The Law of Corresponding States

Fill a tank with methane and a balloon with carbon dioxide. Then compress both toward their critical points. Despite completely different molecular sizes and intermolecular forces, something surprising happens. Their compressibility factors trace out nearly the same curve. This holds when plotted against reduced pressure and temperature. This is not a coincidence. It is one of the great unifying discoveries of nineteenth-century physical chemistry. The law of corresponding states says that gases behave identically under certain conditions. This holds when compared at the same fraction of their own critical temperature, pressure, and volume. It turns a zoo of substance-specific equations of state into one approximately universal chart. This matters for a practical reason. It lets engineers and chemists estimate the properties of gases they have never measured. They can do this using only three critical constants. This article builds reduced variables from the van der Waals equation. It derives why the compressibility factor becomes universal. It also shows how well the law actually works with real numbers. You will see the derivation. Then you will see two worked examples. Finally you will see where the law breaks down.

Key takeaways

Reduced variables rescale $P$ , $V$ , $T$ by the critical constants: $P_r = P/P_c$ , $V_r = V/V_c$ , $T_r = T/T_c$ .
Any two-parameter equation of state (like van der Waals) becomes parameter-free when written in reduced form.
The compressibility factor $Z = PV/RT$ becomes approximately a universal function of $P_r$ and $T_r$ alone.
Generalized compressibility charts, built from real gas data, let you estimate $Z$ for almost any nonpolar gas.
The law fails for polar, hydrogen-bonding, and quantum gases (like He, H₂) where a third parameter is needed.
Key idea: at the same reduced pressure and temperature, all gases have (approximately) the same reduced volume and compressibility factor.

The core insight

When pressure, volume, and temperature are measured as fractions of a gas's own critical constants, the resulting equation of state loses all trace of which specific gas it describes — so a single universal compressibility surface, built once from a handful of reference gases, can predict the behavior of virtually any nonpolar fluid near its critical region.

The physical picture

Near the critical point, molecular identity seems to stop mattering. A gas is a swarm of molecules whose average kinetic energy is set by $T$ , whose average spacing is set by $V$ , and whose interactions are dominated by two features: a size (excluded volume) and a strength (attraction well depth). The critical constants $T_c$ , $P_c$ , $V_c$ already encode both features for any given substance, since they emerge from the same intermolecular potential. The key insight, first stated clearly by Johannes Diderik van der Waals in 1873–1880, is this: if you measure temperature in units of $T_c$ , pressure in units of $P_c$ , and volume in units of $V_c$ , the substance-specific parameters $a$ and $b$ cancel out of the equation of state entirely. What remains is a single equation with no adjustable constants, valid — in principle — for every simple gas. Two gases at the same reduced temperature and reduced pressure are said to be in "corresponding states," and they should have the same reduced volume and the same compressibility factor $Z = PV/RT$ . This is a coarse-graining trick: the messy detail of molecular shape and force is compressed into three critical numbers, and once those numbers are divided out, only the shape of the intermolecular potential matters — and for many simple nonpolar molecules, that shape is similar enough that the law works to a few percent.

Figure

Reduced isotherms collapsing onto one curve

The second picture worth holding in mind is the geometric construction of the reduced variables themselves — how a raw $P$ – $V$ – $T$ surface, unique to each gas, is rescaled onto one universal grid.

Figure

Rescaling raw variables into reduced variables

The core relations

P_r = \frac{P}{P_c}, \qquad V_r = \frac{V}{V_c}, \qquad T_r = \frac{T}{T_c}

where: $P_c, V_c, T_c$ are the critical pressure, molar volume, and temperature of the substance.

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

where: this is the van der Waals equation rewritten purely in reduced variables, with no trace of $a$ , $b$ , or $R$ .

Z = \frac{PV}{RT} = f(P_r, T_r)

where: $Z$ is the compressibility factor; the law of corresponding states asserts $f$ is (approximately) the same function for all gases.

Z_c = \frac{P_c V_c}{R T_c}

where: the critical compressibility factor; van der Waals predicts the universal value $Z_c = 3/8 = 0.375$ for every gas.

The core derivation

Derivation AFrom van der Waals to the universal reduced equation

Goal. Show that substituting the critical constants into the van der Waals equation eliminates $a$ , $b$ , and $R$ , leaving one dimensionless equation valid for any gas obeying van der Waals behavior.

Step 1. Start from the van der Waals equation.

P = \frac{RT}{V - b} - \frac{a}{V^{2}}

Here $a$ measures attractive interactions and $b$ is the excluded molar volume, both specific to the gas.

Step 2. Recall the critical constants in terms of $a$ and $b$ . These follow from the conditions $(\partial P/\partial V)_T = 0$ and $(\partial^2 P/\partial V^2)_T = 0$ at the critical point (derived in the treatment of critical constants):

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

Step 3. Introduce reduced variables. Define $P = P_r P_c$ , $V = V_r V_c$ , $T = T_r T_c$ , and substitute directly into the van der Waals equation:

P_r P_c = \frac{R T_r T_c}{V_r V_c - b} - \frac{a}{(V_r V_c)^2}

Step 4. Substitute the critical-constant expressions. Replace $V_c = 3b$ , $P_c = a/27b^2$ , and $T_c = 8a/27Rb$ :

P_r \frac{a}{27 b^{2}} = \frac{R T_r \cdot \dfrac{8a}{27Rb}}{3 V_r b - b} - \frac{a}{9 V_r^{2} b^{2}}

Step 5. Simplify term by term. The middle term simplifies because $R$ cancels:

\frac{8a T_r}{27 b (3V_r - 1)}

So the whole equation reads

P_r \frac{a}{27b^2} = \frac{8aT_r}{27b(3V_r-1)} - \frac{a}{9V_r^2 b^2}

Step 6. Divide through by $a/27b^2$ . Every remaining term now carries the same prefactor $a/b^2$ , which cancels completely:

P_r = \frac{8T_r}{3V_r - 1} - \frac{3}{V_r^{2}}

Step 7. Rearrange into the compact form. Moving the $3/V_r^2$ term to the left and multiplying both sides by $(3V_r - 1)$ gives the boxed universal equation:

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

No trace of $a$ , $b$ , or even $R$ remains. Every gas that obeys the van der Waals equation, regardless of its actual $a$ and $b$ , obeys this exact same reduced equation. This is the mathematical heart of the law of corresponding states: two gases at equal $P_r$ and $T_r$ must have equal $V_r$ , because they solve the identical equation.

Worked examples

Example 1 — Foundational: checking $Z_c = 3/8$ against a real gas. The law of corresponding states, applied strictly to van der Waals, predicts every gas has the same critical compressibility factor:

Z_c = \frac{P_c V_c}{RT_c} = \frac{3}{8} = 0.375

Test this against argon, whose measured critical constants are $T_c = 150.9$ K, $P_c = 48.1$ atm, $V_c = 74.9$ cm³ mol⁻¹. Convert units so everything is consistent with $R = 0.08206$ L atm K⁻¹ mol⁻¹:

V_c = 74.9\ \text{cm}^3\text{mol}^{-1} = 0.0749\ \text{L mol}^{-1}

Now compute:

Z_c = \frac{(48.1\ \text{atm})(0.0749\ \text{L mol}^{-1})}{(0.08206\ \text{L atm K}^{-1}\text{mol}^{-1})(150.9\ \text{K})}

Z_c = \frac{3.603\ \text{L atm mol}^{-1}}{12.38\ \text{L atm mol}^{-1}} = 0.291

The measured value, $0.291$ , is noticeably below the van der Waals prediction of $0.375$ . This 22% discrepancy is well known: van der Waals systematically overestimates $Z_c$ for real gases, because the two-parameter equation is too crude near the critical point itself. Most real gases cluster near $Z_c \approx 0.27$ – $0.29$ , which is why generalized compressibility charts are built from experimental data rather than from the van der Waals equation directly. The law (universality of $f(P_r, T_r)$ ) survives better than the specific van der Waals numbers do.

Example 2 — Applied: estimating the molar volume of nitrogen using corresponding states. Estimate the molar volume of $\text{N}_2$ gas at $T = 152$ K and $P = 60$ atm, using only its critical constants: $T_c = 126.2$ K, $P_c = 33.5$ atm.

Step 1. Compute the reduced variables.

T_r = \frac{152\ \text{K}}{126.2\ \text{K}} = 1.20

P_r = \frac{60\ \text{atm}}{33.5\ \text{atm}} = 1.79

Step 2. Read $Z$ off the generalized compressibility chart. At $T_r = 1.20$ and $P_r = 1.79$ , standard generalized charts (e.g. Standing–Katz type charts, built from many real gases) give

Z \approx 0.80

This is a chart-reading value; different tabulations give $Z$ in the range 0.78–0.82 here, consistent with the ≈3% scatter typical of the corresponding-states correlation.

Step 3. Solve for molar volume.

V = \frac{ZRT}{P} = \frac{(0.80)(0.08206\ \text{L atm K}^{-1}\text{mol}^{-1})(152\ \text{K})}{60\ \text{atm}}

V = \frac{9.978\ \text{L atm mol}^{-1}}{60\ \text{atm}} = 0.166\ \text{L mol}^{-1} = 166\ \text{cm}^3\text{mol}^{-1}

Sanity check. The ideal-gas prediction would be

V_{\text{ideal}} = \frac{RT}{P} = \frac{(0.08206)(152)}{60} = 0.2078\ \text{L mol}^{-1} = 208\ \text{cm}^3\text{mol}^{-1}

The real volume, 166 cm³/mol, is smaller than ideal by the factor $Z = 0.80$ , exactly as expected: at $P_r \approx 1.8$ and $T_r$ only modestly above 1, attractive forces still pull the gas below ideal-gas volume even though $P$ is fairly high. This is physically sensible — nitrogen at 152 K and 60 atm is well within the regime where compressibility-factor corrections matter, yet not so extreme that $Z$ collapses toward very small values.

Limiting cases & common traps

Situation	What happens	Why
$P_r \to 0$ at any $T_r$	$Z \to 1$	Dilute gas limit; all equations of state reduce to ideal gas law
$T_r \gg 1$ (e.g. $T_r > 2$ )	$Z > 1$ for most $P_r$	Kinetic energy dominates; molecules behave as if repulsive, less compressible than ideal
$T_r \approx 1$ , $P_r \approx 1$	Largest deviations from ideality	Near critical point, attractive and excluded-volume effects both large
Quantum gases: He, H₂, Ne	Law fails without correction	Zero-point quantum effects distort the intermolecular potential; use modified $T_c$ , $P_c$ ("quantum corresponding states")
Polar/associating gases: H₂O, NH₃	Law fails, needs 3rd parameter	Dipole-dipole and hydrogen bonding are not captured by a two-parameter potential; acentric factor $\omega$ added (Pitzer correlation)

The chemist behind it

Johannes Diderik van der Waals introduced his equation of state in his 1873 doctoral thesis, correcting the ideal gas law for finite molecular size and intermolecular attraction. But the deeper insight — that this correction, once rescaled by the critical constants, produces a single universal equation for all substances — took him another several years to fully articulate, culminating around 1880. This was a radical claim in an era when each gas was studied as its own isolated phenomenon. Van der Waals showed that the critical point itself, not any arbitrary reference state, was the natural yardstick for comparing different substances. His corresponding-states principle directly inspired industrial correlations still used today, including generalized compressibility charts and the Pitzer acentric-factor extension for non-simple fluids.

Origins · 1 minJohannes Diderik van der Waals and how this was discovered

Johannes Diderik van der Waals

1837-1923 · Dutch physicist who unified real-gas behavior through the critical point, earning the 1910 Nobel Prize in Physics

Ask the tutor: How does the acentric factor extend the law of corresponding states to real, non-spherical molecules?

Is the law of corresponding states exact

No. It is an approximate empirical regularity, exact only for a hypothetical gas obeying a two-parameter equation of state like van der Waals. Real intermolecular potentials are not perfectly described by two parameters, so real gases deviate from perfect universality, typically by a few percent in $Z$ . The law works best for simple nonpolar molecules of similar shape (Ar, Kr, N₂, CH₄, O₂) and worst for polar or hydrogen-bonding substances such as water and ammonia, where a third parameter, the acentric factor, is usually added.

Why does van der Waals theory predict Zc = 3/8 but real gases show closer to 0.27–0.29

The value 3/8 comes purely from the algebra of the van der Waals equation, which crudely models molecules as hard spheres with a mean-field attraction. Real molecules have more complex, non-spherical potentials, and near the critical point correlations between molecules become long-ranged and strongly non-mean-field. Van der Waals theory is known to fail qualitatively very close to the critical point (it does not correctly predict critical exponents), which is exactly why $Z_c$ is systematically overestimated for essentially every real gas.

How are generalized compressibility charts related to this law

Generalized compressibility charts plot $Z$ against $P_r$ for several fixed values of $T_r$ , using data compiled from many real gases rather than from the van der Waals equation. They are the practical, empirically corrected embodiment of the law of corresponding states. Because real gases only approximately share one universal curve, these charts represent an average behavior, typically accurate to within a few percent for nonpolar and slightly polar gases — good enough for most engineering estimates.

Why is a third parameter like the acentric factor needed for some gases

The two-parameter corresponding-states law assumes all substances have geometrically similar intermolecular potentials, differing only in size and well depth. Real molecules also differ in shape, polarity, and the softness of their repulsive potential, all of which alter their vapor pressure curve. Pitzer's acentric factor $\omega$ quantifies how much a substance's vapor pressure deviates from that of simple spherical molecules like argon at $T_r = 0.7$ , adding one more parameter to restore accuracy for polar and elongated molecules.

Does the law of corresponding states apply to liquids too

Yes, in a limited sense. Because the critical point sits at the boundary between liquid and vapor, the same reduced variables can describe coexisting liquid and vapor densities, surface tension, and viscosity trends across different substances, often through separate empirical corresponding-states correlations. These are less universal than the vapor-phase compressibility factor, but they remain useful for estimating properties of liquids near their critical temperature where direct data are unavailable.