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

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SUMMARY

The discussion focuses on proving the equation (du/dP)|t = Pkv - (cp - cv)K/B, as outlined in problem 4-4 of Carter or 1.14 in Cheng. Key equations include B = 1/v(R/P) = 1/T and k = -1/v(-RT/p^2) = 1/P. The proof begins with the first law of thermodynamics, dq = (du + Pdv), and involves substituting variables to derive the desired relationship. The challenge lies in proving this equation generally, beyond the ideal gas scenario.

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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 [itex]C_P[/itex] and [itex]C_V[/itex]?
 

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