Experiments · Experiment 11

Joule-Thomson Expansion and the Maxwell Relations

Throttling, inversion temperatures, and a direct test of a Maxwell relation.

Objectives

On completing this experiment you should be able to:

Measure the temperature change across a porous plug as a function of the pressure drop and obtain the Joule-Thomson coefficient.
Show that the coefficient is positive below the inversion temperature and negative above it, and locate the inversion temperature of nitrogen.
Show that helium warms on throttling at room temperature and explain why.
Verify the identity mu $C_p = T \left(\frac{dV}{dT}\right)_P - V$ , and hence test the Maxwell relation from which it follows.
Relate the sign of the effect to the second virial coefficient and hence to the balance between molecular attraction and repulsion.
Identify an unlabelled cylinder from its Joule-Thomson coefficient.

Theory

In a throttling process the gas is pushed through a constriction by the upstream pressure and pushes back the downstream gas. The net work per mole is $P_1 V_1$ - $P_2 V_2$ , and since the process is adiabatic, $U_2 - U_1 = P_1 V_1 - P_2 V_2$ , that is $H_1 = H_2$ . The process is isenthalpic but strongly irreversible.

Writing H as a function of T and P,

dH = C_p\,dT + \left[V - T \left(\frac{dV}{dT}\right)_P\right]\,dP ,

where the bracket comes from $dH = T\,dS + V\,dP$ with $\left(\frac{dS}{dP}\right)_T$ = -(dV/dT)_P. That last step is one of the four Maxwell relations, obtained by equating the mixed second derivatives of the Gibbs function. Setting $dH = 0$ gives the Joule-Thomson coefficient. For an ideal gas $T(dV/dT)_P = V$ exactly and mu is zero: any observed effect is a direct measurement of departure from ideality.

Key relations

$\left(\frac{\partial S}{\partial P}\right)_T=-\left(\frac{\partial V}{\partial T}\right)_P$
The Maxwell relation from the Gibbs function.
$\mu=\left(\frac{\partial T}{\partial P}\right)_H=\frac{1}{C_p}\left[T\left(\frac{\partial V}{\partial T}\right)_P-V\right]$
The Joule-Thomson coefficient.
$\mu C_p = T\frac{\mathrm{d}B}{\mathrm{d}T}-B$
Low-density form; inversion where T B' = B.

Procedure by part

Part A - The Joule-Thomson coefficient

Set the inlet temperature and pressure, throttle to the outlet pressure and read the temperature difference.

Select the gas and set the thermostat.
For a fixed inlet pressure, record the temperature drop for a series of outlet pressures.
Plot the temperature change against the pressure drop and take the gradient in the limit of small drop; this is the Joule-Thomson coefficient.
Repeat for at least four gases at room temperature, and note the sign in each case.

Part B - The inversion temperature

Repeat a small throttling step over a wide range of inlet temperatures and find where the effect changes sign.

Use a small pressure drop, so that the coefficient is measured essentially at the inlet temperature.
Step the inlet temperature over the whole available range for the chosen gas.
Plot the coefficient against temperature and locate the zero crossing.
Compare with the accepted maximum inversion temperature and comment on the accuracy of the low-density prediction.
Explain why helium and hydrogen must be pre-cooled before they can be liquefied by throttling.

Part C - Testing the Maxwell relation

Measure the isobaric expansivity of the same gas directly, and compare $T(dV/dT)_P$ - V with mu $C_p$ .

Select the gas and the temperature. The dilatometer reports the molar volume at the stated temperature and pressure.
Record the molar volume at several temperatures around the point of interest at fixed pressure, and obtain $\left(\frac{dV}{dT}\right)_P$ numerically.
Compute T $\left(\frac{dV}{dT}\right)_P$ - V and compare with mu $C_p$ from Part A, using the accepted $C_p$ .
State the Maxwell relation being tested and the potential from which it derives.

Questions to answer in your report

Derive the Joule-Thomson coefficient from $dH = T\,dS + V\,dP$ , identifying the Maxwell relation used and the potential it comes from. Show that it vanishes for an ideal gas. [10]
Explain physically why some gases cool and others warm on throttling at room temperature, and why every gas has an inversion temperature. [8]
Compare mu $C_p$ from Part A with $T(dV/dT)_P$ - V from Part C. Quantify the agreement and discuss the dominant source of uncertainty. [8]
The Joule-Thomson process is isenthalpic but irreversible. Compute the entropy generated per mole for one of your runs and explain why the process cannot be reversed by simply raising the downstream pressure. [6]