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Ok, I feel a bit silly asking this. I'm asking in relation to this question here on the molecular basis of hydrostatic pressure in a gas. There's been quite a bit of discussion and one of the commenters has confused me by suggesting that:

The kinetic energy of an ideal gas is only related to the temperature through the internal energy. Typically all contributions to internal energy are ignored other than kinetic energy, however, in this scenario the gravitation contribution to internal energy is not negligible. I agree that a isothermal profile maximizes entropy and thus is the steady state, however that constant temperature means constant internal energy not kinetic energy

He seems to be suggesting that, because internal energy of a gas should include gravitational potential energy (GPE), gas temperature should be related to gravitational potential (not just kinetic energy of the molecules). That would suggest that a region of gas molecules could have more kinetic energy but the same temperature as another region of molecules higher up, because the molecules have lower GPE.

This seems wrong to me, as my understanding is that temperature is linked to the average kinetic energy of the gas molecules and has nothing to do with gravitational potential. If his theory were correct, then if I had a quantity of gas in a sealed, perfectly insulating container and I took it from sea level up Mount Everest, then its temperature would increase just because the gravitational potential has increased. This would seem to violate the conservation of energy (because now the gas has extra thermal energy as well as increased GPE) as well as the ideal gas law (because the pressure and density have not changed, but the temperature has).

I'm looking for some confirmation one way or the other of which of us is correct. If anyone knows of any reliable references that deal with this or can provide a convincing counter-example, that would be great.

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The internal energy of the gas should not include the $GPE$.

A well-known example of how things work in this way is stars. Protostars have to heat up before they can produce energy through fusion.

The initial temperature increase comes from the transfer of $GPE$ to $KE$ as the particles condense. As the gas loses $GPE$, it gains $KE$ (and therefore temperature).

It's difficult to see this directly in earth's atmosphere because other processes such as heat transfer and adiabatic compression tend to swamp the effect.

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    $\begingroup$ Nice answer BowlOfRed. So, if GPE was included in internal energy/temperature then protostars would not heat up as they condense, because the internal energy would not change, hence stars could not form. Sounds pretty convincing to me! $\endgroup$ – Time4Tea Jan 9 '15 at 20:48
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Even John Tyndall recognised that changes in GPE influenced temperature. It shows up in waterfalls where the temperature of the water at the bottom of a waterfall is higher than that at the top. Also, consistent with this argument, if the falling water is used to generate electricity, the water at the outflow (tailrace) is the same as the inflow, the GPE having been abstracted as electrical energy.

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    $\begingroup$ The waterfall converts GPE to kinetic energy which can dissipate as heat. It doesn't mean that the GPE itself influences temperature. The potential energy of the system can be converted to heat through various methods; but the GPE itself does not cause any change to the heat (see the other answer). $\endgroup$ – JMac Jun 30 '17 at 14:11

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