(Solved): (12) Volumetric thermal expansion of a material under constant pressure const ...

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(12) Volumetric thermal expansion of a material under constant pressure constant temperature: On the basis of statistical thermodynamics, one can define the average of any property, say Volume, $$<V>$$ as follows: $$<V>=\frac{?V\exp \left(?\frac{H}{kT}\right)dVd?}{?\exp \left(?\frac{H}{kT}\right)dVd?}=\frac{?V\exp \left(?\frac{E+PV}{kT}\right)dVd?}{?\exp \left(?\frac{E+PV}{kT}\right)dVd?}$$ $$H=E+PV$$ is the enthalpy of the system and $$d?=\{drdp{\}}^{3N}$$ for $$N$$ body system. The volumetric thermal expansion is defined as: $$?=\frac{1}{?V?}\frac{??V?}{?T}$$. Using the above definition for average volume, derive the expression for volumetric thermal expansion.