User:Retired Pchem Prof/JouleExpansion

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Introduction
old 2nd paragraph

The Joule expansion is also called free expansion. The process is a useful exercise in classical thermodynamics, as it is easy to work out the resulting increase in entropy in an ideal gas, the so-called entropy production. If the gas is not ideal, the process is more complex and is called the Joule–Thomson effect.

revised 2nd paragraph

The Joule expansion, treated as a thought experiment involving ideal gases, is a useful exercise (often called a free expansion) in classical thermodynamics. It provides a convenient example for working out changes in thermodynamic quantities, including the resulting increase in entropy of the universe (entropy production) that results from this inherently irreversible process. An actual Joule expansion experiment necessarily involves real gases; the temperature change in such a process provides a measure of intermolecular forces.

Description
The process begins with gas under some pressure, $$P_{\mathrm{i}}$$, at temperature $$T_{\mathrm{i}}$$, confined to one half of a thermally isolated container (see the top part of the drawing at the beginning of this article). The gas occupies an initial volume $$V_{\mathrm{i}}$$, mechanically separated from the other part of the container, which has a volume $$V_{\mathrm{0}}$$, and is under near zero pressure. The tap (solid line) between the two halves of the container is then suddenly opened, and the gas expands to fill the entire container, which has a total volume of $$V_{\mathrm{f}} = V_{\mathrm{i}} + V_{\mathrm{0}}$$ (see the bottom part of the drawing). A thermometer inserted into the compartment on the left (not shown in the drawing) measures the temperature of the gas before and after the expansion.

The system in this experiment consists of both compartments; that is, the entire region occupied by the gas at the end of the experiment. Because this system is thermally isolated, it cannot exchange heat with its surroundings. Also, since the system's total volume is kept constant, the system cannot perform work on its surroundings. As a result, the change in internal energy, $$\Delta U$$, is zero. Internal energy consists of internal kinetic energy (due to the motion of the molecules) and internal potential energy (due to intermolecular forces). Temperature is the measure of the internal kinetic energy; therefore a change in temperature indicates a change in kinetic energy. Since the total internal energy does not change, there must be an offsetting change in potential energy. Because of this, the Joule expansion provides information on intermolecular forces.

Ideal gases
If the gas is ideal, both the initial ($$T_{\mathrm{i}}$$, $$P_{\mathrm{i}}$$, $$V_{\mathrm{i}}$$) and final ($$T_{\mathrm{f}}$$, $$P_{\mathrm{f}}$$, $$V_{\mathrm{f}}$$) conditions follow the Ideal Gas Law, so that initially


 * $$P_{\mathrm{i}} V_{\mathrm{i}} = n R T_{\mathrm{i}}$$

and then, after the tap is opened,


 * $$P_{\mathrm{f}} V_{\mathrm{f}} = n R T_{\mathrm{f}}$$.

Here $$n$$ is the number of moles of gas and $$R$$ is the molar ideal gas constant. Because the internal energy does not change and the internal energy of an ideal gas is solely a function of temperature, the temperature of the gas does not change; therefore $$T_{\mathrm{i}} = T_{\mathrm{f}}$$. This implies that


 * $$P_{\mathrm{i}} V_{\mathrm{i}} = P_{\mathrm{f}} V_{\mathrm{f}} = n R T_{\mathrm{i}}$$.

Therefore if the volume doubles, the pressure halves.

The fact that the temperature does not change makes it easy to compute the change in entropy of the universe for this process.

Real gases
Unlike ideal gases, the temperature of a real gas will change during a Joule expansion. Empirically, it is found that almost all gases cool during a Joule expansion; the exceptions are helium, at temperatures above about 200 K, and hydrogen, at temperatures above about 400 K. Since internal energy is constant, cooling must be due to the conversion of internal kinetic energy to internal potential energy, with the opposite being the case for warming.

Intermolecular forces are repulsive at short range and attractive at long range (for example, see the Lennard-Jones potential). Since distances between gas molecules are large compared to molecular diameters, the energy of a gas is usually influenced mainly by the attractive part of the potential. As a result, expanding a gas usually increases the potential energy associated with intermolecular forces. Some textbooks say that for gases this must always be the case and that a Joule expansion must always produce cooling. In liquids, where molecules are close together, repulsive interactions are much more important and it is possible to get an increase in temperature during a Joule expansion.

It is theoretically predicted that, at sufficiently high temperature, all gases will warm during a Joule expansion The reason is that at any moment, a very small number of molecules will be undergoing collisions; for those few molecules, repulsive forces will dominate and the potential energy will be positive. As the temperature rises, both the frequency of collisions and the energy involved in the collisions increase, so the positive potential energy associated with collisions increases strongly. If the temperature is high enough, that can make the total potential energy positive, in spite of the much larger number of molecules experiencing weak attractive interactions. When the potential energy is positive, a constant energy expansion reduces potential energy and increases kinetic energy, resulting in an increase in temperature. This behavior has only been observed for hydrogen and helium; which have very weak attractive interactions. For other gases this "Joule inversion temperature" appears to be extremely high.