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.

(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!))

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.,

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

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 T_{c} and P_{c} are the critical
temperature and pressure, and B_{o} and B_{1}
are given by the following expressions:

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

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

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 T_{r}
(P_{r}).

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.

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.)

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:

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

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

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

PV=znRT

or $P\hat V = zRT$

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.

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.)

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":

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

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