Basic Approach

All the macroscopic thermodynamic properties of a fluid (like ethalphy, entropy, etc.) can be calculated if the following two pieces of information are known: (1) An equation of state for the fluid and (2) the heat capacity of the fluid as a function of temperature. Note that you do not need to have a separate relation for the heat capacity as a function of pressure since this information can be derived from the equation of state. For a fluid that is a liquid (that is, a fluid that is incompressible, or nearly so), the equation of state approach is often replaced by the concepts of activity, activity coefficient, and the standard state, along with a model that gives activity coefficients for a specified standard state.

Given the above basic starting point, in practice you need to know the following to determine phase equilibrium in a vapor-liquid system:

    1.   An equation of state for the vapor, such as the ideal gas law or the virial equation of state for multicomponent gas mixtures.  

    2.  A method for estimating parameters used in an equation of state for the vapor, such as the three parameter Pitzer and Curl correlation for the second virial coefficient (which is based on the corresponding states principle). You may also need to have available a multicomponent version of the correlation you are using.

    3.  A method that gives the fugacities and fugacity coefficients for components in a vapor mixture from an equation of state, such as the virial equation of state.

    4.  A method that gives the fugacities, activities, and activity coefficients of components in a liquid mixture from a model for liquid behavior, such as regular solution theory.

    5.  When the liquid phase is represented using an activity coefficient model like regular solution theory (as opposed to an equation of state model for the liquid), you may need some additional information about the liquid. For example, you may need a method for predicting the vapor pressure of a pure liquid as a funtion of temperature such as given by the Antoine equation.

    If two phase are in equilibrium, then the fugacity of a component in one phase can be equated to the fugacity of that component in the other phase.  This relationship among component fugacities, together with the fact that the sum of the mole fractions in any phase is unity, permits the solution of phase equilibrium problems.  More specifically, consider the "bubble T" problem for a five component system where the five mole fractions in the liquid phase are specified along with the pressure.  The objective is to calculate the five mole fractions in the vapor phase and the temperature, i.e., there are six unknowns.  To solve for these unknowns, we can use the five equilibirum expressions for the five components where the fugacities are equated in the two phases.  The fact that the sum of the vapor phase mole fractions is unity is the sixth equation.   This system of equations can be solved using the numerical methods and software discussed in Chapters 2 and 3. Note that this approach is much simpler and more straightforward than the customized direct-substitution method used for this problem in the textbook Introduction to Chemical Engineering Thermodynamics by Smith, van Ness, and Abbott. 

Use of an Enthalpy Balance

    When performing enthalpy balances, a relation giving the heat of vaporization of a pure liquid as a function of temperature and a relation giving the heat capacity of a pure liquid (or alternatively of a pure gas) as a function of temperature may also be needed.  For the former we can use the Watson relation and for the later we can use the Lee-Kesler heat capacity departure function, which is a standard correlation in terms of the reduced temperature and the accentric factor the gives the relation between the heat capacity of a pure liquid cubstance and the heat capacity of that substance in the ideal gas state. See The Properties of Gases and Liquids (4th ed.) by Reid, Prausnitz, and Poling (Wiley, 1987) for more details.  

Simplified Approaches

    The basic phase equilibrium calculation described above can be simplified in several ways. One simplification is to remove the composition dependence of the fugacity coefficients in the vapor phase. You can do this by assuming that the fugacity coefficients for each component in the vapor mixture are equal to the fugacity coefficients of each component as a pure vapor at the temperature and pressure of mixture. This assumption is termed the "Lewis Fugacity Rule" and is equivalent to assuming the real gas mixture can be approximated as an ideal mixture of nonideal gases. This assumption, along with assuming there is no composition dependence for the activity coefficients in the liquid phase, is used to produce DePriester charts, which give the equilibrium ratio (given by K = y/x) for each component in a multicomponent vapor-liquid system as a function solely of the temperature and pressure.  

 

• Detailed notes for Chapter 4.

   

• Description of regular solution theory. This simple description of regular solution theory comes from the book Separation Processes by C. J. King.

 

• NIST Chemistry WebBook. This useful on-line database contains thermodynamic properties for over 7000 organic and inorganic chemical species.

 

• CRC Handbook of Solubility Parameters and other Cohesion Parameters. This handbook is partially available on-line from Google Books at this link. Especially useful parts of the on-line version are Table 2, starting on page 27, which lists the molar cohesive energies (in kJ/mol) at 25 C for 150 chemical species and Table 3, starting on page 34, which lists solubility parameters (in MPa^(1/2)) at 25 C for 75 chemical species. To see the missing parts from the on-line version, you can borrow the printed version from the UMCP library using the call number QD543.B22 1991.