Derive the following relations for a pure substance: (a) (left(frac{partial H}{partial T} ight)_{V}=C_{V}+frac{beta V}{alpha}) (b) (left(frac{partial H}{partial

Derive the following relations for a pure substance:

(a) \(\left(\frac{\partial H}{\partial T}\right)_{V}=C_{V}+\frac{\beta V}{\alpha}\)

(b) \(\left(\frac{\partial H}{\partial P}\right)_{T}=V(1-T \beta)\)

(c) \(\left(\frac{\partial H}{\partial P}\right)_{V}=\frac{C_{V} \alpha}{\beta}+V\)

(d) \(\left(\frac{\partial S}{\partial P}\right)_{T}=T-\frac{1}{\beta}\)

(e) \(\left(\frac{\partial H}{\partial S}\right)_{V}=T\left(1+\frac{V \beta}{C_{V} d}\right)\)

(f) \(\left(\frac{\partial T}{\partial S}\right)_{U}=\frac{T}{C_{V}}\left(1-\frac{T \beta}{P \alpha}\right)\)