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Chemistry Forums for Students => Physical Chemistry Forum => Topic started by: gingi85 on November 17, 2007, 02:28:36 PM

Title: Adiabatic Process
Post by: gingi85 on November 17, 2007, 02:28:36 PM

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?

Title: Re: Adiabatic Process
Post by: Yggdrasil 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 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.