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I thought I could simply Google this question and get a straight, simple answer. Everything but.

Is it simply that the mean velocities of the molecules are increased (so molecular kinetic energy) as pressure increases?

Or is energy going anywhere else internally? I'm discounting possible heat transfer across the considered control volume.

Does some of that energy flow into the molecular structure?

The VanderWalls model is nonlinear, manifesting either attractive or repulsive forces depending on molecular separation. I'm having trouble fitting this model into how energy is stored.

What are the fundamental mechanisms and how does VanderWall fit into the mechanism?

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    $\begingroup$ Energy can be stored in translations, rotations, vibrations and also electrons can be promoted to accommodate new energy. After you've crossed the threshold to allow those modes to activate. At least as I understand it. $\endgroup$ – Charlie Mar 27 at 16:16
  • $\begingroup$ @Charlie after more research I see that it depends allot on the size, complexity of the molecules as to where energy can flow. For small molecules almost all the energy in translation (average kinetic energy). For larger molecules there are more internal modes that can store energy. $\endgroup$ – docscience Mar 31 at 20:36
  • $\begingroup$ Yes larger molecules with more bonds around which the molecule can rotate will have more complex rotational modes for instance (as long as indistinguishability rules will allow it). $\endgroup$ – Charlie Mar 31 at 20:58
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If we assume a monatomic ideal gas then the internal energy of the gas is simply the kinetic energy of the gas particles, and the average kinetic energy of a single gas particle is dependent only on the temperature:

$$ KE = \tfrac32 k_B T \tag{1}$$

If you have some fixed volume $V$ at a fixed temperature $T$ then the number of moles of gas in that volume is given by the ideal gas law:

$$ n = \frac{PV}{RT} $$

And the energy stored in that volume is the number of moles $n$ times Avagadro's number times the energy we calculated in equation (1):

$$ U = \frac{PV}{RT} N_a \tfrac32 k_B T = \tfrac32 PV $$

So at constant volume the energy stored in the gas is simply proportional to pressure, as indeed experience tells us. The mechanism is that the energy is just the KE of gas molecules and increasing the pressure means we have more gas molecules and hence more energy.

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  • $\begingroup$ More molecules in the same space at the same temperature, , more kinetic energy. Yes that makes sense. But when you stuff more molecules into that space at the same temperature does the average molecular velocity increase? I'm guessing no, that the velocity increases defined a temperature increase. So then increase in energy due to geothermal compression strictly due to more moles. Right,? $\endgroup$ – docscience Mar 27 at 19:41
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    $\begingroup$ @docscience correct. The average KE depends only on the temperature and not on the pressure or volume. So it is as simple as more moles per unit volume = higher energy per unit volume. $\endgroup$ – John Rennie Mar 27 at 19:52
  • $\begingroup$ geothermal above should be isothermal $\endgroup$ – docscience Mar 31 at 20:39
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There are two pressure effects on a gas that relate to the temperature and stored energy. The first, is that when applying pressure, one accelerates a few atoms as they bounce off the moving piston. This causes adiabatic heating (Diesel engines and clever little fire-starting devices put this effect to good use).

If, however, you let the heated gas cool back to its original temperature, there is STILL stored energy, in the form of the perturbed (mainly outer electron) atomic state. It is known that pressure broadens spectrum lines, because pressure causes the ground-state of an undisturbed atom or molecule to become a hybrid with excited states becoming partly populated (each collision may be regarded as partly exciting the atoms that collide).

In an ideal, noninteracting gas model, pressure effects are ignored; this is why the stored energy seems mysterious. In a metal spring (not an ideal gas) it is taken for granted that stresses store energy in the material. That metal spring is storing the energy in perturbed ground-state electron wavefunctions, just as a compressed gas does.

The Van der Waals quasi-empirical additions don't really address the statistical range of electron configuration due to pressure, either.

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Compressed gasses do not store energy. When you you use a compressed air tool, for example, the energy is coming from the heat in the surrounding environment. As the compressed air in the cylinder expands and does work, it gets colder. Because heat can enter and rewarm the air, that added heat energy replaces the energy used by the tool. Compressed air can extract a lot of convertable-to-work heat energy from the environment. Uncompressed air cannot do this.

Put mathematically, The internal energy $U$ of a given mass of air at temperature $T$ is independent of its volume $V$. What the compressed air has that the same amount of uncompressed air at the same temperature does not have is free energy. The free energy is $F=U-TS$ where $S$ is the entropy. Compressed air has lower entropy than uncompressed gas, and so $U-TS$ is much larger than that of uncompressed air. The work done by the tool results in a reduction of $F$ because it increases $TS$. If the temperature stays the same, the heat that flows into the compressed air is $Q=T(S_{end}-S_{start})$ and this is equal to the work $P(V_{end}-V_{start})$ done by the expanding air.

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