Ideal and real gas laws: van der Waals equation and compressibility factor

Intuition Beginner

The ideal gas law $PV=nRT$ works well at low pressure and high temperature. Under those conditions, gas molecules are far apart and their individual volumes are negligible compared to the container. Molecules also rarely interact with each other — they fly freely between wall collisions.

Real gases deviate from this picture. Two physical effects break the ideal-gas assumptions. First, molecules occupy real space. At high pressure, the container volume is not fully available because the molecules themselves take up room. Second, molecules attract each other through intermolecular forces. At low temperature, molecules move slowly enough that these attractions pull them together, reducing the pressure they exert on the walls.

The van der Waals equation corrects for both effects with two constants, $a$ and $b$, that differ for every gas. The constant $b$ represents the excluded volume per mole — the space unavailable because molecules have finite size. The constant $a$ represents the strength of intermolecular attraction.

The compressibility factor $Z=PV/(nRT)$ tells you at a glance whether a gas behaves ideally. When $Z=1$, the gas follows the ideal gas law exactly. When $Z<1$, attractive forces dominate and the pressure is lower than predicted. When $Z>1$, repulsive forces dominate and the pressure is higher than predicted.

Visual Beginner

Imagine a container full of gas molecules. In the ideal-gas picture, the molecules are dimensionless points bouncing off the walls. The entire container volume is free space. In the real-gas picture, each molecule occupies a small but nonzero volume, and nearby molecules tug on each other with weak attractive forces.

The diagram shows that every gas follows the ideal gas law at sufficiently low pressure, but the onset and direction of deviation depend on the specific gas and the temperature.

Worked example Beginner

Calculate the pressure of 1.00 mol of ${\text{CO}}_2$ in a 0.250 L container at 350 K using both the ideal gas law and the van der Waals equation. The van der Waals constants for ${\text{CO}}_2$ are $a=0.3658$ Pa m${}^6$/mol${}^2$ and $b=4.286\times 10^{-5}$ m${}^3$/mol.

Ideal gas law:

$$P=\frac{nRT}{V}=\frac{(1.00)(8.314)(350)}{0.000250}=11639600\text{Pa}=11.6\text{MPa}$$

Van der Waals:

$$P=\frac{nRT}{V-nb}-\frac{a{n}^2}{V^2}$$ $$=\frac{(1.00)(8.314)(350)}{0.000250-0.00004286}-\frac{0.3658\times (1.00{)}^2}{(0.000250{)}^2}$$ $$=\frac{2910}{0.000207}-\frac{0.3658}{6.25\times 10^{-8}}=14057971-5852800=8205171\text{Pa}=8.21\text{MPa}$$

The van der Waals pressure (8.21 MPa) is 29% lower than the ideal-gas prediction (11.6 MPa). At this high density, intermolecular attractions significantly reduce the pressure. The compressibility factor is $Z=PV/(nRT)=8.21/11.6=0.71$, confirming that ${\text{CO}}_2$ at these conditions is far from ideal.

Check your understanding Beginner

Formal definition Intermediate+

The ideal gas law relates pressure $P$, volume $V$, amount $n$, and temperature $T$:

$$PV=nRT,R=8.314462618\text{J/(mol K)}.$$

It subsumes three empirical laws. Boyle's law: $P$ is inversely proportional to $V$ at constant $n$ and $T$. Charles's law: $V$ is directly proportional to $T$ at constant $n$ and $P$. Gay-Lussac's law: $P$ is directly proportional to $T$ at constant $n$ and $V$.

Dalton's law of partial pressures. For a mixture of ideal gases, each component exerts a partial pressure $P_i$ proportional to its mole fraction $x_i=n_i/n_{\text{total}}$:

$$P_{\text{total}}=\sum _i{P}_i,P_i=x_i{P}_{\text{total}}.$$

Kinetic molecular theory provides the microscopic foundation for the ideal gas law through five assumptions: (1) gas molecules are point particles with negligible volume; (2) intermolecular forces are absent except during collisions; (3) collisions with walls and other molecules are perfectly elastic; (4) molecular motion is random and continuous; (5) the average kinetic energy is proportional to absolute temperature. Under these assumptions, kinetic theory recovers $PV=nRT$ exactly.

The van der Waals equation relaxes assumptions (1) and (2). For $n$ moles of gas:

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

where $a$ (units Pa m${}^6$/mol${}^2$) quantifies intermolecular attraction and $b$ (units m${}^3$/mol) quantifies excluded volume. The pressure correction $a{n}^2/V^2$ adds to the measured pressure because attractions between molecules reduce the momentum transferred to the walls. The volume correction $V-nb$ subtracts the space occupied by the molecules themselves.

The compressibility factor measures deviation from ideal behaviour:

$$Z=\frac{P{V}_m}{RT},V_m=\frac{V}{n}.$$

For an ideal gas, $Z=1$ at all conditions. Real gases have $Z<1$ when attractive forces dominate (typically at moderate pressures) and $Z>1$ when repulsive forces dominate (typically at very high pressures). The crossover point where $Z=1$ at nonzero pressure defines the Boyle temperature $T_B$.

Counterexamples to common slips

• "The ideal gas law fails because molecules have mass." Molecular mass does not violate any ideal-gas assumption. The ideal gas law fails when molecular volume is non-negligible or intermolecular forces are non-negligible — both consequences of finite molecular size and electronic structure, not mass.

• "Boyle's law and Charles's law are independent of the ideal gas law." They are special cases. Boyle's law is $PV=nRT$ at fixed $n$ and $T$; Charles's law is the same equation at fixed $n$ and $P$. They are not independent empirical facts but consequences of one equation.

• "The van der Waals equation is always more accurate than the ideal gas law." At very high density and near the critical point, the van der Waals equation can give worse predictions than more modern equations of state (Redlich-Kwong, Peng-Robinson). It is qualitatively correct but quantitatively limited.

• "A gas with $Z<1$ is always more compressible than an ideal gas." $Z<1$ means the pressure is lower than the ideal-gas prediction at the same $V_m$ and $T$ — equivalently, the molar volume is lower than the ideal-gas prediction at the same $P$ and $T$. The gas is denser, not necessarily "more compressible" in the sense of being easier to compress further.

Key theorem with proof Intermediate+

Theorem (Critical point from the van der Waals equation). The van der Waals equation for one mole of gas,

$$P=\frac{RT}{V_m-b}-\frac{a}{V_m^2},$$

predicts a unique critical point $(T_c,P_c,V_c)$ at which the gas and liquid phases become indistinguishable. The critical constants are

$$V_c=3b,T_c=\frac{8a}{27Rb},P_c=\frac{a}{27b^2}.$$

Proof. At the critical point, the $P$-versus-$V_m$ isotherm has a horizontal inflection point. This requires two conditions simultaneously:

$$\frac{\partial P}{\partial V_m}=0\text{and}\frac{{\partial }^{2}P}{\partial V_m^2}=0.$$

Computing the first derivative from $P=RT/(V_m-b)-a/V_m^2$:

$$\frac{\partial P}{\partial V_m}=-\frac{RT}{(V_m-b{)}^2}+\frac{2a}{V_m^3}=0\Rightarrow \frac{RT}{(V_m-b{)}^2}=\frac{2a}{V_m^3}.$$

Computing the second derivative:

$$\frac{{\partial }^{2}P}{\partial V_m^2}=\frac{2RT}{(V_m-b{)}^3}-\frac{6a}{V_m^4}=0\Rightarrow \frac{2RT}{(V_m-b{)}^3}=\frac{6a}{V_m^4}.$$

Dividing the second equation by the first:

$$\frac{2RT}{(V_m-b{)}^3}\cdot \frac{(V_m-b{)}^2}{RT}=\frac{6a}{V_m^4}\cdot \frac{V_m^3}{2a}\Rightarrow \frac{2}{V_m-b}=\frac{3}{V_m}.$$

Cross-multiplying: $2V_m=3(V_m-b)$, giving $V_c=3b$.

Substituting $V_c=3b$ into the first-derivative equation:

$$\frac{R{T}_c}{(3b-b{)}^2}=\frac{2a}{(3b{)}^3}\Rightarrow \frac{R{T}_c}{4b^2}=\frac{2a}{27b^3}.$$

Solving: $T_c=8a/(27Rb)$.

Substituting $V_c$ and $T_c$ into the van der Waals equation:

$$P_c=\frac{R\cdot 8a/(27Rb)}{3b-b}-\frac{a}{(3b{)}^2}=\frac{8a/(27b)}{2b}-\frac{a}{9b^2}=\frac{4a}{27b^2}-\frac{a}{9b^2}=\frac{4a-3a}{27b^2}=\frac{a}{27b^2}.\square$$

Corollary. The compressibility factor at the critical point is a universal constant for all van der Waals gases:

$$Z_c=\frac{P_c{V}_c}{R{T}_c}=\frac{(a/27b^2)(3b)}{R\cdot 8a/(27Rb)}=\frac{3a/27b}{8a/(27b)}=\frac{3}{8}=\mathrm{0.375.}$$

Real gases have $Z_c$ between 0.23 and 0.30, indicating that the van der Waals model overestimates $Z_c$ but captures the qualitative universality.

Worked example at intermediate level

The van der Waals constants for nitrogen are $a=0.1370$ Pa m${}^6$/mol${}^2$ and $b=3.870\times 10^{-5}$ m${}^3$/mol. Compute the predicted critical temperature, critical pressure, and critical molar volume. Compare with the experimental values $T_c=126.2$ K, $P_c=3.39$ MPa, $V_c=0.0895$ L/mol.

Predicted: $V_c=3b=3(3.870\times 10^{-5})=1.161\times 10^{-4}$ m${}^3$/mol $=0.1161$ L/mol.

$T_c=8a/(27Rb)=8(0.1370)/(27\times 8.314\times 3.870\times 10^{-5})=1.0960/0.008688=126.2$ K.

$P_c=a/(27b^2)=0.1370/(27\times (3.870\times 10^{-5}{)}^2)=0.1370/(27\times 1.4977\times 10^{-9})=0.1370/4.044\times 10^{-8}=3.389\times 10^6$ Pa $=3.39$ MPa.

The predicted $T_c$ and $P_c$ match the experimental values well. The predicted $V_c$ (0.116 L/mol) overestimates the experimental value (0.0895 L/mol) by 30%, a known weakness of the van der Waals model.

Bridge. The critical-point theorem connects directly to 14.06.01 thermodynamics, where the equation of state determines the Gibbs free energy, phase boundaries, and the Clausius-Clapeyron equation. The compressibility factor $Z$ introduced here becomes the bridge to fugacity, which replaces pressure in the thermodynamic treatment of real-gas equilibria. The van der Waals constants $a$ and $b$ will reappear in the context of the Lennard-Jones potential and intermolecular force theory in 14.09.01 solutions and phase equilibria.

Exercises Intermediate+

The virial expansion Master

The virial equation of state ^{[Kamerlingh Onnes 1901]} provides a systematic framework for real-gas corrections. The compressibility factor is expanded as a power series in $1/V_m$:

$$Z=1+\frac{B(T)}{V_m}+\frac{C(T)}{V_m^2}+\frac{D(T)}{V_m^3}+\cdots$$

The coefficients $B(T)$ (second virial coefficient), $C(T)$ (third), $D(T)$ (fourth), etc., are temperature-dependent functions determined experimentally from PVT measurements. The virial expansion converges at sufficiently low density; it diverges near the critical point where the compressibility diverges.

The physical interpretation is hierarchical. $B(T)$ accounts for pair interactions: it is negative when attractions dominate (low $T$), passes through zero at the Boyle temperature, and becomes positive when repulsions dominate (high $T$). $C(T)$ accounts for three-body interactions. Higher virial coefficients describe successively more complex clustering of molecules.

The van der Waals equation is equivalent to a truncated virial expansion:

$$Z=1+\frac{b-a/(RT)}{V_m}+\frac{b^2}{V_m^2}+\cdots$$

Reading off the coefficients: $B(T)=b-a/(RT)$ and $C(T)=b^2$. The van der Waals model predicts that the third virial coefficient is temperature-independent — a prediction that is incorrect for most real gases. This illustrates a general principle: each successive equation of state improves the representation of higher virial coefficients.

An alternative form of the virial expansion expresses $Z$ as a power series in pressure rather than inverse volume:

$$Z=1+B^{\prime }(T)P+C^{\prime }(T)P^2+\cdots$$

The pressure-explicit coefficients are related to the volume-explicit ones by $B^{\prime }=B/(RT)$, $C^{\prime }=(C-B^2)/(RT{)}^2$, and so on. The pressure form is convenient for engineering calculations because $P$ and $T$ are the controlled variables in most experiments.

Proposition (statistical-mechanical expression for $B(T)$). For a gas of spherical molecules interacting via a pair potential $u(r)$, the second virial coefficient is

$$B(T)=-2\pi N_A{\int }_0^{\infty }\left[e^{-u(r)/(k_{B}T)}-1\right]r^{2}dr.$$

Proof sketch. The configurational partition function for $N$ particles is $Z_N=\int \exp (-\beta U)d{\mathbf{r}}^N$ with $U={\sum }_{i<j}u(r_{ij})$. A cluster expansion reorganises $\ln Z_N$ as a sum over $n$-body clusters. The two-body (Mayer) cluster integral gives the first correction to the ideal-gas pressure. Defining the Mayer $f$-function $f(r)=e^{-\beta u(r)}-1$, the two-body cluster integral is $b_2=(1/2)\int f(r)d\mathbf{r}=2\pi {\int }_0^{\infty }f(r)r^{2}dr$. The virial coefficient is $B=-N_A{b}_2$, yielding the stated expression. $\square$

This result connects the macroscopic equation of state directly to the microscopic intermolecular potential. For the hard-sphere potential ($u=\infty$ for $r<\sigma$, $u=0$ for $r>\sigma$): $B(T)=(2/3)\pi N_A{\sigma }^3=b_{\text{hs}}$, independent of temperature. For a potential with both repulsive and attractive parts (such as the Lennard-Jones potential $u(r)=4ϵ[(\sigma /r{)}^{12}-(\sigma /r{)}^6]$), $B(T)$ interpolates between large positive values at high $T$ (repulsion-dominated) and negative values at low $T$ (attraction-dominated), crossing zero at the Boyle temperature $T_B\approx 3.42ϵ/k_B$.

The law of corresponding states Master

Theorem (Principle of corresponding states). All gases at the same reduced pressure $P_r=P/P_c$ and reduced temperature $T_r=T/T_c$ have approximately the same compressibility factor $Z$. The principle holds to within a few percent for spherical, nonpolar molecules.

The van der Waals equation provides an exact realisation. Substituting $P=P_c{P}_r$, $V_m=V_c{V}_r$, $T=T_c{T}_r$ with the critical constants $V_c=3b$, $T_c=8a/(27Rb)$, $P_c=a/(27b^2)$ yields the reduced van der Waals equation:

$$\left(P_r+\frac{3}{V_r^2}\right)(3V_r-1)=8T_r.$$

This equation contains no substance-specific parameters: $a$ and $b$ have cancelled completely. Every van der Waals gas is described by the same equation in reduced coordinates. The corresponding-states principle is therefore built into the van der Waals model at its foundations.

Real gases do not obey the van der Waals equation exactly, but the corresponding-states principle still holds approximately because the intermolecular potential has a universal shape when scaled by the well depth $ϵ$ and molecular diameter $\sigma$. The generalised compressibility chart plots $Z$ versus $P_r$ at constant $T_r$, and data for many gases collapse onto a single family of curves.

The accuracy of the corresponding-states prediction is improved by the Pitzer acentric factor $\omega =-{\log }_{10}(P_{\text{vp}}/P_c){|}_{T_r=0.7}-1$, which measures the non-sphericity of the intermolecular potential. The Pitzer correlation $Z=Z^{(0)}(T_r,P_r)+\omega Z^{(1)}(T_r,P_r)$ reproduces $Z$ to within 2--3% for most gases. The acentric factor is zero for spherical molecules (argon, krypton, xenon) and increases for elongated or polar molecules (water $\omega =0.344$, n-octane $\omega =0.399$).

Cubic equations of state Master

The van der Waals equation is the simplest member of the family of cubic equations of state — so called because they are cubic in $V_m$ when solved for $P$. Several extensions improve its quantitative accuracy:

Redlich-Kwong equation (1949):

$$P=\frac{RT}{V_m-b}-\frac{a}{\sqrt{T}{V}_m(V_m+b)}.$$

The modified temperature dependence $1/\sqrt{T}$ in the attractive term improves vapour-pressure predictions.

Soave-Redlich-Kwong (SRK) equation (1972):

$$P=\frac{RT}{V_m-b}-\frac{a(T)}{V_m(V_m+b)},$$

where $a(T)=a_c[1+\kappa (1-\sqrt{T_r}){]}^2$ and $\kappa$ is a function of the acentric factor. The SRK equation accurately predicts vapour-liquid equilibria for hydrocarbon mixtures.

Peng-Robinson equation (1976):

$$P=\frac{RT}{V_m-b}-\frac{a(T)}{V_m(V_m+b)+b(V_m-b)}.$$

The Peng-Robinson equation is the standard for hydrocarbon processing because it predicts liquid densities and vapour pressures with typical errors of 1--3%. It improves upon the van der Waals and SRK equations near the critical point and for liquids.

All cubic equations share the critical-point structure of the van der Waals equation: they predict a critical point, a universal $Z_c$, and a corresponding-states form. Their accuracy varies because the functional forms of the repulsive and attractive terms differ. Modern process simulation software uses the Peng-Robinson equation as the default equation of state for nonpolar and weakly polar mixtures.

The virial coefficients of any cubic equation of state can be extracted by expanding in powers of $1/V_m$. For the Peng-Robinson equation: $B(T)=b-a(T)/(RT)$ (same functional form as van der Waals, but with a temperature-dependent $a$), while $C(T)$ differs from the van der Waals prediction $b^2$ and provides a better fit to experimental data.

Connections Master

• Stoichiometry and the ideal gas law 14.03.01 introduced $PV=nRT$, Dalton's law, and the limiting-reagent framework. This unit deepens the gas-law treatment by examining when and why the ideal gas law fails, and how real-gas equations of state correct for intermolecular forces and finite molecular volume. The stoichiometric mole concept carries through: every equation of state in this unit relates $P$, $V$, and $T$ to $n$.

• Chemical thermodynamics 14.06.01 uses the equation of state to compute PV work ($w=-\int PdV$) and defines thermodynamic potentials ($H=U+PV$, $G=H-TS$) that depend on the relationship between $P$, $V$, and $T$. The compressibility factor $Z$ introduced here becomes the basis for fugacity $f=\varphi P$, where $\ln \varphi ={\int }_0^P(Z-1)/PdP$, which replaces pressure in the equilibrium-constant expressions for real gases.

• Solutions and phase equilibria 14.09.01 extends the equation-of-state concept to mixtures and phase diagrams. The critical point derived from the van der Waals equation is the terminus of the liquid-vapour coexistence curve. The corresponding-states principle generalises to mixtures through mixing rules for the equation-of-state parameters.

• Statistical mechanics 14.07.01 derives the virial coefficients from first principles by computing the partition function of an interacting gas. The Mayer cluster expansion connects $B(T)$ to the two-body integral over the intermolecular potential, $C(T)$ to the three-body integral, and so on. This provides the microscopic interpretation of the macroscopic virial coefficients treated here.

• Chemical kinetics 14.08.01 uses the molar concentration $c=n/V=P/(ZRT)$ for real gases. When $Z\ne 1$, the relationship between partial pressure and concentration deviates from the ideal-gas form, affecting the rate constants and equilibrium positions of gas-phase reactions.

Historical and philosophical context Master

The empirical gas laws were discovered across two centuries. Boyle established the inverse pressure-volume relationship in 1662 using a J-tube with mercury. Charles found the linear volume-temperature dependence in 1787; Gay-Lussac published it in 1802 and independently established the pressure-temperature law. These three proportionalities, combined with Avogadro's 1811 hypothesis that equal volumes contain equal numbers of molecules, imply the ideal gas law $PV=nRT$.

The kinetic molecular theory, developed by Clausius (1857) and Maxwell (1860), provided the microscopic explanation: point particles in random motion exert pressure by elastic collisions with the walls. The theory predicted the Maxwell-Boltzmann speed distribution and established that temperature measures average kinetic energy.

Deviations from ideality were observed throughout the nineteenth century. Thomas Andrews's 1869 experiments on carbon dioxide isotherms ^{[Andrews 1869]} revealed the critical point and the continuity between gas and liquid states. His isotherms showed that above a certain temperature, no amount of pressure could liquefy ${\text{CO}}_2$ — a finding that contradicted the then-prevailing view that gases and liquids were fundamentally different states of matter.

Johannes Diderik van der Waals's 1873 doctoral dissertation ^{[van der Waals 1873]} proposed the equation of state that bears his name, providing a single continuous function describing both gas and liquid phases. Van der Waals derived his equation from physical reasoning: molecules have finite size (the $b$ parameter) and attract each other (the $a$ parameter). The equation predicted the existence of the critical point and the law of corresponding states, earning van der Waals the 1910 Nobel Prize. His work demonstrated that macroscopic thermodynamic behaviour could be derived from simple assumptions about molecular properties — a conceptual breakthrough that laid the groundwork for statistical mechanics.

Kamerlingh Onnes introduced the virial expansion in 1901 ^{[Kamerlingh Onnes 1901]} as a systematic empirical representation of PVT data. The statistical-mechanical interpretation came later, through the cluster-expansion work of Ursell and Mayer in the 1920s--1930s, which expressed the virial coefficients as integrals over intermolecular potentials. This completed the conceptual arc: empirical gas laws → phenomenological equation of state → microscopic derivation from molecular interactions.

The Redlich-Kwong equation (1949) and the Peng-Robinson equation (1976) extended the van der Waals framework to achieve the quantitative accuracy required for chemical-engineering process design. These modern equations of state are used in every refinery and petrochemical plant, processing billions of dollars of materials annually. The progression from Boyle's glass tubes to Peng-Robinson software illustrates how fundamental physical insight, refined over centuries, becomes industrial infrastructure.

Bibliography Master

Primary literature:

• Boyle, R., New Experiments Physico-Mechanicall, Touching the Spring of the Air and its Effects (1662). The inverse pressure-volume relationship.

• Charles, J. A. C., unpublished lecture (1787); published by Gay-Lussac (1802). Volume-temperature proportionality.

• Andrews, T., "On the Continuity of the Gaseous and Liquid States of Matter", Phil. Trans. R. Soc. 159 (1869), 575--590. Critical-point isotherms.

• van der Waals, J. D., Over de Continuiteit van den Gas- en Vloeistoftoestand (Leiden, 1873). The equation of state.

• Kamerlingh Onnes, H., "Expression of the Equation of State of Gases and Liquids", Commun. Phys. Lab. Leiden 71 (1901). The virial expansion.

• Redlich, O. & Kwong, J. N. S., "On the Thermodynamics of Solutions", Chem. Rev. 44 (1949), 233--244. The Redlich-Kwong equation.

• Peng, D.-Y. & Robinson, D. B., "A New Two-Constant Equation of State", Ind. Eng. Chem. Fundam. 15 (1976), 59--64. The Peng-Robinson equation.

Modern references:

• Zumdahl, S. S. & DeCoste, D. J., Chemical Principles, 8e (Cengage, 2017), Ch. 5. Introductory gas laws and real-gas behaviour.

• Atkins, P. & de Paula, J., Physical Chemistry, 12e (Oxford, 2023), Ch. 1. Equations of state and the virial expansion.

• McQuarrie, D. A. & Simon, J. D., Physical Chemistry (University Science Books, 1997), Ch. 1. The properties of gases.

• Reif, F., Fundamentals of Statistical and Thermal Physics (McGraw-Hill, 1965), Ch. 7. Statistical mechanics of ideal gases and the virial expansion.

• Hirschfelder, J. O., Curtiss, C. F. & Bird, R. B., Molecular Theory of Gases and Liquids (Wiley, 1954). The definitive reference on intermolecular forces and equations of state.