Question

The following reaction is at equilibrium: $$ \mathrm{CH}_{4}+2 \mathrm{Cl}_{2} \rightleftarrows \mathrm{CH}_{2} \mathrm{Cl}_{2}+2 \mathrm{HCl} $$ If \(\left[\mathrm{CH}_{4}\right]\) is \(0.250 \mathrm{M},\left[\mathrm{Cl}_{2}\right]\) is \(0.150 \mathrm{M},\) and \(\left[\mathrm{CH}_{2} \mathrm{Cl}_{2}\right]\) is \(0.175 \mathrm{M}\) at equilibrium, what is \([\mathrm{HCl}] \mathrm{at}\) equilibrium if the \(K_{\text {eq }}\) is \(2.30 ?\)

Short answer

[HCl] at equilibrium is approximately 0.272 M.

Step-by-step solution

Write the Equilibrium Expression

The equilibrium constant expression for the given chemical reaction is:\[ K_{eq} = \frac{[\mathrm{CH}_{2} \mathrm{Cl}_{2}][\mathrm{HCl}]^2}{[\mathrm{CH}_{4}][\mathrm{Cl}_{2}]^2} \]

Plug Known Concentrations into the Expression

We know the equilibrium concentrations are:\[ [\mathrm{CH}_{4}] = 0.250 \text{ M}, \quad [\mathrm{Cl}_{2}] = 0.150 \text{ M}, \quad [\mathrm{CH}_{2} \mathrm{Cl}_{2}] = 0.175 \text{ M} \] Plug these values into the equilibrium expression:\[ 2.30 = \frac{0.175 \cdot [\mathrm{HCl}]^2}{0.250 \cdot (0.150)^2} \]

Solve for \\[ \mathrm{[HCl] } \]\

First, compute the denominator:\[ 0.250 \times (0.150)^2 = 0.250 \times 0.0225 = 0.005625 \]Now set up the equation:\[ 2.30 = \frac{0.175 \times [\mathrm{HCl}]^2}{0.005625} \]Multiply both sides by 0.005625:\[ 2.30 \times 0.005625 = 0.175 \times [\mathrm{HCl}]^2 \]Solve for \( [\mathrm{HCl}]^2 \):\[ 0.0129375 = 0.175 \cdot [\mathrm{HCl}]^2 \]\[ [\mathrm{HCl}]^2 = \frac{0.0129375}{0.175} \approx 0.07393 \]Finally, take the square root to solve for \( [\mathrm{HCl}] \):\[ [\mathrm{HCl}] = \sqrt{0.07393} \approx 0.272 \text{ M} \]

Key concepts

Equilibrium Constant (K_eq)

The equilibrium constant, denoted as \( K_{eq} \), is a measure of the ratio of products to reactants at equilibrium for a reversible chemical reaction. This constant helps in predicting the direction of the reaction and the extent to which reactants are converted to products. For the given reaction, the equilibrium expression reflects this balance:

• \( K_{eq} = \frac{[\mathrm{CH}_{2} \mathrm{Cl}_{2}][\mathrm{HCl}]^2}{[\mathrm{CH}_{4}][\mathrm{Cl}_{2}]^2} \)

In any reaction at equilibrium, \( K_{eq} \) will remain constant at a given temperature because the reaction rates of the forward and reverse processes are equal.
Understanding \( K_{eq} \) is crucial as it allows chemists to calculate unknown concentrations, which in turn helps in designing chemical processes and understanding natural systems. A high \( K_{eq} \) value much greater than 1 indicates a reaction favoring product formation, while a small \( K_{eq} \) much less than 1 signifies that reactants are favored.

Concentration Calculation

Calculating equilibrium concentrations involves using the known values and the equilibrium constant. For any reversible reaction at equilibrium, you can determine the unknown concentration if you have all other concentrations and the value of \( K_{eq} \).
Let's look at the example problem:

• Given \( K_{eq} = 2.30 \)
• Known concentrations: \([\mathrm{CH}_{4}] = 0.250 \text{ M}, [\mathrm{Cl}_{2}] = 0.150 \text{ M}, [\mathrm{CH}_{2} \mathrm{Cl}_{2}] = 0.175 \text{ M}\)

First, all known concentrations are placed in the equilibrium expression, leaving out the unknown \( [\mathrm{HCl}] \). By rearranging this expression and solving algebraically, you can determine the equilibrium concentration of \( [\mathrm{HCl}] \).
This process requires basic algebra skills, utilizing actions such as multiplication, division, and taking square roots to retrieve the unknown value. This is achieved by isolating \( [\mathrm{HCl}] \) on one side of the equation, calculating the value step by step.

Le Chatelier's Principle

Le Chatelier's Principle is a fundamental concept that describes how a system at equilibrium responds to disturbances or changes. When a system is subjected to a change in concentration, temperature, or pressure, the principle predicts how the balance will shift to relieve that change.
For instance, in the reaction of \(\mathrm{CH}_{4}+2 \mathrm{Cl}_{2} \rightleftarrows \mathrm{CH}_{2} \mathrm{Cl}_{2}+2 \mathrm{HCl}\), if we increase the concentration of \( \mathrm{CH}_{4} \), the system will shift towards producing more products to counteract the disturbance.

• Addition of more reactants drives the equilibrium to form more products.
• Increasing temperature or pressure may also shift the equilibrium based on whether the reaction is exothermic or endothermic.

This principle is incredibly useful in industrial and laboratory processes, helping chemists optimize conditions to maximize product yields or minimize waste.