Question

A gaseous mixture of 30 percent (by mole fraction) methane and 70 percent carbon dioxide is heated at 1 atm pressure to \(1200 \mathrm{K}\). What is the equilibrium composition (by mole fraction) of the resulting mixture? The natural logarithm of the equilibrium constant for the reaction \(\mathrm{C}+2 \mathrm{H}_{2} \rightleftharpoons \mathrm{CH}_{4}\) at \(1200 \mathrm{K}\) is 4.147.

Short answer

Answer: The equilibrium mole fractions for the mixture are approximately: Carbon (C): 5.7% Hydrogen (H2): 21.4% Methane (CH4): 24.3%

Step-by-step solution

Identify the reaction and given information

The given reaction is \(\mathrm{C} + 2\mathrm{H}_{2} \rightleftharpoons \mathrm{CH}_{4}\). We know that the initial mole fraction of methane (CH4) is 30% and that of carbon dioxide (CO2) is 70%. The temperature is 1200 K, and the natural logarithm of the equilibrium constant (K) at this temperature is 4.147.

Calculate the equilibrium constant (K)

To find the equilibrium constant K, we use the given natural logarithm value: ln(K) = 4.147 Therefore, K = exp(4.147) ≈ 63.25

Define the variables for the reaction

Let's define the change in moles for each species involved in the reaction. Let x represent the moles of CH4 at equilibrium. Then, since the initial mole fraction for CH4 is 30%, we can express x as 0.3 - x for carbon (C), 0.7 - 2x for H2, and x for CH4.

Set up the equilibrium expression

Using the reaction and the equilibrium constant, set up the equilibrium expression for the reaction: K = \(\frac{[\mathrm{CH}_4]}{[\mathrm{C}][\mathrm{H}_2]^2} = \frac{x}{(0.3 - x)(0.7 - 2x)^2}\)

Solve for x

Now, we need to solve for x: 63.25 = \(\frac{x}{(0.3 - x)(0.7 - 2x)^2}\) We can solve this equation numerically to find the value of x: x ≈ 0.243

Calculate the equilibrium mole fractions

Since we now have obtained the value of x, we can calculate the equilibrium mole fractions for carbon (C), hydrogen (H2), and methane (CH4): Mole fraction of Carbon (C) = 0.3 - x = 0.3 - 0.243 ≈ 0.057 Mole fraction of Hydrogen (H2) = 0.7 - 2x = 0.7 - 2(0.243) ≈ 0.214 Mole fraction of Methane (CH4) = x ≈ 0.243

Present the final equilibrium mole fractions

The equilibrium composition (by mole fraction) of the resulting mixture is: Carbon (C): ≈ 5.7% Hydrogen (H2): ≈ 21.4% Methane (CH4): ≈ 24.3%