$5\mathrm{.}9.$ Consider a system of $C$ components with internal energy $E(S,V,N_1,N_2,$dots,$N_C).$

Find the differential and integrated forms of the fundamental equation for $e=\frac{E}{N},$

where $N={\displaystyle \underset{i}{\overset{?}{}}}{N}_i.$

Require the step solutions, the expected answer $is$:

$e=Ts-Pv+\backslash$sum$\_($i$)m{u}_i{x}_i$

$de=Ts-Pv+summatio{n}_i^{C-1}(m{u}_i-m{u}_c)d{x}_i$