Thermodynamics filled in class notes_Part_60

# Thermodynamics filled in class notes_Part_60 - 4.7...

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Unformatted text preview: 4.7. COMPRESSIBILITY AND GENERALIZED CHARTS 127 Proceeding as before, we have ∂u ∂v vextendsingle vextendsingle vextendsingle vextendsingle T = T ∂P ∂T vextendsingle vextendsingle vextendsingle vextendsingle v − P, (4.284) = T parenleftbigg R v + a 2 v 2 T 3 / 2 parenrightbigg − parenleftbigg RT v − a v 2 T 1 / 2 parenrightbigg , (4.285) = 3 a 2 v 2 T 1 / 2 . (4.286) Integrating, we find u ( T,v ) = − 3 a 2 vT 1 / 2 + f ( T ) . (4.287) Here f ( T ) is a yet-to-be-specified function of temperature only. Now the specific heat is found by the temperature derivative of u : c v ( T,v ) = ∂u ∂T vextendsingle vextendsingle vextendsingle vextendsingle v = 3 a 4 vT 3 / 2 + df dT . (4.288) Obviously, for this material, c v is a function of both T and v . Let us define c vo ( T ) via df dT ≡ c vo ( T ) . (4.289) Integrating, then one gets f ( T ) = C + integraldisplay T T o c vo ( ˆ T ) d ˆ T. (4.290) Let us take C = u o +3 a/ 2 /v o /T 1 / 2 o . Thus we arrive at the following expressions for....
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Thermodynamics filled in class notes_Part_60 - 4.7...

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