At \(1000 \mathrm{K},\) an equilibrium mixture in the reaction \(\mathrm{CO}_{2}(\mathrm{g})+\mathrm{H}_{2}(\mathrm{g}) \rightleftharpoons \mathrm{CO}(\mathrm{g})+\mathrm{H}_{2} \mathrm{O}(\mathrm{g}) \quad\) contains \(0.276 \mathrm{mol} \quad \mathrm{H}_{2}, 0.276 \mathrm{mol} \mathrm{CO}_{2}, \quad 0.224 \mathrm{mol} \mathrm{CO}, \quad\) and \(0.224 \mathrm{mol} \mathrm{H}_{2} \mathrm{O}\) (a) What is \(K_{\mathrm{p}}\) at \(1000 \mathrm{K} ?\) (b) Calculate \(\Delta G^{\circ}\) at \(1000 \mathrm{K}\). (c) In which direction would a spontaneous reaction occur if the following were brought together at 1000 K: \(0.0750 \mathrm{mol} \mathrm{CO}_{2}, 0.095 \mathrm{mol} \mathrm{H}_{2}, 0.0340 \mathrm{mol} \mathrm{CO},\) and \(0.0650 \mathrm{mol} \mathrm{H}_{2} \mathrm{O} ?\)

Step by step solution

01

Preparing for calculating \(K_p\)

To find the equilibrium constant \(K_p\) for the reaction, we first need to notice that we have 4 gases and for gases, a reaction quotient can be expressed in terms of partial pressures. We are given equilibrium number of moles for all gases.

02

Calculate \(K_p\)

The reaction quotient (Q), is defined as \[ Q_p = \frac{P_{CO} P_{H_2O}}{P_{CO2} P_{H2}} \]. On the equilibrium, Q_p becomes the equilibrium constant \(K_p\). The total number of moles is 0.276 + 0.276 + 0.224 + 0.224 = 1. Using ideal gas law, the total pressure for 1 mol at 1000K can be calculated as follows \[ P_{tot}= nRT/V = R*1000/1 =1000R \] and the partial pressure for each gas would be amount of gas in mol times the total pressure. So, \(P_{H2} = \frac{0.276}{1}*1000R = 276R\), \(P_{CO2} = \frac{0.276}{1}*1000R = 276R\), \(P_{CO} = \frac{0.224}{1}*1000R = 224R\) and \(P_{H2O} = \frac{0.224}{1}*1000R = 224R\). Now, calculating \(K_p\) using these values in the equation \[ K_p = \frac{P_{CO} P_{H2O}}{P_{CO2} P_{H2}} = \frac{224R*224R}{276R*276R} = 0.656814 \]

03

Calculate \(\Delta G^\circ\)

Using the relationship between \(K_p\) and \(\Delta G^\circ\) which is \(\Delta G^\circ = -RT \ln K_p\), with R as 8.314 and T at 1000K, we have: \[\Delta G^\circ = -8.314*1000*ln(0.656814) = 2047.85 \ J/mol\]

04

Predict the direction of spontaneous reaction

Here, we again calculate \(Q_p\) using the given number of moles in the reaction mixture using the formula \[ Q_p = \frac{P_{CO} P_{H2O}}{P_{CO2} P_{H2}} \]. After calculating each pressure using the same process as in step 2, we get \(Q_p = 0.294703\). As \(Q_p < K_p\), according to Le Chatelier's principle, the reaction will shift towards right and hence the reaction is spontaneous in the forward direction.

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