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*From*: John Denker <jsd@av8n.com>*Date*: Fri, 13 Jun 2008 17:30:52 -0700

We know that for an ideal gas, there is a one-to-one

correspondence between the temperature and the kinetic energy of the

gas particles. However, that does not mean that there is a one-to-one

correspondence between kinetic energy and heat energy. (In this

context, heat energy refers to whatever is measured by a heat

capacity experiment.)

To illustrate this point, let’s consider a sample of pure monatomic

nonrelativistic nondegenerate ideal gas in a cylinder of horizontal

radius r and vertical height h at temperature T. The pressure

measured at the bottom of the cylinder is P. Each particle in the gas

has mass m. We wish to know the heat capacity per particle at

constant volume, i.e. Cv/N.

At this point you may already have in mind an answer, a simple answer,

a well-known answer, independent of r, h, m, P, T, and N. But

wait, there’s more to the story: The point of this exercise is that h

is not small. In particular, mgh is not small compared to kT, where g

is the acceleration of gravity. For simplicity, you are encouraged to

start by considering the limit where h goes to infinity, in which

case the exact value of h no longer matters. Gravity holds virtually

all the gas near the bottom of the cylinder, on the scale of a few

kT/mg.

Later, if you want to come back and work the problem a second time,

with a large but finite h, that’s worth doing. Also if you want to

generalize to a polyatomic gas, that’s also worth doing.

You will discover that a distinctly nontrival contribution to the

heat capacity comes from the potential energy of the ideal gas. When

you heat it up, the gas column expands, lifting its center of mass,

doing work against gravity. (Of course, as always, there will be a

contribution from the kinetic energy.)

So, we conclude that in general, heat energy is not just kinetic

energy.

For more on this, see

http://www.av8n.com/physics/thermo-laws.htm#sec-ideal-ke-pe-cv

This reinforces points made at:

http://www.av8n.com/physics/thermo-laws.htm#sec-not-just-kinetic

**References**:**Re: [Phys-l] Phys-l Digest, Vol 41, Issue 9***From:*"David Strasburger-fac" <David_Strasburger-fac@nobles.edu>

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