Question

If a catalyst could be found that would establish an equilibrium between 1,2 -butadiene and 2 -butyne, what would be the ratio of the more stable isomer to the less stable isomer at \(25^{\circ} \mathrm{C}\) ? $$ \mathrm{CH}_{2}=\mathrm{C}=\mathrm{CHCH}_{3} \rightleftharpoons \mathrm{CH}_{3} \mathrm{C} \equiv \mathrm{CCH}_{3} \quad \Delta \mathrm{G}^{0}=-16.7 \mathrm{~kJ}(-4.0 \mathrm{kcal}) / \mathrm{mol} $$

Short answer

Answer: The ratio of the more stable isomer to the less stable isomer at \(25^{\circ} \mathrm{C}\) for the given reaction is approximately 17.1.

Step-by-step solution

Recall the equation relating Gibbs free energy, equilibrium constant, and temperature

The equation relating \(\Delta \mathrm{G}^{0}\), equilibrium constant K, and temperature T is: $$ \Delta \mathrm{G}^{0} = -\mathrm{RT} \ln{K} $$ Where R is the universal gas constant (8.314 J/mol K) and T is the temperature in Kelvin.

Convert the given temperature to Kelvin

To use the temperature in the equation, we must first convert it from Celsius to Kelvin: $$ T(\mathrm{K}) = T(\degree \mathrm{C}) + 273.15 $$ For \(25^{\circ} \mathrm{C}\), we have: $$ T(\mathrm{K})=25+273.15=298.15 \mathrm{K} $$

Calculate the equilibrium constant K

Now, using the given value of \(\Delta \mathrm{G}^{0}=-16.7 \mathrm{kJ/mol}\) and the calculated temperature, solve for K: $$ -\mathrm{RT} \ln{K} =\Delta \mathrm{G}^{0} $$ $$ K=\exp{\left(-\frac{\Delta \mathrm{G}^{0}}{\mathrm{RT}}\right)} $$ First convert -16.7 kJ/mol to J/mol: $$ -16.7 \mathrm{~kJ/mol} \times \frac{1000 \mathrm{~J}}{1 \mathrm{~kJ}} = -16700 \mathrm{~J/mol} $$ Now substitute the values and calculate K: $$ K = \exp{\left(-\frac{-16700 \mathrm{~J/mol}}{(8.314 \mathrm{~J/mol~K})(298.15 \mathrm{~K})}\right)} \approx 17.1 $$

Determine the ratio of the more stable isomer to the less stable isomer

The equilibrium constant (K) represents the ratio of the more stable isomer to the less stable isomer at \(25^{\circ} \mathrm{C}\). Therefore, the ratio is approximately 17.1. In conclusion, at \(25^{\circ} \mathrm{C}\), the ratio of the more stable isomer to the less stable isomer in this reaction is approximately 17.1.