$1$ The following are the total energy expressions for certain molecules:

a$)\frac{?}{\text{b}\text{a}\text{r}}(E{)}_{\text{k}\text{i}\text{n}}=\frac{1}{2}\frac{m}{b}ar(v{)}_x^2+\frac{1}{2}\frac{m}{b}ar(v{)}_y^2+\frac{1}{2}\frac{m}{b}ar(v{)}_z^2$

b$)\frac{?}{\text{b}\text{a}\text{r}}(E{)}_{\text{k}\text{i}\text{n}}=\frac{1}{2}\frac{m}{b}ar(v{)}_x^2+\frac{1}{2}\frac{m}{b}ar(v{)}_y^2+\frac{1}{2}\frac{m}{b}ar(v{)}_z^2+\frac{1}{2}\frac{\text{I}}{b}ar({)}_x^2+\frac{1}{2}\frac{\text{I}}{b}ar({)}_y^2$

c$)\frac{?}{\text{b}\text{a}\text{r}}(E{)}_{\text{k}\text{i}\text{n}}=\frac{1}{2}\frac{m}{b}ar(v{)}_x^2+\frac{1}{2}\frac{m}{b}ar(v{)}_y^2+\frac{1}{2}\frac{m}{b}ar(v{)}_z^2+\frac{1}{2}\frac{\text{I}}{b}ar({)}_x^2+\frac{1}{2}\frac{\text{I}}{b}ar({)}_y^2+\frac{1}{2}\frac{k}{b}ar(z{)}^2+\frac{1}{2}\frac{k}{b}ar(v{)}_{\text{o}\text{s}\text{c}}^2$

d$)\frac{?}{\text{b}\text{a}\text{r}}(E{)}_{\text{k}\text{i}\text{n}}=\frac{1}{2}\frac{m}{b}ar(v{)}_x^2+\frac{1}{2}\frac{m}{b}ar(v{)}_y^2$

e$)\frac{?}{\text{b}\text{a}\text{r}}(E_{kln})=\frac{1}{2}{k}_x{x}^2+\frac{1}{2}{k}_y{y}^2+\frac{1}{2}{k}_z{z}^2+\frac{1}{2}m{v}_{\text{o}\text{s}\text{c}}{x}^2+\frac{1}{2}m{v}_{\text{o}\text{s}\text{c}\text{y}}^2+\frac{1}{2}m{v}_{\text{o}\text{s}\text{c}\text{z}}^2$

i$)$ Find the total average energy of each of these molecules

ii$)$ Identify $C_v$ for gas made of such molecules.

ii$)$ Find the internal energy of $1$ mole of each of these gases