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Jalo
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Homework Statement
Consider a system with a constant number of particles.
Write the total differential dS in terms of the derivarives [itex]\frac{∂S}{∂T}[/itex] and [itex]\frac{∂S}{∂V}[/itex]. Introduce CV (calorific capacity at constant volume).
Next write the total differential of the volume dV in terms of the parcial derivatives [itex]\frac{∂V}{∂T}[/itex] and [itex]\frac{∂V}{∂P}[/itex]. Assume that the pressure is constant. Show that the result comes in the form of:
CP-CV= Expression
Homework Equations
CP=T[itex]\frac{∂S}{∂T}[/itex] , P and N Constant
CV=T[itex]\frac{∂S}{∂T}[/itex] , V and N Constant
The Attempt at a Solution
First I wrote the differential of the entropy as asked:
dS = [itex]\frac{∂S}{∂T}[/itex]dT + [itex]\frac{∂S}{∂V}[/itex]dV
I know that [itex]\frac{∂S}{∂V}[/itex] = CV/T. Substituting I get:
dS = CV/T dT + [itex]\frac{∂S}{∂V}[/itex]dV
Next I found the differential of the volume:
dV = [itex]\frac{∂V}{∂T}[/itex]dT + [itex]\frac{∂V}{∂P}[/itex]dP
Since the pressure is constant it reduces to the form
dV = [itex]\frac{∂V}{∂T}[/itex]dT
Substituting in our dS expression we get:
dS = CV/T dT + [itex]\frac{∂S}{∂V}[/itex][itex]\frac{∂V}{∂T}[/itex]dT =
= CV/T dT + [itex]\frac{∂S}{∂T}[/itex]dT =
= CV/T dT + CP/T dT ⇔
⇔ T[itex]\frac{dS}{dT}[/itex] = CV + CP
I'm making some mistake. If anyone could point me in the right direction I'd appreciate.
Thanks!