12.6 Behaviour of Ideal & Real Gases and Equations of State

When the Ideal Gas Law Fails

The ideal gas equation

$pv = RT$

assumes molecules are point masses with no intermolecular forces. Both assumptions fail when molecules are pressed close together or move slowly, that is at high pressure or low temperature, and especially near the saturation line.

For a Management Trainee this is not academic. Compressed air receivers at 10 bar, refrigerants throughout the vapour-compression cycle, and coal-bed methane stored at pressure all depart measurably from ideal behaviour.

The rule of thumb is that the ideal gas law is acceptable when the pressure is well below the critical pressure and the temperature is well above the critical temperature, typically taken as $T > 2T_c$ or $p < 0.1 p_c$.

The Compressibility Factor

The departure is quantified by a single dimensionless correction:

$\boxed{Z = \frac{pv}{RT}}$

At moderate pressures $Z$ is typically less than one, because attraction pulls molecules together. At very high pressures the finite size of the molecules dominates and $Z$ exceeds one. For hydrogen and helium, whose intermolecular attraction is very weak, $Z$ exceeds one over almost the whole practical range.

The van der Waals Equation

The first and most-examined real-gas equation modifies both terms of the ideal law:

$\boxed{\left(p + \frac{a}{v^2}\right)\left(v - b\right) = RT}$

The constant $b$ is called the co-volume and is roughly four times the actual volume of the molecules.

Critical-point constants

At the critical point the isotherm has a point of inflection with a horizontal tangent, so

$\left(\frac{\partial p}{\partial v}\right)_{T_c} = 0 \quad\text{and}\quad \left(\frac{\partial^2 p}{\partial v^2}\right)_{T_c} = 0$

Applying both conditions to the van der Waals equation yields

$v_c = 3b, \qquad T_c = \frac{8a}{27Rb}, \qquad p_c = \frac{a}{27b^2}$

and inverting,

$a = \frac{27R^2T_c^2}{64p_c}, \qquad b = \frac{RT_c}{8p_c}$

The most striking consequence is the critical compressibility factor:

$Z_c = \frac{p_c v_c}{RT_c} = \frac{3}{8} = 0.375$

This is a universal constant for all van der Waals gases, independent of $a$ and $b$. Real substances give values between about 0.23 and 0.31, so the van der Waals equation overestimates $Z_c$ — accurate qualitatively but not quantitatively. That combination of conceptual correctness and quantitative imprecision is exactly why it survives in teaching but not in design.

Other Equations of State

The virial equation deserves note because its coefficients have a statistical-mechanical meaning: the second virial coefficient $B$ describes two-molecule interactions, $C$ three-molecule interactions, and so on. Truncating after $B$ is adequate at moderate pressures.

The Principle of Corresponding States

Define reduced properties by dividing each by its critical value:

$p_r = \frac{p}{p_c}, \qquad T_r = \frac{T}{T_c}, \qquad v_r = \frac{v}{v_c}$

Substituting into the van der Waals equation eliminates $a$ and $b$ entirely and gives

$\left(p_r + \frac{3}{v_r^2}\right)\left(3v_r - 1\right) = 8T_r$

No substance-specific constants remain. This is the mathematical statement of the principle of corresponding states:

All gases, when compared at the same reduced pressure and reduced temperature, have approximately the same compressibility factor.

The generalised compressibility chart

The practical consequence is a single chart of $Z$ against $p_r$ with $T_r$ as parameter, usable for any gas. The procedure is:

1. Look up $p_c$ and $T_c$ for the gas.
2. Compute $p_r$ and $T_r$.
3. Read $Z$ from the chart.
4. Use $pv = ZRT$.

When the volume is known but not the pressure, the pseudo-reduced specific volume is used instead:

$v_r' = \frac{v\,p_c}{R\,T_c}$

Worked example

Find the specific volume of carbon dioxide at 8 MPa and 320 K. For carbon dioxide, $p_c = 7.39$ MPa, $T_c = 304.2$ K and $R = 0.1889$ kJ/kg K.

$p_r = \frac{8}{7.39} = 1.08, \qquad T_r = \frac{320}{304.2} = 1.05$

From the generalised chart at these reduced conditions, $Z \approx 0.30$. Then

$v = \frac{ZRT}{p} = \frac{0.30\times0.1889\times320}{8000} = 2.27\times10^{-3} \text{ m}^3/\text{kg}$

The ideal gas law would have given $7.56\times10^{-3}$ m$^3$/kg — an error of more than 230%. This is exactly the region, just above the critical point, where ideal-gas assumptions are least defensible, and it is the operating region of supercritical carbon dioxide systems.

Behaviour Near Saturation

A useful qualitative picture of $Z$ against $p_r$:

• At very low pressure ($p_r \to 0$), $Z \to 1$ for all gases and all temperatures. Ideal behaviour is always recovered as pressure falls.
• At high temperature ($T_r > 2$), $Z$ stays near unity across a wide pressure range.
• Near $T_r \approx 1$, $Z$ dips sharply, reaching values as low as 0.2 to 0.3 near the critical pressure. This is the worst region for the ideal gas assumption.
• At very high $p_r$, $Z$ rises above unity as repulsion dominates.

Practical Note

For design work, tabulated property data — steam tables, refrigerant tables, software libraries — are always preferred to any equation of state, because they are fitted directly to measurement. Equations of state are used where tables do not exist, in computational work where a closed form is needed, and in examination questions where the compressibility chart is the intended route.