Basics of Phase Equilibria

     Phase equilibria is on of the foundations of the chemical engineering field.  This physical state and it's associated calculations can be found in countless spreadsheets, freeware programs, and powerful simulators.  At times, it can be easy to forget what is really being calculated with the push of a button.

The vapor pressure of a liquid is defined as the pressure at which the liquid boils at a given temperature.   The vapor pressure relations for liquids that can be considered ideal are the well known Antoine and Clausius-Clapeyron equations.  Special care should be taken when using these relations for vapor pressure calculations.  In particular, there are three cases when they should be avoided.  Table 1 below summarizes these three cases.

Table 1:  Three Cases to Avoid the Antoine or Clausius-Clapeyron Equations

The limitations of the Antoine Equation are supplemented with a property called fugacity.  A detailed discussion of fugacity can be found here at the Resource Page in an article entitled "Validating Your Binary VLE Data".  For binary liquids, fugacity is used to calculate the vapor pressure in a spreadsheet entitled "Vapor Pressure of Binary Mixtures" available here.

     Fugacity relates vapor pressure and temperature to partial molar volumes and partial molar enthalpies.  Since this data is not always available, there was a need to relate experimental data to fugacity and vapor pressure.   This was accomplished with binary interaction parameters.  You may be familiar with popular thermodynamic models such as NTRL, UNIQUAC, and Wilson.  All of these relations relate one component of a mixture to another by the parameters.

     The models also offer an effective means of calculating the K-value for a solution.  By definition, the K-value is the mole fraction of component j in vapor divided by the mole fraction of component j in liquid, or y_j / x_j.  For ideal mixtures, Dalton's Law allows K to equal the vapor pressure of pure component j divided by the vapor pressure of the solution.   Corrections for real thermodynamics include variables such as the fugacity and activity coefficients.  All of the models, vapor pressure calculations, and k-value calculations discussed here are well defined and available in many sources.

     Now, as a quick review of how to use a TXY diagram:

     Let's assume that the above chart is a graph of the phase behavior of a water-methanol solution.  The mole fractions of methanol are graphed above.  At T1, the vapor will have a methanol mole fraction of y1 and a water mole fraction of 1-y1.  At the same temperature, the liquid present will have a mole fraction of methanol being x1 and a mole fraction of water being x1-1.