Question

Express the Internal Energy and Entropy as a Function of T and V for a homogeneous fluid. Develop the same relations using the isothermal compressibility and the volume expansivity coefficients.

Answer 1

`dU=C_(v) dT+(T((\beta )/(\kappa )) -P)dV`

`dS=C_(v) (dT)/(T) +((\beta )/(\kappa ) ) dV`

Step-by-step explanation:

The internal energy is equal to:

`dU=C_(v) dT+(T((\delta P)/(\delta T) )_(v) -P)dV`

The entropy is equal to:

`dS=C_(v) (dT)/(T) +((\delta P)/(\delta T) )_(v) dV`

If we write the pressure derivative in terms of isothermal compresibility and volume expansivity, we have

`(\delta P)/(\delta T)=(\beta )/(\kappa )`

Replacing:

`dU=C_(v) dT+(T((\beta )/(\kappa )) -P)dV`

`dS=C_(v) (dT)/(T) +((\beta )/(\kappa ) ) dV`