The van der Waals equation

Result

The van der Waals equation

T-021Home CU-104Threads thermo

Statement

Real gas molecules have finite size and attract one another — both directly contradicting the idealisations kinetic-theory-pressure used to derive the ideal gas law. Correcting for finite molecular volume (subtracting an excluded volume \(nb\) from the container volume) and for intermolecular attraction (adding a pressure correction \(a(n/V)^2\) to the measured pressure) gives the van der Waals equation, \(\left(P+a\dfrac{n^2}{V^2}\right)(V-nb)=nRT\), with \(a\) and \(b\) empirically determined constants specific to each gas, unlike the universal \(R\).

Why it matters

ideal-gas-law and kinetic-theory-pressure both repeatedly flagged their point-particle, zero-interaction idealisation as an approximation with a known domain of validity, without ever quantifying the correction. This result closes that gap directly, and in doing so accomplishes something the ideal gas law fundamentally cannot: it predicts that every real gas has a critical point (a temperature above which no amount of pressure can liquefy it) and, more generally, a genuine liquid–gas phase transition — behaviour entirely outside the ideal gas law's scope, since an ideal gas by construction never liquefies at all.

Hypotheses

Molecular finite size can be captured by a single excluded-volume constant \(b\) per mole, treating molecules as hard spheres.For two hard spheres of radius \(r\), their centres cannot approach closer than \(2r\), excluding a sphere of radius \(2r\) (volume \(\tfrac43\pi(2r)^3=8\times\tfrac43\pi r^3\), eight times one molecule's own volume) around each molecule from the other's accessible space; since this excluded volume is shared between each interacting pair, the effective excluded volume contributed per molecule is \(4\times\) its own volume, giving \(b=4N_A\left(\tfrac43\pi r^3\right)\) at the molar scale. This is a simplified, single-parameter model of a genuinely more complex finite-size effect, and treats molecules as perfectly rigid spheres, which most are not. Intermolecular attraction contributes a mean-field pressure correction proportional to the square of number density, \((n/V)^2\), rather than a more detailed distance-dependent treatment.Molecules near a container wall feel a net inward pull from the surrounding bulk gas (no gas lies beyond the wall to pull outward), reducing their momentum transfer to the wall and hence the measured pressure below the "true" (attraction-free) value; the size of this reduction depends on both the density of molecules near the wall and the density of molecules pulling them inward, giving the \((n/V)^2\) dependence. This mean-field approach averages over the detailed, generally more complex distance-dependence of real intermolecular potentials, which is why \(a\) and \(b\) are treated as fitted empirical constants rather than derived exactly from first-principles molecular structure.

Proof

V_{\text{available}} = V - nb

Correcting the ideal gas law's implicit assumption of point particles: molecules cannot move through space already occupied by other molecules, so the volume genuinely available for molecular motion is the container volume minus the total excluded volume, \(nb\) (Hypotheses). A

P_{\text{ideal}} = P_{\text{measured}} + a\left(\frac{n}{V}\right)^2

Correcting for intermolecular attraction: the pressure an ideal (non-attracting) gas of the same density and temperature would exert exceeds the actually measured pressure by the mean-field correction from Hypotheses; rearranged, the measured pressure is the "true," attraction-free pressure minus this correction. A

\left(P + a\frac{n^2}{V^2}\right)(V-nb) = nRT

Substituting the corrected pressure (Step 2, rearranged so the "ideal-equivalent" pressure is \(P+a n^2/V^2\)) and the corrected volume (Step 1) into the ideal gas law \(PV=nRT\) in place of the uncorrected \(P\) and \(V\) gives the van der Waals equation of state. A

Result

\left(P+a\frac{n^2}{V^2}\right)(V-nb) = nRT

Reading. Two independent physical corrections — one for finite molecular volume, one for intermolecular attraction — are folded into the ideal gas law's pressure and volume terms respectively, each contributing one gas-specific empirical constant.

Scope. Unlike \(R\), the constants \(a\) and \(b\) are not universal: they differ from gas to gas (generally larger for larger, more polarisable molecules) and are determined empirically for each substance, typically by fitting to measured pressure–volume–temperature data or to the substance's measured critical point (Corollaries).

Corollaries & converses

As \(V\to\infty\) at fixed \(n\) (the low-density, low-pressure limit), \(nb\) becomes negligible compared to \(V\) and \(a(n/V)^2\to0\), and the van der Waals equation reduces exactly to \(PV=nRT\) — confirming the ideal gas law is recovered as the correct low-density limit of the more general van der Waals equation, not an unrelated separate approximation.
Solving the van der Waals equation for the temperature, pressure, and volume at which the equation's characteristic S-shaped isotherm degenerates to a single inflection point (rather than showing the multi-valued behaviour associated with liquid–gas coexistence) gives the critical constants \(T_c=\dfrac{8a}{27Rb}\), \(P_c=\dfrac{a}{27b^2}\), \(V_c=3b\) (per mole) — a genuine theoretical prediction of each substance's critical point directly from \(a\) and \(b\).
Converse: a substance's experimentally measured critical constants can be used to back-calculate \(a\) and \(b\) for that substance (inverting the Corollaries' formulas), an alternative to fitting \(a,b\) directly from pressure–volume–temperature data.

Fails without

Drop the excluded-volume term \(b\) (keep only the attraction correction \(a\)): the equation would permit compressing a gas to exactly zero volume under sufficiently high pressure, since nothing in the remaining equation prevents \(V\to0\) — unphysical, since real molecules occupy finite space and cannot be compressed past close-packing.
Drop the attraction term \(a\) (keep only the excluded-volume correction \(b\)): the resulting equation of state no longer produces the characteristic S-shaped isotherm needed to predict a genuine liquid–gas phase transition or critical point — the \(a\) term specifically is what allows the van der Waals equation to describe condensation at all; a hard-sphere-only correction describes a denser-than-ideal gas, but never a gas that liquefies.

Common errors

Reversing which constant corrects for which effect: \(b\) is fundamentally a repulsive, hard-core (excluded-volume) effect, while \(a\) is the attractive correction — easy to mix up since both appear as positive constants in the equation, but they enter with opposite physical meaning and opposite sign effect on pressure.
Assuming \(a\) and \(b\) are universal constants like \(R\); every gas has its own distinct, empirically fitted \((a,b)\) pair, and using one gas's constants for a different gas gives meaningless results.
Using inconsistent units for \(a\), \(b\), \(R\), \(P\), and \(V\) — tabulated van der Waals constants are commonly given in \(\text{L}^2\cdot\text{atm/mol}^2\) (for \(a\)) and \(\text{L/mol}\) (for \(b\)), which must be paired with \(R\) in \(\text{L}\cdot\text{atm/(mol}\cdot\text{K)}\), not the SI value in \(\text{J/(mol}\cdot\text{K)}\), unless every quantity is first converted to a single consistent unit system.
Assuming the van der Waals correction always predicts a pressure lower (or always higher) than the ideal gas law at the same \(V,T,n\); which correction dominates depends on the density — the attractive term (\(a\)) typically dominates at moderate density, reducing predicted pressure below ideal, while the excluded-volume term (\(b\)) can dominate at very high density, increasing predicted pressure above ideal.

Discussion

Johannes Diderik van der Waals introduced this equation in his 1873 doctoral thesis, work for which he was awarded the 1910 Nobel Prize in Physics. Its significance goes well beyond a numerical correction to the ideal gas law: it was the first equation of state to unify the gas and liquid phases within a single mathematical expression, and to predict, from theory alone, the existence of a critical point — a specific temperature above which a substance cannot be liquefied by pressure alone, however high — a genuinely new theoretical prediction, not merely an explanation of an already-known empirical fact (the more usual pattern seen elsewhere in this network, e.g. VSEPR or molecular orbital theory, where theory explained a phenomenon already observed).

Critical points for essentially every common gas were subsequently measured directly (e.g. carbon dioxide's is at \(304.2\,\text{K}\), \(72.8\,\text{atm}\)) and found to be in reasonable, though not exact, agreement with the van der Waals equation's predictions — the theory captures the qualitative existence and approximate location of the critical point correctly, while more sophisticated modern equations of state (accounting for effects beyond the simple mean-field \(a,b\) picture) achieve substantially better quantitative accuracy, particularly very close to the critical point itself, where van der Waals-type mean-field theories are known to be systematically less accurate.

Common misconception: that the van der Waals equation is simply a more "accurate" version of the ideal gas law in the sense of correcting a numerical error, rather than a genuinely different equation of state describing qualitatively different physics (phase transitions) that the ideal gas law cannot represent at all, at any level of numerical precision, since an ideal gas has no mechanism (no attraction, no excluded volume) capable of producing a liquid phase in the first place.

Worked examples

\text{CO}_2\ (a=3.640\,\text{L}^2\!\cdot\!\text{atm/mol}^2,\ b=0.04267\,\text{L/mol}),\ n=1.00\,\text{mol},\ V=1.00\,\text{L},\ T=300\,\text{K}

A fairly dense sample (\(1\,\text{mol}\) confined to just \(1\,\text{L}\)), chosen specifically because deviations from ideal behaviour are largest at high density, exactly where the van der Waals corrections matter most. A

P_{\text{ideal}}=\frac{nRT}{V}=24.62\,\text{atm}; \qquad P_{\text{vdW}}=\frac{nRT}{V-nb}-a\left(\frac{n}{V}\right)^2\approx22.08\,\text{atm}

The van der Waals prediction is noticeably (about \(10\%\)) lower than the ideal gas law's prediction at this density; here the attractive correction (\(a\)) outweighs the excluded-volume correction (\(b\)), consistent with the Common errors note that the attractive term typically dominates at moderate density. A

P_{\text{ideal}}\approx24.62\,\text{atm}; \qquad P_{\text{vdW}}\approx22.08\,\text{atm}\ (\text{about }10\%\text{ lower})

Reading. At a density where real-gas effects are significant, the van der Waals correction produces a measurably different, and generally more accurate, pressure prediction than the ideal gas law alone.

Scope. At much lower density (larger \(V\) for the same \(n,T\)), the two predictions converge, per the Corollaries' low-density limit.

Problems

Using CO\(_2\)'s van der Waals constants (\(a=3.640\,\text{L}^2\!\cdot\!\text{atm/mol}^2\), \(b=0.04267\,\text{L/mol}\)) and \(R=0.08206\,\text{L}\cdot\text{atm/(mol}\cdot\text{K)}\), compute the predicted critical temperature \(T_c\) and critical pressure \(P_c\) using the Corollaries' formulas, and compare with the experimentally measured values (\(304.2\,\text{K}\), \(72.8\,\text{atm}\)).
Solution
\(T_c=\dfrac{8a}{27Rb}=\dfrac{8(3.640)}{27(0.08206)(0.04267)}\approx308.0\,\text{K}\) (about \(1.3\%\) above the experimental \(304.2\,\text{K}\)). \(P_c=\dfrac{a}{27b^2}=\dfrac{3.640}{27(0.04267)^2}\approx74.0\,\text{atm}\) (about \(1.7\%\) above the experimental \(72.8\,\text{atm}\)). Both predictions are close but not exact, consistent with the Discussion's note that the van der Waals equation captures critical behaviour only approximately.
Using CO\(_2\)'s value of \(b=0.04267\,\text{L/mol}\) and the Hypotheses' geometric relationship \(b=4N_A\left(\tfrac43\pi r^3\right)\), estimate the implied molecular radius \(r\) of a CO\(_2\) molecule.
Solution
Converting \(b\) to \(\text{m}^3/\text{mol}\): \(4.267\times10^{-5}\,\text{m}^3/\text{mol}\). Solving for \(r\): \(r=\left(\dfrac{b}{4N_A\cdot\frac43\pi}\right)^{1/3}\approx1.62\times10^{-10}\,\text{m}=162\,\text{pm}\) — a physically reasonable value, consistent with CO\(_2\)'s known molecular size (roughly \(150\)–\(200\,\text{pm}\) depending on the specific measure used), confirming the excluded-volume model gives a sensible molecular-scale answer.
Explain, using the Fails without discussion, what physically unrealistic prediction would result from a hypothetical equation of state that included the excluded-volume correction \(b\) but omitted the attractive correction \(a\) entirely.
Solution
Without the attractive term, the equation would reduce to \(P(V-nb)=nRT\), which describes a gas of finite-sized, purely repulsive (non-attracting) hard spheres. Such a gas becomes increasingly difficult to compress as \(V\) approaches \(nb\) (correctly reflecting excluded volume), but nothing in the equation ever produces the S-shaped isotherm or multi-valued pressure behaviour needed for a liquid–gas phase transition; a purely repulsive hard-sphere gas, in this simplified picture, would never condense into a liquid at any temperature or pressure, contradicting the observed behaviour of every real gas when sufficiently cooled and compressed.
Two different real gases, A and B, have the same molar mass but gas A has a substantially larger van der Waals constant \(a\) than gas B. Predict, without further calculation, which gas is likely to have the higher critical temperature, and explain your reasoning using the Corollaries' formula for \(T_c\).
Solution
Since \(T_c=\dfrac{8a}{27Rb}\) increases directly with \(a\) (holding \(b\) roughly comparable between the two gases), gas A (larger \(a\), meaning stronger intermolecular attraction) is predicted to have the higher critical temperature. This is physically sensible: stronger intermolecular attraction makes a substance easier to condense into a liquid, so a higher temperature is required before thermal motion overcomes that attraction enough to prevent liquefaction at any pressure — consistent with the general chemical trend that more polarisable or more strongly interacting molecules have higher critical (and boiling) temperatures.