Appendix 4A: Equations of State
There are about 20 different equations of state in use. It is beyond the scope of this text to discuss all of them. Most have been incorporated into computer simulations when needed. They have been used in calculations for advanced theoretical work that depend on the calculus of PVT, but they do exist and do provide results with an accuracy of 1% to 2%, which is significantly better than the Ideal Gas Law.
Some of the better-known equations of state are the Benedict-Webb-Rubin, the Clausius, the Holborn, the Kammerlingh-Onnes, the Peng-Robison, the Redlich-Kwong, and the Soave-Redlich-Kwong equations. One of the most commonly used equations is the van der Waals Equation of State.
The van der Waals Equation of State
One of the most frequently used equations of state is the van der Waals equation shown in Equation 4.23.
Here, a is a measure of the attraction between the particles and b is the exclusion volume of a mole of particles. As can be seen in Equation 4.23, the form of the van der Waals equation is very similar to that of the Ideal Gas Law seen in Equation 4.11. Values for a and b have been experimentally determined and published for a number of different gases.
In general, the attraction constant a in the first set of terms shows that a lower pressure p will be required for a given volume and temperature. Similarly, in the second set of terms, the quantity nb permits a larger volume V for a given pressure p and temperature T.
The other equations of state mentioned previously have structures similar to the van der Waals equation, but with additional terms. They all include some type of relationship of a modified pressure term multiplying a modified volume that equates to the number of moles of gas, the gas constant, and the absolute temperature.
While widely used, the van der Waals Equation of State has many limitations. The equation tends to break down when operating near the “critical temperature” and the “critical pressure” of the gas. Unfortunately, those temperatures and pressures are near the temperatures and pressures of some process operations, particularly for natural-gas plants.
Perhaps the best correlation of PVT for real gases uses a compressibility factor modified by an acentric factor that was developed by Kenneth Pitzer (“The Volumetric and Thermodynamic Properties of Fluids. I. Theoretical Basis and Virial Coefficients,” Journal of the American Chemical Society 77, no. 13, 1955: 3427) based on the critical temperature and critical pressure of a gas. The critical point is defined as the temperature and pressure at which a gas phase of a pure substance becomes indistinguishable from its liquid. There will be further discussion about the critical state in Chapter 6, “Phase Equilibria.” All you currently need to know is that the critical temperature and critical pressure of a pure gas is a unique physical property of that gas.
Using the compressibility factor, the ideal gas law equation of state becomes Equation 4.24.
Here, z is the compressibility factor that corrects the Ideal Gas Law equation of state to a generalized equation of state. The modified compressibility factor z is given by Equation 4.25.
Here, ω is the Pitzer acentric factor. Tables have been developed that correlate values for z0 and z1 with the reduced temperature and reduced pressure of the gas that are calculated by dividing the absolute temperature and absolute partial pressure of the gas by its critical temperature and critical pressure respectively, as shown in Equations 4.26 and 4.27.
Here, TC and pC are the critical temperature and pressure for a given gas.
Originally, only the first term, z0, was correlated with the reduced temperature and pressure. However, it was determined that there still remained a slight error. Later, the Pitzer acentric factor ω was developed that has a constant value unique to the chemical, which multiplies the second compressibility term z1 in Equation 4.25 and is a function of the reduced temperature and pressure. When z0 and z1 are added in Equation 4.25, the overall compressibility factor z for the pure gas can then be used in the generalized equation of state of Equation 4.24.
The value of the compressibility factor’s use is that once the critical temperature and critical pressure for any gas is known, we can then use the reduced temperature and reduced pressure to obtain values for z0 and z1 without having to perform laboratory PVT experiments. Most gases have already had their critical temperatures and pressures determined. Unfortunately, some PVT experiments may still be required to determine the value of the Pitzer acentric factor ω for gases not already quantified. However, it would not be necessary to carry out an exhaustive number of experiments over a wide range of temperatures and pressures as would be needed for the other equations of state.
The conditions of the preceding example are not that different from what might be encountered in a process operation, so be aware of compressed gases. Ideal gases can be dangerous, but real gases can be even more dangerous.
In addition to Pitzer’s work cited above, more information about the use of compressibility factors for mixtures can be found in O.A. Hougen, K.M. Watson, and R.A. Ragatz, Chemical Process Principals, Part II, Thermodynamics, Second Edition. (New York: John Wiley & Sons, 1959): 856–861.
Process gases are generally mixtures. For those gases, we can calculate values for a pseudocritical temperature and a pseudocritical pressure by adding the critical temperature and pressure of each component multiplied by its mole fraction in the gas mixture. In essence, the result is a molar average critical temperature and molar average critical pressure. Though not exact, they are within engineering accuracy.