Thermal Phys: Show (du/dP)|t = Pkv - (cp - cv)K/B

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The discussion revolves around proving the equation (du/dP)|t = Pkv - (cp - cv)K/B, which relates internal energy change to pressure at constant temperature. The user is struggling with the proof, particularly after applying the first law of thermodynamics and substituting variables. They emphasize the need for a general proof rather than one limited to ideal gases. The conversation also touches on the definitions of specific heat capacities, C_P and C_V, which are crucial for the proof. Clarifying these definitions and their implications may help resolve the user's confusion.
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For some reason this proof in thermal is tripping me up.

Homework Statement



Show that (du/dP)|t = Pkv - (cp - cv)K/B (problem 4-4 in Carter or 1.14 in Cheng)

Homework Equations



B = 1/v(R/P) = 1/T
k= -1/v(-RT/p^2) = 1/P

The Attempt at a Solution



Start with 1st law dq=(du + Pdv) and substitute...eventually:
Get to du/dP|T dP - (du/dt)|p dt - PBvdT +PkvdP
Divide by dP

Now is where I'm stuck...
 
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You want to prove this generally, not just for the case of the ideal gas. Do you know the definitions of C_P and C_V?
 

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