Joule expansion

The Joule expansion, in which a volume V_i = V₀ is expanded to a volume V_f = 2V₀ in a thermally isolated chamber.

The Joule expansion is an irreversible process in thermodynamics in which a volume of gas is kept in one side of a thermally isolated container (via a small partition), with the other side of the container being evacuated. The partition between the two parts of the container is then opened, and the gas fills the whole container.

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 calculating 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.

This type of expansion is named after James Prescott Joule who used this expansion, in 1845, in his study for the mechanical equivalent of heat, but this expansion was known long before Joule e.g. by John Leslie, in the beginning of the 19th century, and studied by Joseph-Louis Gay-Lussac in 1807 with similar results as obtained by Joule.^[1][2]

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.^[3] 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. ^{[4] [5]} 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.^[6][7] 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.^[8]

It is theoretically predicted that, at sufficiently high temperature, all gases will warm during a Joule expansion^[4] 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.^[5]

Entropy production

It is awkward to calculate the entropy production in this process directly, because between the time the partition is opened and the time equilibrium is reached, the system passes through states that are far from thermal equilibrium. However, entropy is a function of state, and therefore the entropy change can be computed directly from the knowledge of the final and initial equilibrium states. For an ideal monatomic gas, the entropy as a function of the internal energy U, volume V, and number of moles n is given by the Sackur–Tetrode equation:^[9]

$S = nR \ln \left[\left(\frac VN\right) \left(\frac{4\pi m}{3h^2}\frac UN \right)^{\frac 32}\right]+ {\frac 52}nR.$

In this expression m the particle mass and h Planck's constant. For a monatomic ideal gas U = (3/2)nRT = nC_VT, with C_V the molar heat capacity at constant volume. In terms of classical thermodynamics the entropy of an ideal gas is given by

$S(V,T)=S_0+nR\ln\left(\frac{V}{V_0}\right)+nC_V\ln\left(\frac{T}{T_0}\right)$

where S₀ is the, arbitrary chosen, value of the entropy at volume V₀ and temperature T₀.^[10] It is seen that a doubling of the volume at constant U or T leads to an entropy increase of ΔS = nR ln(2). This result is also valid if the gas is not monatomic, as the volume dependence of the entropy is the same for all ideal gases.

A second way to evaluate the entropy change is to choose a route from the initial state to the final state where all the intermediate states are in equilibrium. Such a route can only be realized in the limit where the changes happen infinitely slowly. Such routes are also referred to as quasistatic routes. In some books one demands that a quasistatic route has to be reversible, here we don't add this extra condition. The net entropy change from the initial state to the final state is independent of the particular choice of the quasistatic route, as the entropy is a function of state.

Here is how we can effect the quasistatic route. Instead of letting the gas undergo a free expansion in which the volume is doubled, a free expansion is allowed in which the volume expands by a very small amount δV. After thermal equilibrium is reached, we then let the gas undergo another free expansion by δV and wait until thermal equilibrium is reached. We repeat this until the volume has been doubled. In the limit δV to zero, this becomes an ideal quasistatic process, albeit an irreversible one. Now, according to the fundamental thermodynamic relation, we have:

$\mathrm{d}U = T \mathrm{d}S - P \mathrm{d}V.$

As this equation relates changes in thermodynamic state variables, it is valid for any quasistatic change, regardless of whether it is irreversible or reversible. For the above defined path we have that dU = 0 and thus TdS=PdV, and hence the increase in entropy for the Joule expansion is

$\Delta S=\int_i^f\mathrm{d}S=\int_{V_{0}}^{2V_{0}} \frac{P\,\mathrm{d}V}{T}=\int_{V_0}^{2V_0} \frac{n R\,\mathrm{d}V}{V}=n R\ln 2.$

A third way to compute the entropy change involves a route consisting of reversible adiabatic expansion followed by heating. We first let the system undergo a reversible adiabatic expansion in which the volume is doubled. During the expansion, the system performs work and the gas temperature goes down, so we have to supply heat to the system equal to the work performed to bring it to the same final state as in case of Joule expansion.

During the reversible adiabatic expansion, we have dS = 0. From the classical expression for the entropy it can be derived that the temperature after the doubling of the volume at constant entropy is given as:

$T = T_i2^{-R/C_V} = T_i2^{-2/3}$

for the monatomic ideal gas. Heating the gas up to the initial temperature T_i increases the entropy by the amount

$\Delta S = n\int_{T}^{T_i}C_{V}\frac{\mathrm{d}T'}{T'}= nR \ln 2.$

We might ask what the work would be if, once the Joule expansion has occurred, the gas is put back into the left-hand side by compressing it. The best method (i.e. the method involving the least work) is that of a reversible isothermal compression, which would take work W given by

$W=-\int_{2V_0}^{V_0} P\,\mathrm{d}V =- \int_{2V_0}^{V_0} \frac{nRT}{V} \mathrm{d}V=nRT\ln 2=T\Delta S_{gas}.$

During the Joule expansion the surroundings do not change, so the entropy of the surroundings is constant. So the entropy change of the so-called "universe" is equal to the entropy change of the gas which is nR ln 2.

Real-gas effect

Joule performed his experiment with air at room temperature which was expanded from a pressure of about 22 bar. Air, under these conditions, is almost an ideal gas, but not quite. As a result the real temperature change will not be exactly zero. With our present knowledge of the thermodynamic properties of air ^[11] we can calculate that the temperature of the air should drop by about 3 degrees Celsius when the volume is doubled under adiabatic conditions. However, due to the low heat capacity of the air and the high heat capacity of the strong copper containers and the water of the calorimeter, the observed temperature drop is much smaller, so Joule found that the temperature change was zero within his measuring accuracy.

References

The majority of good undergraduate textbooks deal with this expansion in great depth; see e.g. Concepts in Thermal Physics, Blundell & Blundell, OUP ISBN 0-19-856770-7

1. D.S.L. Cardwell, From Watt to Clausius, Heinemann, London (1957)
2. M.J. Klein, Principles of the theory of heat, D. Reidel Pub.Cy., Dordrecht (1986)
3. Note that the fact that the gas expands in a vacuum and thus against zero pressure is irrelevant. The work done by the system would also be zero if the right hand side of the chamber were not evacuated, but is instead filled with a gas at a lower pressure. While the expanding gas would then do work against the gas in the right-hand side of the container, the whole system doesn't do any work against the environment.
4. Goussard, J.-O.; Roulet, B. (1993). "Free expansion for real gases". Am. J. Phys. 61: 845–848.
5. Albarrán-Zavala, E.; Espinoza-Elizarraraz, B.A.; Angulo-Brown, F. (2009). "Joule inversion temperatures for some simple real gases". The Open Thermodynamics Journal 3: 17–22.
6. Pippard, A. B. (1957). Elements of Classical Thermodynamics, p. 73. Cambridge University Press, Cambridge, U.K.
7. Tabor, D. (1991). Gases, liquids and solids, p. 148. Cambridge University Press, Cambridge, U.K. ISBN 0 521 40667 6.
8. Keenan, J. H. (1970). Thermodynamics, p. 414. M.I.T. Press, Cambridge, Massachusetts.
9. K. Huang, Introduction to Statistical Physics, Taylor and Francis, London, 2001
10. M.W. Zemansky, Heat and Thermodynamics, McGraw-Hill Pub.Cy. New York (1951), page 177.
11. Refprop, software package developed by National Institute of Standards and Technology (NIST)