Blundell Thermal Physics · Problem 27.2

Cooling real gases

Physics — Statistical Mechanics · joule-kelvin-expansion-inversion-temperature

Problem

In a Joule–Kelvin liquefier, gas is cooled by expansion through an insulated throttle – a simple but inefficient process with no moving parts at low temperature. Explain why enthalpy is conserved in this process. Deduce that (∂T∂P)H=1CP[T(∂V∂T)P−V].\left(\frac{\partial T}{\partial P}\right)_H = \frac{1}{C_P}\left[T\left(\frac{\partial V}{\partial T}\right)_P - V\right]. Estimate the highest starting temperature at which the process will work for helium at low densities, on the following assumptions: (i) The pressure is given at low densities by a virial expansion of the form PVRT=1+(b−aRT)(1V)+⋯ ,\frac{PV}{RT} = 1 + \left(b - \frac{a}{RT}\right)\left(\frac{1}{V}\right) + \cdots, and (ii) The Boyle temperature a/bRa/bR (the temperature at which the second virial coefficient vanishes) is known from experiment to be 19 K for helium. [Hint: One method of solving this problem is to remember that pp is easily made the subject of the equation of state and one can then use (∂V/∂T)p=−(∂p/∂T)V/(∂p/∂V)T(\partial V/\partial T)_p = -(\partial p/\partial T)_V/(\partial p/\partial V)_T.]
Intuitive explanation

The Joule–Kelvin process throttles gas through a valve with no heat exchange or work done on the surroundings, so enthalpy HH remains constant. To find how temperature changes with pressure at constant HH, use the chain rule on the total differential of enthalpy and the Maxwell relation connecting entropy to volume—this reveals that the Joule–Kelvin coefficient depends on how volume responds to temperature changes. For cooling to occur, you need (∂T/∂P)H<0(\partial T/\partial P)_H < 0, which requires the virial equation's temperature-dependent terms to dominate; this sets a maximum starting temperature (the inversion temperature) above which expansion actually warms the gas.

Intuitive explanation

The Joule–Kelvin process throttles gas through a valve with no heat exchange or work done on the surroundings, so enthalpy HH remains constant. To find how temperature changes with pressure at constant HH, use the chain rule on the total differential of enthalpy and the Maxwell relation connecting entropy to volume—this reveals that the Joule–Kelvin coefficient depends on how volume responds to temperature changes. For cooling to occur, you need (∂T/∂P)H<0(\partial T/\partial P)_H < 0, which requires the virial equation's temperature-dependent terms to dominate; this sets a maximum starting temperature (the inversion temperature) above which expansion actually warms the gas.

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