Question

For the reaction $$ \mathrm{CO}(g)+3 \mathrm{H}_{2}(g) \rightleftharpoons \mathrm{CH}_{4}(g)+\mathrm{H}_{2} \mathrm{O}(g) $$ \(K=2.2 \times 10^{11}\) at \(473 \mathrm{~K}\) and \(4.6 \times 10^{8}\) at \(533 \mathrm{~K} .\) Calculate \(\Delta G^{\circ}\) at both temperatures.

Short answer

Answer: The standard change in Gibbs free energy at 473 K is approximately -1.527 x 10^4 J/mol and at 533 K is approximately -1.229 x 10^4 J/mol.

Step-by-step solution

Write down the given constants and equation

We are given the equilibrium constants (\(K\)) for two temperatures and we have the gas constant (\(R\)). We are also given the relationship between the standard change in Gibbs free energy and the equilibrium constant. $$ K_{1} = 2.2 \times 10^{11} \,\,\text{at}\,\, T_{1} = 473\,\mathrm{K} $$ $$ K_{2} = 4.6 \times 10^{8} \,\,\text{at}\,\, T_{2} = 533\,\mathrm{K} $$ $$ R = 8.314\,\mathrm{J\, mol^{-1} K^{-1}} $$ $$ \Delta G^{\circ} = -RT\ln K $$

Calculate ΔG° at T₁ = 473 K

Now, using the equation, we can plug in the values of \(R\), \(T_{1}\), and \(K_{1}\) to find the standard change in Gibbs free energy at \(473\,\mathrm{K}\): $$ \Delta G_{1}^{\circ} = -R \times T_{1} \times \ln K_{1} $$ $$ \Delta G_{1}^{\circ} = -(8.314\,\mathrm{J\, mol^{-1} K^{-1}}) \times (473\,\mathrm{K}) \times \ln (2.2 \times 10^{11}) $$ Now, compute the result: $$ \Delta G_{1}^{\circ} \approx -1.527 \times 10^{4}\,\mathrm{J\, mol^{-1}} $$

Calculate ΔG° at T₂ = 533 K

Using the same equation, we can now plug in the values of \(R\), \(T_{2}\), and \(K_{2}\) to find the standard change in Gibbs free energy at \(533\,\mathrm{K}\): $$ \Delta G_{2}^{\circ} = -R \times T_{2} \times \ln K_{2} $$ $$ \Delta G_{2}^{\circ} = -(8.314\,\mathrm{J\, mol^{-1} K^{-1}}) \times (533\,\mathrm{K}) \times \ln (4.6 \times 10^{8}) $$ Compute the result: $$ \Delta G_{2}^{\circ} \approx -1.229 \times 10^{4}\,\mathrm{J\, mol^{-1}} $$

Present the final results

Now, we have calculated the standard change in Gibbs free energy at both temperatures. At \(473\,\mathrm{K}\): $$ \Delta G_{1}^{\circ} \approx -1.527 \times 10^{4}\,\mathrm{J\, mol^{-1}} $$ At \(533\,\mathrm{K}\): $$ \Delta G_{2}^{\circ} \approx -1.229 \times 10^{4}\,\mathrm{J\, mol^{-1}} $$