Question

For the reaction \( P + Q \rightleftharpoons R + S \), carried out at 298 K, the equilibrium constant was found to be 169 and the initial concentrations of all reactants and products was 1.0 M. What is the equilibrium concentration of the reactants?

• A. 0.143 M
• B. 0.462 M
• C. 0.857 M
• D. 0.698 M

Hint:

When solving equilibrium problems, the ICE method is helpful for tracking concentration changes, and equilibrium constants can be used to find unknown concentrations.

Answer 1

The Correct Option is A

Step 1: Write the expression for the equilibrium constant.
The equilibrium constant \( K \) is given by:
\[ K = \frac{[R][S]}{[P][Q]}. \]
Given that \( K = 169 \) and the initial concentrations of all reactants and products are 1.0 M, we can use the ICE (Initial, Change, Equilibrium) method to find the equilibrium concentrations.

Step 2: Set up the ICE table.
Let the change in concentration for \( P \) and \( Q \) be \( -x \) and the change for \( R \) and \( S \) be \( +x \). Thus, at equilibrium:
- \( [P] = 1 - x \)
- \( [Q] = 1 - x \)
- \( [R] = x \)
- \( [S] = x \)

Step 3: Use the equilibrium expression.
Substitute the equilibrium concentrations into the equilibrium expression:
\[ 169 = \frac{x^2}{(1 - x)^2}. \]

Step 4: Solve the quadratic equation.
Take the square root of both sides:
\[ \sqrt{169} = \frac{x}{1 - x} \quad \Rightarrow \quad 13 = \frac{x}{1 - x}. \]
Solving for \( x \): \[ 13(1 - x) = x \quad \Rightarrow \quad 13 - 13x = x \quad \Rightarrow \quad 13 = 14x \quad \Rightarrow \quad x = \frac{13}{14} = 0.143. \]

Step 5: Calculate the equilibrium concentration of the reactants.
The equilibrium concentration of the reactants is \( [P] = 1 - x = 1 - 0.143 = 0.857 \, \text{M} \). Thus, the equilibrium concentration of the reactants is 0.143 M.

Step 6: Conclusion.
The equilibrium concentration of the reactants is 0.143 M, so the correct answer is option (A).