The heat of formation of $C_{2} H_{5} OH (l) is - 66 kcal / mol .$ The heat of combustion of ${CH}_{3} {OCH}_{3} (g) is - 348 kcal / mol . ∆ H_{f} for H_{2} O and {CO}_{2} are - 68 kcal / mol and - 94 kcal / mol$ respectively. Then, the $∆ H$ for the isomerisatin reaction $C_{2} H_{5} OH (l) \to {CH}_{3} {OCH}_{3} (g), and ∆ E$ for the same are at $T = 25^{0} C$

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$∆ H = 18 kcal / mol, ∆ E = 17.301 kcal / mol$

$∆ H = 22 kcal / mol, ∆ E = 21.408 kcal / mol$

$∆ H = 26 kcal / mol, ∆ E = 25.709 kcal / mol$

$∆ H = 30 kcal / mol, ∆ E = 28.522 kcal / mol$

answer is B.

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Detailed Solution

$∆ H_{f}^{0} C_{2} H_{5} OH (l) = - 66 kcal / mol 2 C + 3 H_{2} + \frac{1}{2} O_{2} \to C_{2} H_{5} OH; ∆ H = - 66 kcal / mol . . . . . . (1) {CH}_{3} - O - {CH}_{3} + 3 O_{2} \to 2 {CO}_{2} + 3 H_{2} O ∆ H = - 348 kcal / mol . . . . . . (2) [H_{2} (g) + O_{2} (g) \to H_{2} O] ∆ H_{3} = - 68 kcal / mol . . . (3) [C + \frac{1}{2} O_{2} \to {CO}_{2}] ∆ H_{4} = - 9 kcal / mol . . . . . . (4)$

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

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