# Relationship between pressure and internal energy

andieje used Ask the Experts™
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Hi

If you heat a gas in constant volume the temperature will go up according to pv = nrt. The change in the internal energy of the gas that relates to that temperature change can be found from the specific heat capacity of the gas.

What if you wanted to look at this from the point of view of pressure and how pressure affects internal energy. I could examine this indirectly by looking at temperature again...If you increase the pressure the temperature will go up. You can find the change in internal energy again from the heat capacity of the gas and teh temperature change. Is there an equation to tell you the change in internal energy from the pressure directly?

thanks
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Commented:
The third formula on the link below ?
http://en.wikipedia.org/wiki/Ideal_gas_law

Commented:
thats the equation in my question

Commented:
I'm not sure what are we trying to find. What else do you mean, to know the internal energy from pressure "directly" ? You can't skip volume, temperature and gas constants...

Commented:
ref: http://hyperphysics.phy-astr.gsu.edu/Hbase/kinetic/kintem.html
pv = nrt = 2/3 * N * avg(1/2 m v^2)
Kinetic Energy = avg(1/2 m v^2)

so,
pv =  2/3 * N * Kinetic-Energy

Since the potential energy of the gas does not change when pressure changes, then change in Internal Energy = change in KE.

So, for your scenario of only changing pressure:
vdp = 2/3 * N * dKE

change in Internal Energy = [3/(2N) ] *V * (change in pressure)

Commented:
That result, of course, is holding V and N constant.

Commented:
Hi phoffric

What you have answered is, I believe, what i described in my question. In my mind that is looking at the effect of pressure on internal energy 'indirectly' by looking at the change in temperature obtained from the pressure change and then looking at the change in internal energy from the heat capacities of the gases. I didnt specify the equations to get  the heat capcacities of the gas but you have kindly given the heat capacity for an ideal gas.

What i was trying to ask, obviousy unsuccessfully, is if you can answer the same question by another route. I was wondering if there was another equation i did not know of which relates pressure directly to internal energy rather than having to 'look via the temperature' and via the formula you suggested (or the other formulas for monoatomic. polyatomic gases etc)

Perhaps there is no such thing and that is the answer to the question.
Commented:
DU = [3/(2N) ] *V * DP, ideal gas. This is as far as I can take you at this time. Sorry :(
Please reference this question in other related questions to get further assistance.

Commented:
That might be the only formula there is. I was only wondering if there was another

Commented:
I am sure there are others for gases in general. These equations are for ideal gases (and is there such a thing?). To come up with these responses, I had to refresh a little, and showed you some links I looked at. Before accepting what I said at face value, I recommend that you hit the "Request attention" link, and specify those experts who are on top of their thermodynamics game to add further insight.

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