Ideal and real gases

Reading a property table is slow. You interpolate twice, you check which side of the saturation line you are on, and you carry four numbers through the rest of the problem. So there is a strong temptation to use the one-line substitute wherever it will fit, and the substitute is right often enough to make the habit stick. What separates a working engineer from a student here is not knowing the equation. It is knowing, before writing it down, roughly how wrong it is about to be.

The ideal gas equation

Any relation that ties a substance’s pressure, temperature and specific volume together is an equation of state. The simplest useful one is

and it comes in whichever dress the problem is wearing. Multiply through by mass and it is \(PV = mRT\). Work in moles instead and it is \(PV = N R_u T\), with \(R_u = 8.314\ \mathrm{kJ/(kmol\cdot K)}\) the same for every gas there is. Take a fixed mass between two states and the mass and the constant cancel, leaving the combined form that turns up in half of all first-year problems:

The gas constant of a particular gas is the universal one divided by the molar mass, \(R = R_u / M\). Air has \(M = 28.97\ \mathrm{kg/kmol}\), so \(R = 0.287\ \mathrm{kJ/(kg\cdot K)}\); water vapour has \(M = 18.015\ \mathrm{kg/kmol}\) and \(R = 0.4615\ \mathrm{kJ/(kg\cdot K)}\). Two arithmetic traps live here and both are expensive. \(T\) must be absolute — kelvin, never celsius, because the equation says the volume goes to zero when the temperature does. So must \(P\), which means adding atmospheric pressure to whatever the gauge reads, as in basic concepts and definitions.

Where the equation comes from matters more than where it is used. It is exactly what you get from the kinetic picture of a gas once two things have been thrown away: molecules are given no volume of their own, and they are given no forces between them except at the instant they collide. Everything the equation predicts is a consequence of those two deletions, and everywhere it fails, it fails because one of them was not true.

Which is the whole point of calling it an equation of state rather than a law. The first law and the second law are statements about the universe that no substance is permitted to violate. \(Pv = RT\) is a correlation that fits some substances in some regions and quietly stops fitting elsewhere. It is exact only in a limit that no real gas ever quite reaches — vanishing pressure — and everywhere else it is an approximation whose error you are responsible for estimating.

The good news is that the range is generous. Air, nitrogen, oxygen, hydrogen, helium, argon, neon and even carbon dioxide can be treated as ideal gases at ordinary temperatures and pressures with errors under one per cent, which is smaller than most of the other uncertainties in an engineering calculation. The trouble is that the list is a list of gases, not of substances, and the exception is the one you meet most often.

When the model fails

The honest way to talk about the error is to make it a number. Compare the specific volume a substance actually has with the volume the ideal equation would have predicted at the same pressure and temperature, and call the ratio the compressibility factor:

For an ideal gas \(Z = 1\) by construction, and for a real one \(Z\) measures the distance from the model directly: \(Z = 0.6\) means the substance occupies sixty per cent of the volume the equation promised, and using \(Pv = RT\) there would put you forty per cent out on a quantity that is about to be multiplied by everything else in the problem.

Two limits push \(Z\) back towards one, and they are the two deletions running in reverse. Drop the pressure and the molecules move far enough apart that neither their own volume nor the attraction between them counts for anything; every gas approaches \(Z = 1\) as \(P \to 0\), whatever its temperature. Raise the temperature well above the critical temperature and the molecules carry so much kinetic energy that the weak attraction between them is a rounding error on the way past.

What ruins it is density. Approach the saturation dome and the molecules are close enough to pull on one another seriously, so the real pressure falls below what the equation says and \(Z\) drops hard. Near the critical point it is worst of all — for most substances \(Z\) is somewhere around \(0.27\) there, which is not an error but a different answer. And inside the dome the question does not even make sense: a saturated mixture’s pressure is fixed by its temperature alone, so no equation of state in \(P\), \(v\) and \(T\) can locate the state.

Steam is the trap, and it is a trap because water is so familiar that it feels like it ought to behave. It does not. Below about \(10\ \mathrm{kPa}\) water vapour is an ideal gas to better than a tenth of a per cent, which is why the moisture in air is treated that way in every air-conditioning calculation without a second thought. In a steam plant the pressures are thousands of times higher and sit near the saturation line by design, and there the ideal equation is not slightly optimistic — it is useless. Use the tables, as in properties of pure substances. Refrigerants have the same problem for the same reason: a refrigeration cycle spends most of its time within touching distance of the dome.

Reduced coordinates

Saying “near the critical point” is only useful if you say near it compared to what. Ten megapascals is a crushing pressure for nitrogen, whose critical pressure is \(3.39\ \mathrm{MPa}\), and an unremarkable one for water, whose critical pressure is \(22.06\ \mathrm{MPa}\). So measure every pressure and temperature against the substance’s own critical values:

These are the reduced pressure and temperature, and they are pure numbers. The remarkable experimental fact — the principle of corresponding states — is that when \(Z\) is plotted against them, gases as different as methane, water and nitrogen fall very nearly on the same set of curves. The deviation from ideal behaviour is not a property of the molecule so much as a property of how close that molecule has been pushed to its own critical point.

That single family of curves is the generalised compressibility chart, and reading it is three steps. Look up \(P_{cr}\) and \(T_{cr}\) for the substance — water is \(22.06\ \mathrm{MPa}\) and \(647.1\ \mathrm{K}\), carbon dioxide \(7.39\ \mathrm{MPa}\) and \(304.2\ \mathrm{K}\). Form \(P_R\) and \(T_R\). Find where the vertical at \(P_R\) crosses the \(T_R\) curve, read \(Z\) off the vertical axis, and use \(Pv = ZRT\) instead. The chart cannot do better than a few per cent, but a few per cent on a quantity that would otherwise have been forty per cent wrong is a different kind of answer.

The chart also settles arguments about the model’s range without appeal to intuition. The whole low-pressure edge sits on \(Z = 1\). The \(T_R = 2\) curve barely leaves it. The \(T_R = 1\) curve plunges to about a quarter and comes back. Whether your state is safe is a question you can answer by looking, which is exactly what a chart is for.

One awkward case remains. Sometimes you know \(v\) and \(T\) and want \(P\), so there is no \(P_R\) to enter with, and iterating on \(Z\) is tedious. The chart carries a second family of lines for this, labelled with the pseudo-reduced specific volume

which is built from \(R T_{cr}/P_{cr}\) rather than from the measured critical volume — deliberately, because \(v_{cr}\) is one of the hardest properties to measure accurately and the substitute costs nothing. Note that \(v_R\) is not \(v/v_{cr}\), and the two are not interchangeable. Enter on \(v_R\) and \(T_R\), read \(Z\) or \(P_R\), and the iteration disappears.

Equations of state for real gases

A chart is fine for homework and useless inside a simulation. For that you want an equation, and the first honest attempt at one is still the best thing to learn from, because it corrects exactly the two assumptions that were thrown away in the first place. Van der Waals proposed it in 1873:

Read it as two edits to \(Pv = RT\). The \(b\) is the volume the molecules themselves occupy, so the space they are free to move in is \(v - b\) rather than \(v\); squeeze hard enough and \(v\) cannot go below \(b\) however much pressure you apply, which is a crude but real account of why liquids resist compression. The \(a/v^2\) is the attraction. A molecule about to strike the wall is pulled back by the ones behind it, so it arrives with less momentum and the measured pressure is lower than it would otherwise be; the term goes as \(1/v^{2}\) because the effect depends on density twice over, once for the molecule leaving and once for those doing the pulling.

The two constants are not fitted to a table of data. They come from the one place a real substance advertises its own molecular behaviour — the critical point, where the critical isotherm has a horizontal inflection. Imposing \((\partial P/\partial v)_T = (\partial^{2} P/\partial v^{2})_T = 0\) there gives

so two numbers from a table of critical properties fix the whole equation for that substance. The result is qualitatively right in a way the ideal equation never is: below the critical temperature the isotherms develop a maximum and a minimum, and the loop between them is the equation’s attempt at the flat saturation line it cannot actually draw. That it tries at all is remarkable for two constants.

Quantitatively it is not good enough for design work, and it is worst exactly where you most want help. Put the critical values back into the equation and it predicts \(Z_{cr} = 3/8 = 0.375\) for every substance, whereas real ones sit between about \(0.23\) and \(0.33\). Near the critical point van der Waals is out by a third.

So the field did the obvious thing and added constants. The Beattie-Bridgeman equation of 1928 uses five and is reasonably accurate up to about eighty per cent of the critical density. Benedict-Webb-Rubin of 1940 extends it to eight and holds to something like two and a half times the critical density. Later correlations run to dozens of coefficients and fit better still. It is worth being clear about what that buys: every one of them is a curve fit, tuned to measured data for one substance over one range, and none of them contains an idea that van der Waals did not already have. More constants mean more accuracy and not one bit more insight, which is why the two-constant equation is the one still taught.

Specific heats of an ideal gas

The equation of state locates a state, but energy balances need internal energy and enthalpy, and for those you need to know how much energy it takes to warm the substance up. That quantity depends on how you warm it, which is why there are two specific heats:

Both are properties in their own right, defined by those derivatives and not by any particular process. Calling \(c_v\) “the specific heat at constant volume” is a description of the easiest experiment for measuring it, not a restriction on where it may be used.

For an ideal gas something unusually convenient happens. Joule found, and the equation of state confirms, that the internal energy of an ideal gas depends on temperature alone. It makes sense from the model: with no forces between molecules there is no potential energy to store in their separation, so pulling them further apart at fixed temperature changes nothing. Enthalpy inherits the property immediately, since \(h = u + Pv = u(T) + RT\) is a function of \(T\) and nothing else. Two states at the same temperature but wildly different pressures have identical \(u\) and identical \(h\).

The partial derivatives therefore become ordinary ones, and the differentials lose their subscripts:

These hold for any process an ideal gas undergoes, not only for constant-volume and constant-pressure ones. Using \(\Delta u = c_v \Delta T\) across an expansion in which the volume changes is not an approximation and not a trick; it is the direct consequence of \(u\) depending on \(T\) alone. Students routinely refuse to believe this and hunt for a different formula whenever the volume moves.

The two specific heats are also tied together. Differentiate \(h = u + RT\) with respect to temperature and the relation falls out in one line:

on a mass basis, or \(\bar{c}_p = \bar{c}_v + R_u\) per mole. The physical reading is worth keeping: heating a gas at constant pressure costs more than heating it at constant volume, because it also has to push the boundary outwards, and that boundary work comes to exactly \(P\,\Delta v = R\,\Delta T\) — the difference between the two specific heats, and nothing else. For air at room temperature \(c_p = 1.005\) and \(c_v = 0.718\ \mathrm{kJ/(kg\cdot K)}\), and the difference is \(0.287\), which is \(R\) for air to three figures. The ratio

is the other combination worth memorising, because it governs isentropic processes and the speed of sound. It is \(1.4\) for air at room temperature and \(1.667\) for the monatomic gases, for which it barely varies with temperature at all.

What does vary is the specific heats themselves. They rise with temperature, because hotter molecules store energy in vibration as well as in translation and rotation, and for diatomic gases the rise over a few hundred kelvin is not negligible. Treating them as constants is legitimate when the temperature change is modest, and the standard dodge is to evaluate both at the average of the two end temperatures rather than at either end, which halves the error for no extra work. Over a swing of several hundred kelvin — a combustion chamber, a gas turbine — that will not do, and you go back to tabulated \(u(T)\) and \(h(T)\) which have the variation already integrated in. The judgement of which regime you are in is the same judgement this whole topic has been about: know the model, and know its range.