The first correction to the gas law

Pressure is a rate of arrival, and if molecules never met, the counting would be exact and $pV = nRT$ would be a law rather than a limit. Molecules do meet, and the first correction to the counting is computable from what happens when two of them do — nothing else, and no fitted constants.

The correction is the second virial coefficient, and it is one integral:

$$B_2(T) = -2\pi\int_0^\infty \left(e^{-u(r)/kT} - 1\right) r^2\,\mathrm{d}r.$$

Everything about a gas’s first departure from ideality is in that expression, and its structure explains the departure’s most surprising feature — that the sign changes.

Two effects with opposite signs #

The integrand is the Mayer function, and reading its two regions is the whole of the physics.

At short separation the potential rises steeply, so the exponential is essentially zero and the function is minus one. That region contributes positively to the coefficient, by an amount equal to the excluded volume: two molecules cannot be in the same place, so the volume available to each is less than the container’s, and the pressure is higher than ideal. That is the effect everyone reaches for first.

At larger separation the potential is a shallow attractive well, so the exponential is slightly above one and the function is positive. That region contributes negatively: molecules spend a little more time near one another than chance would give, which reduces the rate at which they arrive at the walls, and the pressure is lower than ideal.

The two compete, and which wins depends on the temperature — because the well’s contribution has the temperature in the exponent and the core’s does not. At low temperature the attraction dominates, because $kT$ is small compared with the well depth and the exponential is large. At high temperature the well is irrelevant and only the core is left.

Where the integral comes from #

The formula is not a guess dressed as a definition, and the route to it is short enough to give.

The pressure of a gas follows from its partition function, and the only hard part of the partition function is the configuration integral — the integral of the Boltzmann factor of the total potential energy over every position of every molecule. For an ideal gas that energy is zero and the integral is the volume raised to the number of molecules, which returns the ideal law immediately.

For a real gas the total energy is a sum over pairs, so its Boltzmann factor is a product over pairs. The trick, due to Mayer, is to write each factor as one plus the Mayer function and multiply out. The leading term is the ideal gas. The terms with one Mayer function are the pairs, and there are as many of them as there are pairs; each contributes the same integral, which is the one above. The terms with two Mayer functions involve triples and give the third coefficient, and so on down.

That is what makes the expansion an expansion in the density rather than in the strength of the forces. Nothing has been assumed small about the potential — the Mayer function is exact, and it is bounded even where the potential is infinite, which is precisely why the expansion works for a hard core that no expansion in $u/kT$ could touch. What is assumed small is the chance of finding three molecules close together at once, which is a statement about how much gas is in the box.

The structure also explains why the coefficients get hard fast. The third involves an integral over the shape of a triangle rather than over one distance, and only for the simplest potentials has anyone written it down in closed form.

The temperature where they cancel #

At one temperature the two contributions cancel exactly and the coefficient vanishes. There, a real gas obeys $pV = nRT$ to first order in its density — while having molecules of definite size that attract each other strongly.

That is worth dwelling on because it inverts the usual reading of the ideal gas law. Agreement with $pV = nRT$ is normally taken as evidence that a gas is behaving ideally, meaning that its molecules are neither taking up room nor attracting. At the Boyle temperature the agreement is evidence of nothing of the kind: it is evidence that two large effects have cancelled, and the second-order term — which does not vanish there — is the same size it was.

The number is 3.42 times the well depth for a Lennard-Jones gas, which puts it at about 325 kelvin for nitrogen and about 660 for carbon dioxide. Nitrogen at room temperature is therefore very nearly at its Boyle temperature and behaves almost ideally; carbon dioxide is far below its and does not. The familiar ranking of “nearly ideal” gases is a ranking by how far each is from its own Boyle point.

Helium’s is 23 kelvin, which is why helium is nearly ideal at every temperature anybody works at and why it is the working fluid of a gas thermometer.

What departing looks like #

Plotting $pV/nRT$ against density is the standard way of showing how far a gas is from ideal, and the virial expansion says what the plot means. The intercept is one always. The initial slope is the second coefficient. The curvature is the third, and so on.

That makes the second coefficient measurable rather than merely computable: measure the compressibility at several low densities, take the slope as the density goes to zero, and the coefficient follows. The measurement is delicate because it is an extrapolation to a limit where the effect vanishes, and it has been done carefully for a few dozen substances over wide temperature ranges.

The value of those measurements is what they say about the potential. The coefficient at one temperature is one number from an integral over a whole function, so it does not determine the potential. The coefficient over a range of temperatures samples the integral with different weightings and constrains the potential’s shape substantially — and for the simple gases, the potentials in use were largely determined this way before any of them could be computed.

The two limits are worth having separately because each is a familiar approximation in disguise.

At high temperature the coefficient approaches four times the volume of a molecule — the excluded volume per pair, halved because each pair is counted once. That is the hard-sphere result, and it is what van der Waals’ constant $b$ is meant to be. A gas hot enough is a gas of billiard balls.

At low temperature the coefficient is large and negative and dominated by the well, and its magnitude grows roughly exponentially as the temperature falls. That is the regime in which a gas is close to condensing, and the coefficient diverging is the expansion warning that a phase transition is approaching — a warning it gives without being able to describe what happens.

Between them the coefficient rises through zero. A quantity that changes sign has told more than a quantity that is merely small, because a sign is a statement about which of two mechanisms is winning and a magnitude is not.

Where van der Waals sits #

Van der Waals’ equation is the famous first correction and it is worth saying exactly how it relates to this one.

Expanding van der Waals’ equation for small density gives a second virial coefficient of $b - a/RT$: a constant from the excluded volume and a term falling as one over the temperature from the attraction. That has the right structure — a positive piece and a negative piece competing, with the negative one dominating at low temperature — and it produces a Boyle temperature at $a/Rb$.

What it gets wrong is the shape. The true coefficient rises, reaches a broad maximum, and falls slowly at very high temperature, because the repulsive core is soft rather than hard and its effective size shrinks as molecules are thrown together harder. Van der Waals’ $b$ is a constant, so his coefficient rises to $b$ and stays there.

The comparison is instructive about what kind of theory each is. Van der Waals’ equation is a two-parameter fit with the right qualitative behaviour everywhere, including a critical point and a liquid phase; the virial expansion is exact order by order and has nothing to say about a liquid at all, because the expansion diverges before it gets there. Neither replaces the other, and the honest summary is that the virial coefficient is what a potential predicts, and van der Waals’ constants are what a fit reports.

Where the expansion stops #

The virial series is an expansion in the density and it does not converge everywhere. Each coefficient involves interactions among one more molecule than the last — the third involves triples, the fourth quadruples — and they become rapidly harder to compute and are known for only a few potentials.

More importantly, the series has a finite radius of convergence, and it fails before the liquid. A gas near its condensation point is not described by any number of virial terms, and the phase transition is invisible in the expansion: no finite truncation of an analytic series can produce the sharp corner a first-order transition has. That is a general fact about perturbation series and phase transitions, and it is the reason a theory of liquids is a separate subject rather than a continuation of this one.

What the expansion does cover is genuinely useful. At a density where the second term is a per cent correction and the third is a hundredth of that, the gas is describable exactly, and that regime includes almost all of atmospheric science, most vacuum technology and every gas thermometer.

The gas thermometer’s difficulty, and its answer #

There is a practical problem the second virial coefficient was invented to solve, and it is worth ending on because it shows what an exact first correction is worth.

A gas thermometer defines temperature by $pV = nRT$, so it needs a gas that obeys the law — and no gas does. The classical answer was to extrapolate: measure the pressure of a fixed amount of gas at several densities and take the limit as the density goes to zero, where every gas is ideal.

That works and it is slow, since it needs a whole series of measurements for every temperature. The virial answer is better: measure at one convenient density and correct using the known second coefficient, which is tabulated for helium to a precision far beyond what the thermometer needs. The correction is a fraction of a per cent and the corrected reading is as good as the extrapolation.

Helium is chosen for two reasons the essay has already given. Its Boyle temperature is low, so its coefficient is small and positive over almost the whole useful range; and it is spherical, light and simple enough that its coefficient can be computed from first principles to better than experiment. The modern realisation of the kelvin uses computed helium properties rather than measured ones, which makes the first correction to the gas law a part of the definition of temperature.

The sign a valve knows about #

The same competition shows up in a place with no obvious connection to a gas law, and the connection is worth drawing because it is the same integral doing the work.

A gas forced through a porous plug at constant enthalpy either cools or warms, and which it does has a sign that changes with temperature. Cooling is what every refrigerator and every air liquefier is built on, and warming is what happens if the gas is started too hot — hydrogen at room temperature warms on expansion, which is why liquefying it required cooling it first and why several early attempts failed with the apparatus working perfectly.

The temperature at which the effect changes sign is the inversion temperature, and to first order in the density it is where the coefficient’s temperature derivative satisfies $T\,\mathrm{d}B_2/\mathrm{d}T = B_2$ rather than where the coefficient vanishes. Both are readings of the same curve: one asks where it crosses zero, the other where its tangent passes through the origin. For a Lennard-Jones gas the second sits at about 6.4 times the well depth, roughly twice the Boyle temperature, which is the ratio the real gases show.

So the same integral that decides whether a gas is more or less compressible than an ideal one also decides whether it cools when throttled, and it decides both by way of the same balance between a core and a well. That is the general reason a coefficient with a sign is worth more than a coefficient with a size: once the balance is located, everything that depends on it inherits the crossing, at its own temperature and for its own reason.

Where the model stops #

Only pairs are counted. The second coefficient is exactly the two-body contribution, so it is exact as the density goes to zero and says nothing about a gas dense enough for three molecules to be close at once. Where three-body forces matter — and they do, at a per cent level, even in the noble gases — they enter at the third coefficient and not before.

The potential is assumed spherical. That is fair for the noble gases and poor for anything with a shape: nitrogen, carbon dioxide and water all have orientation-dependent forces, and the integral has to be taken over orientations as well as separations. The structure of the argument survives and the arithmetic becomes much heavier.

Everything here is classical. At low enough temperature the molecules’ own wavelength becomes comparable with the range of the potential and the integral has to be replaced by a quantum calculation — which matters for helium below about 50 kelvin and for hydrogen below about 100, and which is why those two gases have virial coefficients that no classical potential reproduces.

And a Lennard-Jones potential is a model. Its twelfth power is chosen for convenience rather than derived, and real repulsions are closer to exponential. The Boyle temperature in units of the well depth depends on the shape, so the number 3.42 belongs to the model rather than to nature; what belongs to nature is that there is such a temperature and that it is a few times the well depth.

What the pictures cannot show #

The Mayer function is drawn against separation and the integral is over volume, so the visual impression is misleading in a specific way: the region beyond the well contributes with a weight of $r^2$, so the shallow attractive tail at large separation counts for far more than its depth suggests. That weighting is why an attraction of a few hundredths of an electronvolt can outweigh a repulsion of several.

The compressibility figure draws only the first correction, so its curves are straight lines with the second coefficient as the slope. A real gas at any visible density has higher terms and its curves bend — and the bending is exactly what the figure has thrown away in order to make the point about the slope.

Where the ladder goes next #

The kinetic-theory ladder began with the speeds in a still room, went through pressure as a rate of arrival, why the air thins with height, how far a molecule gets, the viscosity that does not care how much gas there is and the gas that leaves is not the gas inside. This rung asks what the first departure from ideality is and where it comes from. The rungs after it: the third coefficient and the three-body forces it contains; the relation between the virial coefficients and the critical point, which the series can locate but not describe; and the transport coefficients computed from the same potential, where the same integral appears with a different weighting.

The habit worth carrying away is that a small correction can have a sign worth knowing. Two effects competing with opposite signs produce a temperature at which they cancel, and agreement with the uncorrected law at that point is the least informative agreement there is — it says that two things balanced, not that either was absent.