The Joule-Thomson experiment

The Joule-Thomson experiment, an improved version of the Joule experiment, was conducted by James Joule and William Thomson in 1852 to study the thermodynamic properties of a gas expanding into a vacuum.

The experiment involves a system that is adiabatically insulated from its surroundings (see diagram above). The system consists of a double-piston cylinder that is separated into two compartments by a rigid porous plug. Constant pressures $p_1$ and $p_2$ , where $p_1>p_2$ , are applied to the left piston and the right piston respectively. As a result, the gas in the left compartment is slowly throttled irreversibly to the right compartment. Assuming that pressure is well defined in each compartment, $w_1=-\int_{V_1}^0p_1dV=p_1V_1$ and $w_2=-\int_{0}^{V_2}p_2dV=-p_2V_2$ . The total work done on the system is

$w=w_1+w_2=p_1V_1-p_2V_2$

According to the first law of thermodynamics for the entire system,

$\Delta U=U_2-U_1=q+w=p_1V_1-p_2V_2$

$U_2+p_2V_2=U_1+p_2V_2$

So, $H_2=H_1$ or $\Delta H=0$ . If there is a change in temperature of the system, the experiment measures the change in gas temperature with the change in gas pressure at constant enthalpy, i.e. $\left(\frac{\partial T}{\partial p}\right)_H$ , which is defined as the Joule-Thomson coefficient $\mu_{JT}$ :

$\mu_{JT}=\left(\frac{\partial T}{\partial p}\right)_H\;\;\;\;\;\;\;\;(90)$

The experiment is conducted multiple times by lowering $p_2$ for a constant $p_1$ and $T_1$ . A temperature-pressure plot of the results gives an isenthalpic curve (see diagram above). The gradient at a particular point of the curve is the Joule-Thomson coefficient with respect to $T$ and $p$ at that point. Since the change in pressure of the system is always negative, points on the curve that are associated with negative gradients correspond to the gas warming on expansion. Conversely, points on the curve that are associated with positive gradients correspond to the gas cooling on expansion. The point at which the gradient is zero is called the inversion point. Repeating the experiment by holding $T_1$ at new constant values and varying $p_2$ , we have multiple isenthapic curves:

The line connecting all inversion points is the inversion curve. The area on the left of the inversion curve is where cooling occurs for a gas when it expands into a vacuum, while the area on the right is where heating occurs. This is known as the Joule-Thomson effect, which is exploited in refrigeration processes.

Modern methods of measuring $\mu_{JT}$ involve evaluating $\left(\frac{\partial H}{\partial p}\right)_T$ , which is known as the isothermal Joule-Thomson coefficient. To determine the relation between $\mu_{JT}$ and $\left(\frac{\partial H}{\partial p}\right)_T$ , we refer to eq16, where $\left(\frac{\partial H}{\partial p}\right)_T=-\frac{1}{\left(\frac{\partial T}{\partial H}\right)_p\left(\frac{\partial p}{\partial T}\right)_H}$ . Using eq15, $\left(\frac{\partial H}{\partial p}\right)_T=-\left(\frac{\partial H}{\partial T}\right)_p\left(\frac{\partial T}{\partial p}\right)_H$ . Substituting eq46 and eq90 in this expression,