# Heat capacity at constant volume and energy

So, my understanding of heat capacity at constant volume is that

$$C_V = \bigg(\frac{\delta Q}{dT}\bigg)_V$$

So far, so good. But according to my professor,

$$dU = C_VdT$$

for an ideal gas, even when work is performed, that is, when $$\delta Q\neq dU$$. I see that there are a lot of explanations in this forum as to why the above formula is true, so please note that I am NOT asking that again. What I am asking for is some help understanding how can $$C_V$$ be equal to two different things at the same time... That looks like a contradiction. Any help would be appreciated!

– rob
Sep 23, 2021 at 18:34
• I still think it's unclear as to how the other questions were not duplicates Sep 24, 2021 at 17:54
• @Cross Sorry, but it doesn't. The question I'm interested in is a little bit different... Sep 25, 2021 at 6:39
• I believe you. I think more explanation is needed in the question though. Right now I just can't see it, but I'm not trying to say you're wrong. Sep 25, 2021 at 8:38
• That comment was posted for a previous version of the question. Now my comment is irrelevant, but I suggest you add a little more explanation in the question, and make what you're asking more apparent to the others. Sep 25, 2021 at 11:00

I can feel the confusion, at first it seems contradictory, but you actually need to know about nature of $$\Delta U$$.

There are two functions in thermodynamics:

1. State Functions: These functions depend only on state variables of system (which describe state of the system like $$P,V,T$$). $$\Delta U$$ belongs here.

2. Path Functions: These functions depend on path taken to reach one state of system to other state of the system. Path can be Isobaric, Isothermal, Adiabatic, Isochoric, Polytropic or other complex paths. $$\Delta Q,\Delta W$$ belong here.

Now as you said: $$C_v=\left(\frac{\Delta Q}{n\Delta T}\right)_v$$

Now from first law of thermodynamics:$$\Delta U=\Delta Q-\Delta W$$ In isochoric process (volume is constant as described by our first equation), $$\Delta W=0$$ (Path function). Hence:$$\Delta U=\Delta Q$$

Now: $$C_v=\left(\frac{\Delta U}{n\Delta T}\right)$$ Hence: $$\Delta U=nC_v\Delta T$$

So while we found this equation in isochoric situation, but $$\Delta U$$ is state function, it doesn't care which path system is going through! so this established equation is valid for any ideal gas system taking any kind of path.