[gingi85]

In the derivation of the continuous formula for an adiabatic process we say:

dU=dW=C_vdT

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

dU = dQ + dW = dQ -P_exdV

Since the volume remains constant,

dV = 0

dU = dQ

We then define,

C_v = dQ/dT]_v

and therefore

dU = C_vdT

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

[Yggdrasil]

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 C_v = (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 = C_vdT

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

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