Corresponding

Law of Corresponding States

Back to previous section: Compressibility
Continue to next section: Single Component in Multiphase System

Principle of Corresponding States
We have seen that the critical point of a fluid is where the liquid and vapor molar volumes become equal; i.e., there is no distinction between the liquid and vapor phases. It is the temperature above which the two phases can no longer coexits. Each compound is characterized by its own unique critical temperature (Tc), critical pressure (Pc) and critical volume (Vc). We have also seen that the van der Waals eos can be recast in terms of these critical properties. By requiring that (dP/dV)_T = 0 = (d²P/dV²)_T at the critical point, we were able to solve for the vdw constants a and b in terms of Tc and Pc. If we replace these constants with their corresponding values in terms of critical constants, we can rearrange the equation in terms of dimensionless temperature, pressure, and volume variables. The dimensionless variables used are called the reduced temperature, pressure, and volume; defined as:

Tr = T/Tc Pr = P/Pc Vr = V/Vc

The vdw eos in terms of these reduced variables becomes (we won't do the algebra here):

Pr = 8Tr/(3Vr - 1) - 3/Vr²

Note that in terms of these dimensionless or reduced variables the vdw eos is independent of the specific fluid. No information characteristic of the fluid appears in the equation. The value of Pr is the same for a given Tr and Vr regardless of whether the fluid is methane or hexane. This same procedure with a similar result could be applied to any cubic equation of state. What we are doing is scaling the behavior of the fluid to the critical point; i.e., all fluids at their critical point obviously have Tr = 1, Pr = 1, and Vr = 1.

This doesn't mean that all fluids obey the vdw eos, but it does mean that all fluids that do obey it would behave identically in terms of their reduced variables. This is a very powerful concept. It's extension to all fluids is called the Corresponding States Principle (CSP). We will state the CSP in this manner:

CSP: All fluids behave similarly when described in terms of their reduced temperature and pressure

General correlations have been developed based on CSP both in the form of tables and graphs. Generally these are given in terms of the compressibility factor, Z, as a function of Pr and Tr. For example, Figures 5.4-2, 5.4-3, and 5.4-4 in your text book are reproductions of the graphical form of CSP. Tabular values of the compressibility factor as a funtion of the reduced temperature and pressure are contained as one of the tools in this web course.

Let's try an example and see how we can obtain volumetric information from pressure and temperature using CSP.

Example: What is the molar volume, v, of CO₂ at 456 K and 72.9 atm?

Solution:
We can obtain critical properties from the DIPPR database. For CO₂, T_c = 304 K and P_c = 72.9 atm. We then calculate the reduced temperature and pressure. At the specified conditions,

T_r = 456/304 = 1.5 P_r= 72.9/72.9 =1.0

We can now use either the chart in the text or the tables in the tools portion of this web course. For example, using Figure 5.4.2, we can find Z from the intersection of the Tr = 1.5 line (interpolated on the figure below as the red line) with the coordinate Pr = 1.0 (vertical blue line) as shown in the figure below.

Reading horizontally across the graph (green line), we find that Z is therefore about 0.92. We can then complete the problem using the definition of Z. Thus,

z = 0.92 = Pv/RT

Thus, v = zRT/P = (0.92)[82.06 cm³*atm/(mol*K)](456 K)/(72.9 atm) = 472 cm³/mol

The student should try the same example using the compressibility table in the tools section.

Mixtures
So far, we have only used equations of state and CSP to find pure-component volumetric properties. What about mixture calculations? Generally, we apply exactly the same equations for mixtures by treating the mixture as a hypothetical "pure" component whose properties are some combination of the actual pure components that comprise it. We call this the one-fluid theory. Thus, to apply the eos, we determine a single value of the constants for the mixture; to apply CSP, we use the same plot or table as before but we make the temperature and pressure dimensionless with pseudocriticals for the hypothetical pure fluid instead of any one set of values as scaling variables from pure component values.

In this class we will use Kay's mixing rules, the simplest possible, to obtain the pseudocritical for the hypothetical pure component. Many other mixing rules are commonly used and provide more accuracy, but we choose Kay's rules here for simplicity sake. You see other mixing rules in your thermodynamics class. Mixing rules form the pseudocritical of the hypothetical pure component (the mixture) by taking some composition average of each component's critical properties. Kay's mixing rules use a simple mole fraction average for both Tc and Pc:

Kay's rules: T_c,m = S_ix_iT_c,i

P_c,m = S_ix_iP_c,i

w_c,m = S_ix_iw_c,i

Please note that mixtures have a critical temperature and a critical pressure just like pure fluids. The pseudocriticals computed from a mixing rule are not very good estimators of mixture critical points. They are not the true criticals of the mixture, but pseudocriticals that are useful for use in equations of state that have been designed for pure fluids. In fact, CSP does not work very well when true mixture critical are used as the reducing parameters. Pseudocriticals are instead a way to make the mixture conform to the pure-component CSP and eos.

Using the mixture pseudocriticals, one can make the temperature and pressure of the mixture dimensionless and apply CSP. Thus, T_r = T/T_c,m and P_r = P/P_c,m. The same charts and tables for z can then be used for the mixture as for pure components. Likewise, one can use this mixing rule with equations of state by using pseudocritical values in the equations that relate the equation of state constants to critical constants. This is a better procedure than calculating the volume of each component independently from the equation of state and then taking a compositional average of the molar volumes.

Example: Find v for a 30% CO₂ and 70% CO mixture at 202.73 K and 92.04 atm.

Solution:
First we obtain appropriate properties for the pure compounds from the DIPPR database:

component M (g/mol) T_c (K) P_c (atm)

CO₂ 44 304 72.9

CO 28 133 34.5

Next, we find the pseudocriticals for the given mixture:

T_c,m = (0.3)(304) + (0.7)(133) = 184.3 K
P_c,m = (0.3)(72.9) + (0.7)(34.5) = 46.02 atm
M = (0.3)(44) + (0.7)(28) = 32.8 g/mol

Then we apply CSP or use an eos. Here we will use CSP to obtain the compressibility factor:

T_r = 202.73/184.3 = 1.1 P_r = 92.04/46.02 = 2.0 z = 0.4

The compressibility factor was obtained from the table of compressibility factors in the tools section. The molar volume is then obtained from the definition of Z.

v = zRT/P = (0.4)(0.08206 cm³atm/mol*K)(202.73 K)/(92.04 atm) = 0.072 cm³/mol

Notice how bad the ideal gas law is at this point! If we had simply applied the ideal gas equation to this problem, we would have been off by (1-0.4)/0.4 = 150%!

Continue to next section: Single Component in Multiphase System

component	M (g/mol)	T_c (K)	P_c (atm)
CO₂	44	304	72.9
CO	28	133	34.5