Advanced Theory of Ideal and Real Gases: The van der Waals Equation

Every introductory course states the perfect gas law, $pV = nRT$, and moves on. This chapter asks the question a physical chemist cannot avoid: why does no real gas obey it exactly, and how far can a corrected equation of state be pushed before it, too, breaks down? The perfect gas law is the zeroth-order description of a gas — molecules with no size and no mutual attraction, bouncing elastically off the container walls. Real molecules occupy volume and attract one another at short range, and both effects become significant precisely where chemistry gets interesting: at high pressure, low temperature, and near a phase transition.

You should already have to hand the kinetic model of gases and the ideal gas equation of state from your prior studies — this chapter does not re-derive $pV = \tfrac{1}{3}Nm\langle c^2\rangle$ from first principles, but builds on it. What follows is the toolkit physical chemists use to quantify non-ideality: the compression factor, the virial expansion, the van der Waals equation and its molecular justification, the critical constants it predicts, the principle of corresponding states, and the Maxwell–Boltzmann distribution of molecular speeds that underlies the kinetic model itself. The next chapter takes the equations of state developed here and puts them to work computing the thermodynamic functions of real reacting systems.

1. The Perfect Gas: A Recap of the Model and Its Limits

The perfect gas equation of state,

follows from a model with two defining idealisations: molecules are point particles (zero volume) that exert no force on one another except during instantaneous, perfectly elastic collisions. Neither idealisation survives contact with a real substance. Real molecules have a finite size, so the volume actually available for them to move in is less than the container volume $V$; and real molecules attract one another through van der Waals and dipolar forces, so a molecule approaching the container wall is pulled backward by its neighbours, striking the wall less often and less hard than the perfect-gas model predicts. Both effects vanish in the limit of low pressure and high temperature, where molecules are far apart and moving fast — which is exactly why the perfect gas law is the correct limiting law, approached by every real gas as $p \to 0$.

Key distinction. "Ideal" and "real" describe the same substance under different conditions, not two different kinds of gas. Every real gas behaves ideally in the low-pressure limit; the departure from ideality is a quantitative question, answered by the tools of this chapter.

2. Quantifying Non-Ideality: The Compression Factor

The standard measure of how far a real gas departs from perfect behaviour is the compression factor (or compressibility factor),

where $V_m = V/n$ is the molar volume and $V_m^{\circ} = RT/p$ is the molar volume a perfect gas would occupy at the same $p$ and $T$. By construction $Z = 1$ for a perfect gas at every condition. For a real gas, $Z$ is a function of $p$ and $T$ that measures the net effect of molecular size and attraction:

• At low to moderate pressure, intermolecular attraction dominates: molecules pull each other inward, the gas occupies less volume than expected, and $Z < 1$.
• At high pressure, the finite size of the molecules dominates: there is simply less empty space to compress into, the gas resists further compression more than a perfect gas would, and $Z > 1$.
• At some intermediate pressure — and exactly at the Boyle temperature, defined precisely in Section 5.4 — the two effects cancel to first order and $Z \approx 1$ over an extended range of pressure, even though the gas is not actually ideal.

Worked example. At $500~\text{K}$ and $100~\text{atm}$, one mole of carbon dioxide occupies a molar volume of $V_m = 0.366~\text{dm}^3$. Using $R = 0.08206~\dfrac{\text{dm}^3\,\text{atm}}{\text{mol}\,\text{K}}$,

$Z < 1$: CO$_2$'s Boyle temperature is far above $500~\text{K}$ (experimentally $715~\text{K}$; the van der Waals estimate of Section 5.4 is $1.0\times10^3~\text{K}$), so at this condition attractive forces still outweigh the excluded-volume effect, and the gas is noticeably more compressible than the perfect gas law predicts. N$_2$, whose Boyle temperature is much lower (experimentally $327~\text{K}$; van der Waals estimate $426~\text{K}$), has already passed its own crossover by $500~\text{K}$ and instead shows $Z>1$ there — a reminder that "attraction dominates" is a statement about where a gas sits relative to its own Boyle temperature, not a universal rule at a given $T$.

3. The Virial Equation of State

A compression factor alone only reports a deviation; it does not predict one. The virial equation of state expresses $Z$ as a power series in $1/V_m$ (equivalently, in $p$), fitted empirically to real-gas data:

$B(T)$ and $C(T)$ are the second and third virial coefficients, temperature-dependent constants specific to each gas. The virial equation is the rigorous, general description of a real gas — it is exact order by order, with no underlying molecular model assumed — but each coefficient must be measured or computed from statistical mechanics. Its chief theoretical importance here is as a reference: any proposed equation of state, including the van der Waals equation below, can be tested by expanding it in powers of $1/V_m$ and comparing the resulting $B(T)$ with experiment.

4. Molecular Origins of Non-Ideality

Two physically distinct effects, acting in opposite directions, are responsible for everything in Section 2:

Repulsion at short range (excluded volume). Real molecules cannot overlap; each has an effective hard-core volume. If a single molecule excludes a volume $b$ (per mole) from the space available to the others, the gas resists compression more strongly than the point-particle model predicts, pushing $Z$ above $1$ at high density.

Attraction at longer range. Induced-dipole (London dispersion), permanent-dipole, and hydrogen-bonding forces pull neighbouring molecules together. A molecule in the bulk of the gas is pulled equally in all directions and feels no net force, but a molecule about to strike the container wall is pulled backward by the molecules behind it, reducing both the frequency and the force of wall collisions. This lowers the pressure actually exerted below the perfect-gas prediction, pushing $Z$ below $1$ at low-to-moderate density — precisely the correction that the perfect gas model, with no intermolecular forces at all, cannot capture.

These two molecular pictures motivate, term by term, the correction that van der Waals proposed in 1873.

5. The van der Waals Equation of State

5.1 Correcting the volume

Replace the container volume $V$ (available to point particles) with the volume actually available to real molecules, $V - nb$, where $b$ is the excluded molar volume — approximately four times the volume of a single molecule, since a pair of hard spheres of diameter $d$ cannot approach closer than centre-to-centre distance $d$, excluding a sphere of volume $\tfrac{4}{3}\pi d^3$ shared between the pair. The perfect gas law with this correction alone, $p(V-nb) = nRT$, already predicts $Z > 1$ at all pressures — the excluded-volume effect in isolation.

5.2 Correcting the pressure

The attractive force pulling a wall-bound molecule backward is proportional to the local concentration of neighbours doing the pulling, $n/V$, and the rate at which molecules arrive at the wall is proportional to their own concentration, $n/V$. The reduction in pressure is therefore proportional to the product of these two concentrations:

where $a$ is a positive constant specific to the gas, large for strongly attracting molecules (polar, hydrogen-bonding) and small for weakly attracting ones (noble gases, small non-polar molecules). The pressure that would be measured, $p$, is therefore less than the "ideal" pressure $p_{\text{ideal}}$ that the kinetic model would predict for the same $T$ and the corrected volume $V-nb$:

5.3 Combining both corrections

Substituting $p_{\text{ideal}}$ for $p$ in the excluded-volume equation $p_{\text{ideal}}(V-nb) = nRT$ gives the van der Waals equation of state:

or, per mole ($V_m = V/n$),

Both $a$ and $b$ are empirical constants, fitted to the real gas's measured behaviour (most directly, to its critical constants — Section 6) and tabulated for each substance. Typical literature values, in units of $\text{dm}^6\,\text{atm}\,\text{mol}^{-2}$ for $a$ and $\text{dm}^3\,\text{mol}^{-1}$ for $b$: helium ($a = 0.0341$, $b = 0.0237$), nitrogen ($a=1.352$, $b=0.0387$), and carbon dioxide ($a = 3.610$, $b = 0.0429$) — the small $a$ of helium reflecting its weak dispersion forces, the comparatively large $a$ of CO$_2$ reflecting its greater polarisability.

Worked example. Estimate the pressure exerted by $1.00~\text{mol}$ of CO$_2$ confined to $V = 0.500~\text{dm}^3$ at $T = 300~\text{K}$, comparing the perfect-gas and van der Waals predictions.

Perfect gas: $p = \dfrac{nRT}{V} = \dfrac{(1.00)(0.08206)(300)}{0.500} = 49.24~\text{atm}$.

Van der Waals, rearranged for $p$:

The van der Waals pressure ($39.4~\text{atm}$) is markedly lower than the perfect-gas value ($49.2~\text{atm}$): at this molar volume ($0.500~\text{dm}^3\,\text{mol}^{-1}$, only about twelve times $b$), the attractive term dominates heavily over the excluded-volume term, and CO$_2$ is substantially easier to compress than the perfect gas model allows.

5.4 The van der Waals second virial coefficient and the Boyle temperature

Writing the molar van der Waals equation as $p = \dfrac{RT}{V_m-b} - \dfrac{a}{V_m^2}$ and expanding $\dfrac{1}{V_m-b} = \dfrac{1}{V_m}\left(1 + \dfrac{b}{V_m} + \dfrac{b^2}{V_m^2} + \cdots\right)$ for $b \ll V_m$ gives

Comparing with the virial equation of Section 3 identifies the van der Waals second virial coefficient, $B(T) = b - \dfrac{a}{RT}$. It is negative at low $T$ (attraction dominates, $Z<1$), positive at high $T$ (excluded volume dominates, $Z>1$), and vanishes at the Boyle temperature

at which $Z \approx 1$ over an extended range of pressure even though the gas is not ideal. Within the model, $T_B/T_c = (a/Rb)\big/(8a/27Rb) = 27/8 \approx 3.375$. For N$_2$ the van der Waals estimate is $T_B = 1.352/[(0.08206)(0.0387)] = 426~\text{K}$ (experimental: $327~\text{K}$); for CO$_2$ it is $1.03\times10^3~\text{K}$ (experimental: $715~\text{K}$). The model reproduces the trend but overestimates $T_B$ by $30$–$45\%$ — a reminder that $a$ and $b$ fitted to the critical point are not optimal elsewhere.

6. Critical Phenomena and the Critical Constants

Plotting $p$ against $V_m$ at fixed $T$ from the van der Waals equation (an isotherm) produces, at sufficiently low $T$, a curve with a local maximum and a local minimum — an oscillation with no physical meaning in the region where $(\partial p/\partial V_m)_T > 0$, since pressure cannot increase as a gas expands at constant temperature. This region is the mathematical signature of the liquid–gas phase transition the equation cannot describe directly — the physically correct replacement, the Maxwell equal-area construction, is not derived here. As $T$ rises, the maximum and minimum move together and merge at a single point of inflection: the critical point $(p_c, V_{m,c}, T_c)$, above which no liquid–gas distinction exists at all.

At the critical point, the isotherm has a horizontal tangent with zero curvature:

Writing the van der Waals equation as $p = \dfrac{RT}{V_m - b} - \dfrac{a}{V_m^2}$ and differentiating,

Solving the first equation for $RT$ and substituting into the second (both evaluated at the critical point) eliminates $RT$:

Back-substituting $V_{m,c} = 3b$ into $RT = 2a(V_m-b)^2/V_m^3$ gives $RT_c = \dfrac{2a(2b)^2}{27b^3} = \dfrac{8a}{27b}$, and substituting $V_{m,c}$ and $T_c$ into the equation of state itself gives $p_c$. Collected together, the van der Waals critical constants are:

These three relations can be inverted to extract $a$ and $b$ from a gas's measured critical temperature and pressure — the standard route by which the tabulated constants of Section 5.3 are obtained in the first place:

Worked example — critical constants of CO$_2$. Using $a = 3.610~\text{dm}^6\,\text{atm}\,\text{mol}^{-2}$, $b = 0.0429~\text{dm}^3\,\text{mol}^{-1}$ and $R = 0.08206~\dfrac{\text{dm}^3\,\text{atm}}{\text{mol}\,\text{K}}$:

The predicted $T_c = 304~\text{K}$ and $p_c = 72.6~\text{atm}$ are strikingly close to the experimental values ($304.2~\text{K}$, $72.8~\text{atm}$) — but $V_{m,c}$ is a much poorer match, as the next section's compression factor comparison makes quantitative.

6.1 The critical compression factor: a universal — and flawed — prediction

Combining the three boxed relations gives the critical compression factor,

Because every gas's $a$ and $b$ cancel out of this ratio, the van der Waals equation predicts the same value of $Z_c$ for every substance — a direct consequence of there being only two adjustable constants fitted to three critical-point conditions. Real gases do not cooperate: measured critical compression factors cluster around $0.27$–$0.29$ (CO$_2$: $Z_c = p_cV_{m,c}/RT_c = (72.8)(0.0940)/[(0.08206)(304.2)] = 0.274$, using the experimental $V_{m,c} = 0.0940~\text{dm}^3\,\text{mol}^{-1}$). The van der Waals equation's universal $0.375$ overshoots every real gas, which is exactly why it reproduces $T_c$ and $p_c$ tolerably but $V_{m,c}$ badly — a two-parameter model cannot be forced to fit three independent numbers.

7. The Principle of Corresponding States and Reduced Variables

Define the reduced variables $p_r = p/p_c$, $V_r = V_m/V_{m,c}$, $T_r = T/T_c$ — each physical quantity measured as a multiple of its own critical value. Substituting $p = p_c p_r$, $V_m = 3bV_r$ and $T = T_c T_r$ into the van der Waals equation and using the boxed critical-constant relations to eliminate $a$, $b$ and $R$ entirely (a short algebraic exercise left as Exercise 12 of this chapter) yields the reduced van der Waals equation:

This equation contains no reference to the identity of the gas at all: $a$ and $b$ have disappeared completely. Two different gases held at the same reduced temperature and reduced pressure therefore have the same reduced volume — the principle of corresponding states. It is only an approximation (it holds exactly within the van der Waals model and only approximately for real gases, better for chemically similar, non-polar molecules of comparable shape), but it is a powerful one: a single reduced equation of state, and a single reduced compression-factor chart built from it, can be applied to any gas once its own $T_c$ and $p_c$ are known, without needing separate $a$ and $b$ values at all.

8. The Maxwell–Boltzmann Distribution of Molecular Speeds

The equations of state above describe the macroscopic consequences of molecular motion; the Maxwell–Boltzmann distribution describes the motion itself. In thermal equilibrium, the probability that a molecule's velocity component along any one Cartesian axis lies in a narrow range $dv_x$ follows a Gaussian (from the Boltzmann distribution applied to translational kinetic energy $\tfrac{1}{2}mv_x^2$):

with identical, independent distributions for $v_y$ and $v_z$. The probability of a given speed $c = \sqrt{v_x^2+v_y^2+v_z^2}$, regardless of direction, is obtained by multiplying the three Cartesian distributions and summing over all directions at fixed $c$ — geometrically, over the surface of a sphere of radius $c$ in velocity space, of area $4\pi c^2$:

This is the Maxwell–Boltzmann speed distribution. Three characteristic speeds follow from it by standard calculus:

• The most probable speed, where $F(c)$ is a maximum ($dF/dc=0$): $c_{\text{mp}} = \sqrt{\dfrac{2RT}{M}}$ (using the molar mass $M = N_Am$ and $R = N_Ak_B$).
• The mean speed, $\bar{c} = \displaystyle\int_0^\infty c\,F(c)\,dc = \sqrt{\dfrac{8RT}{\pi M}}$.
• The root-mean-square speed, $c_{\text{rms}} = \sqrt{\langle c^2\rangle} = \sqrt{\dfrac{3RT}{M}}$ — the same quantity that enters the kinetic-theory pressure formula $pV = \tfrac{1}{3}nMc_{\text{rms}}^2$.

These satisfy $c_{\text{mp}} < \bar{c} < c_{\text{rms}}$ always, in the fixed ratio $\sqrt{2}:\sqrt{8/\pi}:\sqrt{3} \approx 1.000 : 1.128 : 1.225$, because the speed distribution is skewed toward high $c$ by the growing phase-space factor $4\pi c^2$ even as the exponential cuts it off.

Worked example. Find $c_{\text{rms}}$, $\bar{c}$ and $c_{\text{mp}}$ for N$_2$ ($M = 28.02~\text{g}\,\text{mol}^{-1} = 0.02802~\text{kg}\,\text{mol}^{-1}$) at $298~\text{K}$, using $R = 8.314~\dfrac{\text{J}}{\text{mol}\,\text{K}}$.

As required, $421 < 475 < 515~\dfrac{\text{m}}{\text{s}}$, in the ratio $1.000 : 1.128 : 1.224$ — matching $1:\sqrt{8/\pi}:\sqrt{3}$ to rounding.

9. Common Exam Pitfalls

• Writing the van der Waals correction with the wrong sign or the wrong power of $n$. The pressure term is added inside the bracket, $p + an^2/V^2$, because the true ideal pressure needed in the excluded-volume factor is larger than the measured $p$; students frequently subtract it instead, or forget the $n^2$ (it is $n^2$, not $n$, because the attraction depends on the product of two concentrations, $n/V$ each).
• Treating $a$ and $b$ as universal constants. They are gas-specific empirical parameters, fitted (usually) to each substance's own critical constants; a value for CO$_2$ must never be substituted into a calculation for N$_2$.
• Assuming $Z = 1$ means "ideal behaviour" at that single pressure, full stop. $Z$ can pass through $1$ at an intermediate pressure purely by cancellation of the attractive and excluded-volume effects (notably at the Boyle temperature, where this holds over an extended range); the gas is not thereby obeying the perfect gas law exactly at neighbouring pressures.
• Confusing $V_{m,c}$ predicted by van der Waals with the experimentally measured critical molar volume. The model fits $T_c$ and $p_c$ tolerably (two conditions, two parameters $a,b$) but $Z_c = 3/8$ is a fixed, substance-independent output that is systematically too large — $V_{m,c}$ inherits that error.
• Using the critical-constant formulas with mismatched units for $R$. $a = 27R^2T_c^2/(64p_c)$ and $b = RT_c/(8p_c)$ require $R$ expressed in the same unit system as $p_c$ (e.g. $\text{dm}^3\,\text{atm}\,\text{mol}^{-1}\text{K}^{-1}$ if $p_c$ is in atmospheres); mixing SI pressure with non-SI $R$ is a common source of an order-of-magnitude error.
• Reporting a speed too small by a factor of $\sqrt{1000} \approx 31.6$ by mishandling the units of $M$ — these formulas require the molar mass in $\text{kg}\,\text{mol}^{-1}$ when $R$ is in $\text{J}\,\text{mol}^{-1}\text{K}^{-1}$; leaving the molar mass in $\text{g}\,\text{mol}^{-1}$ makes the denominator $1000$ times too large and so shrinks every characteristic speed by $\sqrt{1000}$ (e.g. $16~\text{m}\,\text{s}^{-1}$ instead of $515~\text{m}\,\text{s}^{-1}$ for N$_2$ — an obviously unphysical result that should prompt a units check).

Summary

• The compression factor $Z = pV_m/RT$ measures departure from ideality: $Z<1$ signals net attraction dominating, $Z>1$ signals net excluded-volume (repulsion) dominating.
• The virial equation, $pV_m = RT(1 + B/V_m + C/V_m^2 + \cdots)$, is the general, model-free description of a real gas, used as the benchmark against which any proposed equation of state is tested.
• The van der Waals equation, $\left(p + \dfrac{an^2}{V^2}\right)(V-nb) = nRT$, corrects the perfect gas law for excluded molecular volume ($b$) and intermolecular attraction ($a$), both fitted empirically per substance.
• The critical constants predicted by van der Waals are $V_{m,c}=3b$, $T_c = 8a/27Rb$, $p_c = a/27b^2$, invertible to give $b = RT_c/8p_c$ and $a = 27R^2T_c^2/64p_c$; the resulting critical compression factor $Z_c = 3/8 = 0.375$ is universal within the model but systematically higher than real gases' $Z_c \approx 0.27$–$0.29$.
• The principle of corresponding states, $\left(p_r + 3/V_r^2\right)(3V_r-1) = 8T_r$, removes all substance-specific constants by expressing $p$, $V_m$, $T$ relative to their critical values, allowing one reduced equation of state to describe (approximately) every gas.
• The Maxwell–Boltzmann speed distribution, $F(c) = 4\pi(m/2\pi k_BT)^{3/2}c^2e^{-mc^2/2k_BT}$, yields three characteristic speeds in the fixed ratio $c_{\text{mp}} : \bar{c} : c_{\text{rms}} = \sqrt{2} : \sqrt{8/\pi} : \sqrt{3}$, with $c_{\text{rms}} = \sqrt{3RT/M}$ reproducing the kinetic-theory pressure formula.
• Equations of state and critical behaviour developed here are the foundation for the next chapter's thermodynamic treatment of chemical reactions in real, non-ideal systems.