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#### gingi85

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« on: November 17, 2007, 02:28:36 PM »
In the derivation of the continuous formula for an adiabatic process we say:

dU=dW=CvdT

I'm having trouble understanding this, because, as far as I understood, dU=CvdT is only true of a process where the volume remains constant. As follows:

dU = dQ + dW = dQ -PexdV

Since the volume remains constant,

dV = 0

dU = dQ

We then define,

Cv = dQ/dT]v

and therefore

dU = CvdT

But this should only hold true for a process of constant volume, no? What am I missig?

#### Yggdrasil

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« Reply #1 on: November 17, 2007, 04:44:28 PM »
For an ideal gas, we can write the internal energy, U, as a function of temperature and volume.

U = U(T,V)

From this expression, we can obtain the following differential:

dU = (dU/dT)vdT + (dU/dV)TdV

Note that Cv = (dU/dT)v by definition.
In addition, (dU/dV)T = 0 because the internal energy of an ideal gas depends only on its temperature.

Therefore, we get:

dU = CvdT

This equation works in all cases (but only for ideal gases.  If (dU/dV)T is not zero, this does not hold).

dq = CvdT is only valid for constant pressure processes, however.