A gaseous mixture contains 0.30 moles , 0.10 moles , and 0.03 moles vapour and an unknown amount of — Chemical Equilibrium Chemistry Question

💡 Solution & Explanation

Step 1 - Express the Equilibrium Constant ($K_c$) The balanced chemical equation represents the reaction between carbon monoxide (\ce{CO}) and hydrogen (\ce{H2}) to produce methane (\ce{CH4}) and water vapor (\ce{H2O}): \[\ce{CO(g) + 3H2(g) <=> CH4(g) + H2O(g)}\] According to the law of chemical equilibrium, the concentration-based equilibrium constant ($K_c$) is written as: \[K_c = \frac{[\ce{CH4}][\ce{H2O}]}{[\ce{CO}][\ce{H2}]^3}\] Step 2 - Identify the Equilibrium Concentrations Since the volume of the mixture is $1\text{ L}$ (indicated by "per litre" in the question statement), the molar concentration ($[\text{C}] = \frac{n}{V}$) of each species is numerically equal to its number of moles: * Concentration of carbon monoxide, $[\ce{CO}] = \frac{0.30\text{ mol}}{1\text{ L}} = 0.30\text{ M}$ * Concentration of hydrogen, $[\ce{H2}] = \frac{0.10\text{ mol}}{1\text{ L}} = 0.10\text{ M}$ * Concentration of water vapor, $[\ce{H2O}] = \frac{0.03\text{ mol}}{1\text{ L}} = 0.03\text{ M}$ * Concentration of methane, $[\ce{CH4}] = x\text{ M}$ (unknown) Step 3 - Substitute Values and Calculate the Concentration of \ce{CH4} We are given the equilibrium constant $K_c = 3.9$ at $1200\text{ K}$. Substituting these values into our equilibrium constant expression: \[3.9 = \frac{x \times 0.03}{0.30 \times (0.10)^3}\] Now, let us simplify the calculation: 1. Calculate the cube of the hydrogen concentration: \[(0.10)^3 = 0.001\text{ M}^3\] 2. Simplify the denominator: \[0.30 \times (0.10)^3 = 0.30 \times 0.001 = 0.0003\text{ M}^4\] 3. Set up the simplified equation: \[3.9 = \frac{0.03 \cdot x}{0.0003}\] 4. Calculate the ratio in the fraction: \[\frac{0.03}{0.0003} = 100\] 5. Solve for $x$: \[3.9 = 100x\] \[x = \frac{3.9}{100} = 0.039\text{ M}\] Thus, the equilibrium concentration of methane in the mixture is: \[[\ce{CH4}] = \boxed{0.039\text{ M}}\] Step 4 - Evaluate the Options * **Option (A) $0.39\text{ M}$**: Incorrect. This represents an arithmetic error, likely due to a miscalculation by a factor of 10 during division. * **Option (B) $0.039\text{ M}$**: Correct. As calculated, substituting the correct molar concentrations and solving the equation yields exactly $0.039\text{ M}$. * **Option (C) $0.78\text{ M}$**: Incorrect. This value represents a scale error of doubling the correct answer. * **Option (D) $0.078\text{ M}$**: Incorrect. This value represents an arithmetic error.