Interpreting Diagrams: Why is C_v Used in Isobaric Process?

[yklin_tux]

Hello All, I am interpreting a diagram from the following question.
(I didnt have a chance to take thermo and I learned it by myself so I might have some problems)

http://www.physicsgrad.com/pgre/9677-15

I understand that I can just do ΔW and I get the right answer,
but upon doing it with the specific heats, I am a little confused.

The isobaric process (B->C) specifically:

I looked at this solution:
http://grephysics.net/ans/9677/15

Why, in the calculation of U, is there C_v present?
I thought the pressure was constant, and the volume changes,
then why is U = C_v(ΔT)??

Any advise appreciated.

 

[Philip Wood]

You ask a good question.

For n moles of an ideal gas, we can show that $U = n\frac{\nu}{2}RT,$
in which $\nu$ is 3 for a monatomic gas (such as helium) and approximately 5 for a diatomic gas such as oxygen.

Thus we have $\Delta U = n\frac{\nu}{2}R\Delta T$.

Thus there is a proportionality constant, $n\frac{\nu}{2}R,$ between $\Delta U$ and $\Delta T$ which is fixed for any particular sample of gas, and is independent of whether the gas experiences changes at constant volume, constant pressure, or under any other conditions.

Now, for a constant volume change the heat flow Q is equal to the rise in internal energy, since no work is done.
So $Q = \Delta U$ [constant volume!]

But, by definition of the molar heat capacity, $C_v,$ $Q = nC_v \Delta T$ [constant volume!]

So $\Delta U = nC_v \Delta T$

But we showed at the beginning that the proportionality constant between $\Delta U$ and $\Delta T$ is fixed for any particular sample of gas, and is independent of whether the gas experiences changes at constant volume, constant pressure, or under any other conditions. So having shown for constant volume that the proportionality constant is $nC_v$, this must be the proportionality constant for all changes.

In other words, $nC_v$ is an alternative way of writing $n\frac{\nu}{2}R$.

 

[yklin_tux]

Philip,

Thank you for that explanation.

I understand everything you say when there is constant volume, etc,
but in that problem (first link), the B->C process has constant pressure,
and volume changes, so work is being done...

What you are saying is that for ΔU calculation for that specific process I can use
C_v?

 

[Chestermiller]

A characteristic of an ideal gas is that the internal energy is a function only of temperature, irrespective of the process.

 

[Philip Wood]

yklyn Yes, that's just what I'm saying. nC_v is the proportionality constant between ΔU and ΔT.

The key passage in my earlier post was the penultimate paragraph, staring "But we showed...".