The heat of formation of ${\mathrm{C}}_2{\mathrm{H}}_5\mathrm{OH}\left(\mathrm{l}\right) \mathrm{is} -66 \mathrm{kcal}/\mathrm{mol}.$ The heat of combustion of ${\mathrm{CH}}_3{\mathrm{OCH}}_3\left(\mathrm{g}\right) \mathrm{is} -348 \mathrm{kcal}/\mathrm{mol}. ∆{\mathrm{H}}_{\mathrm{f}} \mathrm{for} {\mathrm{H}}_2\mathrm{O} \mathrm{and} {\mathrm{CO}}_2 \mathrm{are} -68 \mathrm{kcal}/\mathrm{mol} \mathrm{and} -94 \mathrm{kcal}/\mathrm{mol}$ respectively. Then, the $∆\mathrm{H}$ for the isomerisatin reaction ${\mathrm{C}}_2{\mathrm{H}}_5\mathrm{OH}\left(\mathrm{l}\right) \to {\mathrm{CH}}_3{\mathrm{OCH}}_3\left(\mathrm{g}\right), \mathrm{and} ∆\mathrm{E}$ for the same are at $\mathrm{T}={25}^0\mathrm{C}$

$$\begin{gathered} ∆{\mathrm{H}}_{\mathrm{f}}^0{\mathrm{C}}_2{\mathrm{H}}_5\mathrm{OH}\left(\mathrm{l}\right) = -66 \mathrm{kcal}/\mathrm{mol} \\ 2\mathrm{C}+3{\mathrm{H}}_2+\frac{1}{2}{\mathrm{O}}_2 \to {\mathrm{C}}_2{\mathrm{H}}_5\mathrm{OH}; \\ ∆\mathrm{H}=-66 \mathrm{kcal}/\mathrm{mol} ......\left(1\right) \\ {\mathrm{CH}}_3-\mathrm{O}-{\mathrm{CH}}_3+3{\mathrm{O}}_2 \to 2{\mathrm{CO}}_2+3{\mathrm{H}}_2\mathrm{O} \\ ∆\mathrm{H}=-348 \mathrm{kcal}/\mathrm{mol} ......\left(2\right) \\ \left[{\mathrm{H}}_2\left(\mathrm{g}\right)+{\mathrm{O}}_2\left(\mathrm{g}\right) \to {\mathrm{H}}_2\mathrm{O}\right]∆{\mathrm{H}}_3=-68 \mathrm{kcal}/\mathrm{mol} ...\left(3\right) \\ \left[\mathrm{C}+\frac{1}{2}{\mathrm{O}}_2 \to {\mathrm{CO}}_2\right] \\ ∆{\mathrm{H}}_4=-9 \mathrm{kcal}/\mathrm{mol} ......\left(4\right) \end{gathered}$$

Target equation = -eq. 1 - eq. 2 + 3 eq. 3 + 2eq. 2

$$\begin{gathered} ∆\mathrm{H}=+66+348-3 \times 68-2\times 94=+66+348-204-188; ∆\mathrm{H}=22 \mathrm{kcal}/\mathrm{mol} \\ ∆\mathrm{H}=∆\mathrm{E}+∆{\mathrm{n}}_{\mathrm{g}}\mathrm{RT}\Rightarrow 22 = ∆\mathrm{E} +1 \times 2 \times 298 \times {10}^{-3}; ∆\mathrm{E} = 21.4 \mathrm{kcal}/\mathrm{mol} \end{gathered}$$