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Paul Hewitt writes in his book

Expansion of real gases lead to cooling as average translational kinetic energy per molecule decreases.

The reason given is:

During Expansion molecules collide with more receding molecules than with approaching so that net work done on a molecule is negative due to which speed decreases and hence temperature decreases.

I am not able to understand how a molecule can collide more with molecules ahead than hits received from molecules behind. To me it seems that both density and speed of molecules approaching a particular molecule from behind will be more.

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  • $\begingroup$ All collisions are between molecules that are approaching each other. $\endgroup$
    – causative
    Dec 10, 2023 at 5:42

5 Answers 5

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There are various conditions which might apply while a gas expands.

  1. adiabatic (isentropic) expansion. Gas does work on its surroundings (pushing on the boundary as the boundary moves) so loses energy. No heat comes in. So internal energy goes down. Temperature goes down.

  2. isothermal. Obviously the temperature does not change. This is because heat flows in while the gas does work in expanding.

  3. constant enthalpy (Joule-Kelvin process, also known as throttling). The temperature may go up or down, depending on the initial conditions.

  4. free expansion (expansion into vacuum). This is a process at constant internal energy. Temperature of an ideal gas stays constant, temperature of a real gas falls a little. This is because the molecules attract one another so as they move apart they gain potential energy. But no energy has come in so they must lose kinetic energy. The kinetic energy is directly related to the temperature.

I can't tell whether the original question is about case (1) or (4). In both cases I already gave the reason for the temperature change, and here I will say a little more about case (1).

The above explains why there is a loss of internal energy of the gas as a whole. If you look at the situation from the point of view of individual molecules and their collisions, you see that while the volume is increasing, the molecules must be, on average, moving away from one another more than they are moving towards one another. But this does not tell you much about the collisions. The collisions inside the bulk of the gas conserve energy so do not tell you much about change of temperature. It is the collisions with the walls you should think about. When it collides with a wall the molecule sticks for a short time (called the dwell time) and then it is re-emitted. If the wall is moving away from the centre of mass of the gas then the average speed of the molecule after re-emission (or desorption), relative to the centre of mass of the gas, will be smaller than what it was when it approached the wall.

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The reason given in the book stays on a kinetic theory level of description. I think it is not the simplest level to explain the phenomenon. It is much better to start acknowledging that in a free expansion, the energy of the gas is constant. Typical potential energies at large interparticle separations are dominated by attraction. Therefore, the average potential energy has to increase under the gas expansion, consequently decreasing the kinetic energy.

At a more complex level of explanation, and going to kinetic energy, I would introduce collisions with expanding walls (every collision reduces the kinetic energy). Finally, discussing the phenomenon by taking into account inter-particle collisions would require an analysis of the effect of the expansion on the molecular velocity distribution. By far the most complex task. In my opinion, not justified if the aim is to provide an intuitive justification for the cooling effect.

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  • $\begingroup$ I appreciate how your answer focuses on what you consider to be the most important level of description for the context. Upvoted. $\endgroup$ Dec 10, 2023 at 11:52
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Let us consider gas molecules collision with the piston to be fairly elastic in nature. While we expand the gas by lifting the piston with some velocity u. And let the gas molecules be traveling towards the piston with velocity v and with velocity v' after the collision. Since the collisions are elastic
$v' + u = v - u$ ;
$v' = v - 2u $ ; (here the values only represent their magnitudes)
The kinetic energy of the molecules is less after collision which leads to cooling.

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Consider expansion into vacuum, where no external force acts on the gas to slow it down.

Gas temperature of a gas element(say, 1mm$^3$ cube) is a measure of chaotic motion of the gas molecules belonging to that element, in the reference frame where the element is at rest (the rest frame). More precisely, it is proportional to average kinetic energy of molecules in that rest frame.

Gas expansion means macroscopic elements of gas start moving and accelerate and flow appears. If the gas is ideal, kinetic energy of all molecules is conserved during such flow, it just changes its distribution on the 3 space directions in the velocity space: more kinetic energy will be in the one direction the flow has, less kinetic energy will be in the other two directions. The motion of the molecules becomes more ordered and less chaotic in the lab frame. It becomes less chaotic in the rest frame as well, so average kinetic energy there decreases. And decrease of average kinetic energy in the rest frame means lower temperature.

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collisions don't lead to cooling (except if it causes bonding and release of a photon, or increase in molecular vibrational energy), otherwise this would violate energy conservation (if the gas is an isolated system). However, if the gas has intra-molecular potential energy, expansion would lead to cooling because you need energy to compensate for the lower negative potential energy overall. Since on average the total energy is $\sim T - P$, with $T$ the thermal energy and $P$ the potential energy. Upon expansion $P$ becomes less negative, as such $T$ has to decrease to conserve energy.

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