*Historical background*

If you received formal tuition in physical chemistry at school, then it’s likely that among the first things you learned were the 17th/18th century gas laws of Mariotte and Gay-Lussac (Boyle and Charles in the English-speaking world) and the equation that expresses them: PV = kT.

It may be that the historical aspects of what is now known as the ideal (perfect) gas equation were not covered as part of your science education, in which case you may be surprised to learn that it took 174 years to advance from the pressure-volume law PV = k to the combined gas law PV = kT.

The lengthy timescale indicates that putting together closely associated observations wasn’t regarded as a must-do in this particular era of scientific enquiry. The French physicist and mining engineer Émile Clapeyron eventually created the combined gas equation, not for its own sake, but because he needed an analytical expression for the pressure-volume work done in the cycle of reversible heat engine operations we know today as the Carnot cycle.

Students sometimes get in a muddle about combining the gas laws, so for the sake of completeness I will set out the procedure. Beginning with a quantity of gas at an arbitrary initial pressure P_{1} and volume V_{1}, we suppose the pressure is changed to P_{2} while the temperature is maintained at T_{1}. Applying the Mariotte relation (PV)_{T} = k, we write

The pressure being kept constant at P_{2} we now suppose the temperature changed to T_{2}; the volume will then change from V_{x} to the final volume V_{2}. Applying the Gay-Lussac relation (V/T)_{P} = k, we write

Substituting V_{x} in the original equation:

whence

**– – – –**

*Differences of opinion*

In the mid-19th century, the ideal gas equation – or rather the ideal gas itself – was the cause of no end of trouble among those involved in developing the new science of thermodynamics. The argument went along the lines that since no real gas was ever perfect, was it legitimate to base thermodynamic theory on the use of a perfect gas as the working substance in the Carnot cycle? Joule, Clausius, Rankine, Maxwell and van der Waals said yes it was, while Mach and Thomson said no it wasn’t.

With thermometry on his mind, Thomson actually got quite upset. Here’s a sample outpouring from the *Encyclopaedia Britannica*:

*“… a mere quicksand has been given as a foundation of thermometry, by building from the beginning on an ideal substance called a perfect gas, with none of its properties realized rigorously by any real substance, and with some of them unknown, and utterly unassignable, even by guess.”*

It seems strange that the notion of an ideal gas, as a theoretical convenience at least, caused this violent division into believers and disbelievers, when everyone agreed that the behavior of all real gases approaches a limit as the pressure approaches zero. This is indeed how the universal gas constant R was computed – by extrapolation from pressure-volume measurements made on real gases. There is no discontinuity between the measured and limiting state, as the following diagram demonstrates:

Experiments on real gases show that

where v is the molar volume and *i* signifies ice-point. The universal gas constant is defined by the equation

so for real gases

The behavior of n moles of any gas as the pressure approaches zero may thus be represented by

The notion of an ideal gas is founded on this limiting state, and is defined as a gas that obeys this equation at all pressures. The equation of state of an ideal gas is therefore

**– – – –**

*Testing Mayer’s assumption*

The notion of an ideal gas was not the only thing troubling William Thomson at the start of the 1850s. He also had a problem with real gases. This was because he was simultaneously engaged in a quest for a scale of thermodynamic temperature that was independent of the properties of any particular substance.

What he needed was to find a property of a real gas that would enable him to

a) prove by thermodynamic argument that real gases do not obey the ideal gas law

b) calculate the absolute temperature from a temperature measured on a (real) gas scale

And he found such a property, or at least he thought he had found it, in the thermodynamic function (∂U/∂V)_{T}.

In the final part of his landmark paper, ** On the Dynamical Theory of Heat**, which was read before the Royal Society of Edinburgh on Monday 15 December 1851, Thomson presented an equation which served his purpose. In modern notation it reads:

This is a powerful equation indeed, since it enables * any* equation of state of a PVT system to be tested by relating the mechanical properties of a gas to a thermodynamic function of state which can be experimentally determined.

If the equation of state is that of an ideal gas (PV = nRT), then

This defining property of an ideal gas, that its internal energy is independent of volume in an isothermal process, was an assumption made in the early 1840s by Julius Robert Mayer of Heilbronn, Germany in developing what we now call Mayer’s relation (C_{p} – C_{V} = PΔV). Thomson was keen to disprove this assumption, and with it the notion of the ideal gas, by demonstrating non-zero values for (∂U/∂V)_{T}.

In 1845 James Joule had tried to verify Mayer’s assumption in the famous experiment involving the expansion of air into an evacuated cylinder, but the results Joule obtained – although appearing to support Mayer’s claim – were deemed unreliable due to experimental design weaknesses.

Thomson had meanwhile been working on an alternative approach to testing Mayer’s assumption. By 1852 he had a design for an apparatus and had arranged with Joule to start work in Manchester in May of that year. This was to be the Joule-Thomson experiment, which for the first time demonstrated decisive differences from ideal behavior in the behavior of real gases.

Mayer’s assumption was eventually shown to be incorrect – to the extent of about 3 parts in a thousand. But this was an insignificant finding in the context of Joule and Thomson’s wider endeavors, which would propel experimental research into the modern era and herald the ** birth of big science**.

Curiously, it was not the fact that (∂U/∂V)_{T} = 0 for an ideal gas that enabled the differences in real gas behavior to be shown in the Joule-Thomson experiment. It was the other defining property of an ideal gas, that its enthalpy H is independent of pressure P in an isothermal process. By parallel reasoning

If the equation of state is that of an ideal gas (PV = nRT), then

Since the Joule-Thomson coefficient (μ_{JT}) is defined

and the second term on the right is zero for an ideal gas, μ_{JT} must also be zero. Unlike a real gas therefore, an ideal gas cannot exhibit Joule-Thomson cooling or heating.

**– – – –**

*Finding a way to define absolute temperature*

But to return to Thomson and his quest for a scale of absolute temperature. The equation he arrived at in his 1851 paper,

besides enabling any equation of state of a PVT system to be tested, also makes it possible to give an exact definition of absolute temperature independently of the behavior of any particular substance.

The argument runs as follows. Given the temperature readings, t, of any arbitrary thermometer (mercury thermometer, bolometer, whatever..) the task is to express the absolute temperature T as a function of t. By direct measurement, it may be found how the behavior of some appropriate substance, e.g. a gas, depends on t and either V or P. Introducing t and V as the independent variables in the above equation instead of T and V, we have

where (∂U/∂V)_{t}, (∂P/∂t)_{V} and P represent functions of t and V, which can be experimentally determined. Separating the variables so that both terms in T are on the left, the equation can then be integrated:

Integrating between the ice point and the steam point

This completely determines T as a function of t.

But as we have already seen, there was a catch to this argumentation – namely that (∂U/∂V) could not be experimentally determined under isothermal conditions with sufficient accuracy.

**– – – –**

*The Joule-Thomson coefficient provides the key*

Thomson’s means of circumventing this problem was the steady state Joule-Thomson experiment, which measured upstream and downstream temperature and pressure, and enabled the Joule-Thomson coefficient, μ_{JT} = (∂T/∂P)_{H}, to be computed.

It should be borne in mind however that when Joule and Thomson began their work in 1852, they were not aware that their cleverly-designed experiment was subject to isenthalpic conditions. It was the Scottish engineer and mathematician William Rankine who first proved in 1854 that the equation of the curve of free expansion in the Joule-Thomson experiment was d(U+PV) = 0.

As for the Joule-Thomson coefficient itself, it was the crowning achievement of a decade of collaboration, appearing in an appendix to Joule and Thomson’s final joint paper published in the Philosophical Transactions of the Royal Society in 1862. They wrote it in the form

where the upper symbol in the derivative denotes “thermal effect”, and K denotes thermal capacity at constant pressure of a unit mass of fluid.

The equation is now usually written

By the method applied previously, this equation can be expressed in terms of an empirical t-scale and the absolute T-scale:

where C’_{P} is the heat capacity of the gas as measured on the empirical t-scale, i.e. C’_{P} = C_{P}(dT/dt). Cancelling (dT/dt) and separating the variables so that both terms in T are on the left, the equation becomes:

Integrating between the ice point and the steam point

This completely determines T as a function of t, with all the terms under the integral capable of experimental determination to a sufficient level of accuracy.