T$ can be used to explicitly calculate $P^{sat}$ for each component.

If we add the first two equations and combine this with the third, we get:

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

This can easily be solved for $P_{tot}$ and then $P_{tot}$ can be plugged into either of the first two equations to yield $y_1$ and $y_2$.

If, instead, we know $P_{tot}$ and the vapor phase composition we now have five unknowns in a binary mixture: $x_1, x_2, P_1^{sat}, P_2^{sat}$ and $T$. Luckily, we can also write five equations (explicitly counting the Antoine equations now):

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

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

$P_1^{sat} = \exp\left [ A_1 - \frac{B_1}{T-C_1} \right ]$

$P_2^{sat} = \exp\left [ A_2 - \frac{B_2}{T-C_2} \right ]$

$x_1 + x_2 = 1$

Recall that $T$ is **not** the boiling temperature
of either species. Instead, it is the system temperature of
interest (so it is the same in the 3rd and 4th equations).

If we rearrange the first two equations to isolate $x_i$ on the
left and *then* add them together, we get:

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

Plugging in the Antoine equations into the $P_i^{sat}$ relations on the right we get an equation with $T$ as the only unknown:

$1 = y_1\frac{P_{tot}}{\exp\left [ A_1 - \frac{B_1}{T-C_1} \right ]} + y_2\frac{P_{tot}}{\exp\left [ A_2 - \frac{B_2}{T-C_2} \right ]}$

Once we identify $T$ from this equation, we can plug it into the two $P_i^{sat}$ relations and use those results in the first two equations to yield $x_1$ and $x_2$.

Perform bubble-point and dew point calculations using Raoult's Law

Small variations of these procedures would be used if we knew values or expressions for $\hat \phi_i$ and $\gamma_i$, rather than assuming that they were equal to 1 (ideal). Also, a similar procedure would be used for equilibrium between phases other than vapor and liquid (for example, liquid and liquid or even multiple phases).

Perform bubble-point and dew point calculations using complete fugacity relations (assuming known fugacity coefficients and activity coefficients)