Question

A mixture of 2 mol of \(\mathrm{H}_{2} \mathrm{O}\) and \(3 \mathrm{mol}\) of \(\mathrm{O}_{2}\) is heated to \(3600 \mathrm{K}\) at a pressure of 8 atm. Determine the equilibrium composition of the mixture, assuming that only \(\mathrm{H}_{2} \mathrm{O}, \mathrm{OH}, \mathrm{O}_{2}\), and \(\mathrm{H}_{2}\) are present.

Short answer

Based on the given data and the assumption that Reaction 1 is the most dominant, we could determine the equilibrium composition of the mixture if we had the temperature dependence of the equilibrium constant for Reaction 1. However, due to the lack of this information, we cannot solve the problem completely, and thus cannot determine the final equilibrium composition of the mixture containing H2O, OH, O2, and H2.

Step-by-step solution

Determine the chemical reactions and equilibrium constants

To begin, let's write the dissociation reactions involving the four given species, OH, O2, H2, and H2O: 1. \(\mathrm{H}_{2}\mathrm{O}\rightleftharpoons\mathrm{OH}+\mathrm{H}_{2}\) (Reaction 1) 2. \(\mathrm{H}_{2}\mathrm{O}\rightleftharpoons\mathrm{O}_{2}+\mathrm{H}_{2}\) (Reaction 2) Since the process takes place at high temperature, let's assume that dissociation of water molecule into its component particles is negligible. Furthermore, let's assume that reaction 1 is most dominant. Therefore, we will focus on Reaction 1. The equilibrium constant \(K_1\) for Reaction 1 can be expressed as follows: \(K_1 = \frac{[\mathrm{OH}][\mathrm{H}_2]}{[\mathrm{H}_2\mathrm{O}]}\) Next, we need to find the temperature dependence of the equilibrium constant. For that, we can use the van't Hoff equation: \(\Delta G^{\circ} = -RT\ln{K}\) where \(\Delta G^{\circ}\) is the standard Gibbs free energy of the reaction, \(R\) is the ideal gas constant, and \(T\) is the temperature in Kelvin. However, we need \(\Delta G^{\circ}\) for reaction 1 at \(3600K\), which isn't provided. Therefore, we need to make a simplifying assumption. Let's assume that the system is nearly ideal and the fugacities can be replaced with the partial pressures, then we can determine \(K_1\) by using \(KP\) instead of fugacity coefficients.

Set up a system of equations

First, let's convert the initial total number of moles into the partial pressures: \(P_{\mathrm{H}_2\mathrm{O}} = \frac{2\ mol}{8\ atm} = 0.25\ atm\) \(P_{\mathrm{O}_2} = \frac{3\ mol}{8\ atm} = 0.375\ atm\) Now, let's set up the equilibrium constant expression for Reaction 1: \(K_1 = \frac{(P_{\mathrm{OH}})(P_{\mathrm{H}_2})}{P_{\mathrm{H}_2\mathrm{O}}}\) Since only \(\mathrm{H}_{2}\mathrm{O}\) is consumed during reaction, we can write the change of partial pressures as follows: \(P_{\mathrm{H}_2\mathrm{O}} = 0.25 - x\) \(P_{\mathrm{OH}} = x\) \(P_{\mathrm{H}_2} = x\) Now, substitute these relationships into the equilibrium constant expression: \(K_1 = \frac{x^2}{0.25 - x}\)

Solve the equations for the equilibrium composition

In this case, we don't have the value of \(K_{1}\), given the lack of information on the temperature dependence of the equilibrium constant for Reaction 1. Nonetheless, if we had a numerical value for \(K_{1}\), we could proceed as follows: 1. Substitute the numerical value of \(K_{1}\) into the equation: \(K_{1} = \frac{x^2}{0.25 - x}\) 2. Solve this equation for \(x\). This can be done through algebraic manipulation or with the help of a numerical solver. 3. Obtain the final equilibrium composition of the mixture. The equilibrium partial pressures of \(\mathrm{H}_2\mathrm{O}\), \(\mathrm{OH}\), and \(\mathrm{H}_2\) can then be determined using the relationships \(P_{\mathrm{H}_2\mathrm{O}} = 0.25 - x\), \(P_{\mathrm{OH}} = x\), and \(P_{\mathrm{H}_2} = x\) This method allows us to determine the equilibrium composition of the mixture of species \(\mathrm{H}_{2}\mathrm{O}\), \(\mathrm{OH}\), \(\mathrm{O}_{2}\), and \(\mathrm{H}_{2}\) at the given temperature and pressure, assuming that Reaction 1 is the most dominant reaction in the system.