Question

One mole of \(\mathrm{H}_{2} \mathrm{O}\) is heated to \(3400 \mathrm{K}\) at a pressure of 1 atm. Determine the equilibrium composition, assuming that only \(\mathrm{H}_{2} \mathrm{O}, \mathrm{OH}, \mathrm{O}_{2},\) and \(\mathrm{H}_{2}\) are present.

Short answer

Question: Determine the equilibrium composition of a mixture of H2O, OH, O2, and H2 when one mole of H2O is heated at a temperature of 3400 K and at a pressure of 1 atm. Answer: To determine the equilibrium composition of the mixture, follow these steps: 1. Write down the chemical reactions, given species, and initial conditions. 2. Write down the equilibrium constant expressions. 3. Define the changes in concentration. 4. Substitute the final concentrations into the equilibrium expressions and solve for the unknowns. 5. Calculate the equilibrium composition using the solved values. Once you have calculated the equilibrium concentrations of each species, the equilibrium composition of the mixture at 3400 K and 1 atm can be determined.

Step-by-step solution

Write down the chemical reactions, given species and initial conditions

We have the following two reactions happening in this system: 1. \(\mathrm{H}_{2}\mathrm{O} \rightleftharpoons \mathrm{OH} + \mathrm{H}\) 2. \(\mathrm{H}_{2} \rightleftharpoons 2\mathrm{H}\) Given species: H2O, OH, O2, and H2 Initial conditions: 1 mole of H2O, at 3400 K and 1 atm pressure

Write down the equilibrium constant expressions

Using the equilibrium constants for each reaction, we can write the equilibrium expressions: For reaction 1: \[K_1 = \frac{[\mathrm{OH}][\mathrm{H}]}{[\mathrm{H}_2\mathrm{O}]}\] For reaction 2: \[K_2 = \frac{[\mathrm{H}]^2}{[\mathrm{H}_2]}\]

Define the changes in concentration

Let x moles of H2O dissociate into OH and H. Then the initial and final concentrations will be: Initial: [H2O] = 1, [OH] = 0, [H] = 0 Final: [H2O] = 1 - x, [OH] = x, [H] = x for reaction 1 Final: [H] = 2y, [H2] = 1 - y for reaction 2

Substitute final concentrations into the equilibrium expressions and solve

We can now substitute the final concentrations into the equilibrium expressions: For reaction 1: \[K_1 = \frac{x^2}{1 - x}\] For reaction 2: \[K_2 = \frac{(2y)^2}{1 - y}\] Since H is involved in both reactions, we can write a relation between x and y: \[x = 2y\] Now, we need to find the values of K1 and K2 at 3400 K using thermodynamic tables or data. (Note: The actual values depend on the reference materials and are not included in this explanation.) With K1 and K2 known, substitute the values into the expressions and use the relation between x and y to solve for x and y.

Calculate the equilibrium composition

Once the values of x and y are obtained, we can find the equilibrium concentrations of all species: [H2O] = 1 - x [OH] = x [H] = x or 2y [H2] = 1 - y These concentrations represent the equilibrium composition of the mixture when 1 mole of H2O is heated to 3400 K at a pressure of 1 atm.