Question

Use the Gibbs function to determine the equilibrium constant of the \(\mathrm{H}_{2} \mathrm{O} \rightleftharpoons \mathrm{H}_{2}+\frac{1}{2} \mathrm{O}_{2}\) reaction at \((a) 1000 \mathrm{K}\) and \((b) 2000 \mathrm{K} .\) How do these compare to the equilibrium constants of Table \(\mathrm{A}-28 ?\)

Short answer

Question: Calculate the equilibrium constants, \(K_P\), for the reaction \(\mathrm{H}_{2} \mathrm{O} \rightleftharpoons \mathrm{H}_{2}+\frac{1}{2} \mathrm{O}_{2}\) at \(1000 \mathrm{K}\) and \(2000 \mathrm{K}\) using the Gibbs free energy values from Table A-28 and compare them to the equilibrium constants provided in the table.

Step-by-step solution

Write down the Gibbs Free Energy equation for the reaction

The change in Gibbs free energy for the given reaction can be calculated using the following equation: \(\Delta G = G_{products} - G_{reactants}.\)

Obtain the Gibbs Free Energy values for the individual substances at the given temperatures from Table A-28

Before calculating the change in Gibbs free energy, we must obtain the Gibbs free energy (\(G\)) for each substance involved in the reaction (i.e., \(\mathrm{H}_{2} \mathrm{O}, \mathrm{H}_{2}\) and \(\frac{1}{2} \mathrm{O}_{2}\)) at the given temperatures (1000 K and 2000 K) from Table A-28.

Calculate the change in Gibbs Free Energy, \(\Delta G\) ,for each temperature

Using the Gibbs free energy values obtained from Table A-28, we can calculate the \(\Delta G\) for the reaction at each temperature: a) For \(T = 1000 \mathrm{K}:\) \(\Delta G_{1000} = G_{\mathrm{H}_{2}} + \frac{1}{2} G_{\mathrm{O}_{2}} - G_{\mathrm{H}_{2}\mathrm{O}}\) b) For \(T = 2000 \mathrm{K}:\) \(\Delta G_{2000} = G_{\mathrm{H}_{2}} + \frac{1}{2} G_{\mathrm{O}_{2}} - G_{\mathrm{H}_{2}\mathrm{O}}\)

Calculate the equilibrium constant, \(K_P\), using the Gibbs Free Energy values and given temperatures

Now, we can find the equilibrium constant (\(K_P\)) for each temperature using the equation: \(K_P = e^{-\frac{\Delta G}{RT}}.\) a) For \(T = 1000 \mathrm{K}:\) \(K_{P_{1000}} = e^{-\frac{\Delta G_{1000}}{R(1000K)}}\) b) For \(T = 2000 \mathrm{K}:\) \(K_{P_{2000}} = e^{-\frac{\Delta G_{2000}}{R(2000K)}}\) Here, \(R\) is the universal gas constant and must be in the appropriate units to match those used for the Gibbs free energy values.

Compare the calculated equilibrium constants to the ones in Table A-28

Once you have calculated the equilibrium constants \(K_{P_{1000}}\) and \(K_{P_{2000}}\) at the given temperatures, you can compare them to the equilibrium constants found in Table A-28 to check your results.