Th: Use one (or all) of the covered non-ideal equations

Non-ideal Gases

As mentioned, the ideal gas law is only applicable at high specific volumes and becomes essentially useless at lower specific volumes. (Sometimes too high, sometimes too low ... )

For gases under non-ideal conditions, something else needs to be done.

Non-Ideal Equations of State

DEFINITION
(reminder)

An equation of state relates the molar density (or specific molar volume) ot a fluid to the temperature and pressure of that fluid. (Note that I am using the term fluid rather than gas!)

Ideal gas law works very well when fluids are very "gassy" (high specific volume). Need others when we are closer to "liquidy" conditions.

In general, as fluids become less "gassy" the assumptions of the ideal equation of state become incorrect (we assume that individual gases molecules take up NO VOLUME and that they do NOT INTERACT WITH EACH OTHER when we derive the ideal gas law).

Since the non-ideal equations that we will be talking about here relax these assumptions, you should always "see" a way to reduce them to the ideal gas law by making these assumptions again!

As we will discuss, these more complex equations of state are sometimes motivated by mathematics and sometimes by theory. (Just in case you might wonder, "where on earth did they get that?!", we will mention it if I know (or maybe I will just make something up!))

Virial Equations of State

A primarily mathematics motivated (theory is from statistical mechanics) equations of state, the virial equation uses a power series in the form:

$\frac{P\hat V}{RT} = 1 + \frac{B}{\hat V}+ \frac{C}{\hat V^2}+ \frac{D}{\hat V^2} + ...$

where B, C, and D (etc.) are material dependent constants.

These constants are sometimes difficult to determine (theoretically) so the expression is typically truncated after the B (so there is only one "fitting" parameter! (i.e.,

$\frac{P\hat V}{RT} = 1 + \frac{B}{\hat V}$

NOTE

This expression is good for non-polar gases.

The constant, B, can be obtained from the following relation:

$B=\frac{RT_c}{P_c}(B_o+\omega B_1)$

where Tc and Pc are the critical temperature and pressure, and Bo and B1 are given by the following expressions:

$B_o = 0.083 - \frac{0.422}{T_r^{1.6}$

$B_1 = 0.139 - \frac{0.172}{T_r^{4.2}$

DEFINITION

The reduced temperature (pressure) is a dimensionless temperature (pressure) given as the ratio of the actual temperature (pressure) to the critical temperature (pressure) and is denoted by Tr (Pr).

DEFINITION

The Pitzer ascentric factor ($\omega$) is a material parameter that reflects the polarity and "shape" of the molecule (i.e., it is different for every molecule and therefore must be looked up!)

There are other expressions similar to this that are ALSO called virial equations of state. We will ignore them entirely.

Cubic Equations of State

Another important class of equations of state are the cubic ones. They are called this because, mathematically, the equations are third order polynomials (cubic equations) in $\hat V$

(Note that the truncated virial equation is quadratic!)

Why are cubic equations "good" as equations of state?! Look at the shape of the phase envelope!

The two most important cubic equations of state are the van der Waals equation and the Soave-Redlich-Kwong (SRK) equation. We will use only the van der Wants equation. (They are very similar to use and SRK just makes our life more tedious at this point.)

van der Waals Equation

The van der Waals equation is interesting to examine because it is easy to qualitatively discuss the origin of its deviation from ideality.

$P = \frac{RT}{\hat V-b}-\frac{a}{\hat V^2}$

The symbols, b and a are material dependent constants (however, you can calculate them from a set of material independent. equations!). They represent (roughly) the volume of the gas molecules (b) and the "interaction" of the gas molecules (a).

As usual, the constants are correlated to the critical constants of the materials:

$a = \frac{27R^2T_c^2}{64P_c}$

$b = \frac{RT_c}{8P_c}$

Compressibility Factor and Corresponding States

A simpler (mathematically) way of representing non- ideality is the compressibility factor.

PV=znRT
or $P\hat V = zRT$

DEFINITION

The compressibility factor, z, is a dimensionless number that represents a material's deviation from ideal gas behavior.

While this equation (being linear) is very simple to use, obtaining values of z is not! Like our previous examples, it would be nice to generate material-independent correlations for this parameter.

DEFINITION

The "law" at corresponding states suggests that gases behave similarly depending on how far from their critical point they are. (In other words, if we use the reduced temperature and pressure instead of the actual T and P, all gases behave similarly.)

NOTE

For Hydrogen and Helium it is necessary to add 8K and 8atm to the critical temperature and pressure respectively in order for them to behave like a "corresponding states" material.

Charts are available for the (generalized) compressibility factor, z, as a function of the reduced temperature and pressure.

What happens if we don't know the T or P? (but are given $\hat V$ instead)? Use the (ideal) "reduced volume":

$\hat V_r^{ideal} = \frac{\hat V}{RT'_c/P'_c}$

Outcome:

Use one (or all) of the covered non-ideal equations (SRK, compressibility factor, van der Waals, virial equation) to determine P, V, or T