Question

For the reaction \(A + B \rightleftharpoons 2C\) the value of equilibrium constant is 100 at 298 K. If the initial concentration of all the three species is 1 M each, then the equilibrium concentration of C is \(x \times 10^{-1}\) M. The value of \(x\) is ________. (Nearest integer)

Hint:

If the initial concentrations are the same and \(K_c\) is a perfect square, taking the square root simplifies the calculation significantly.

Answer 1

Correct Answer: 25

Step 1: Understanding the Concept:
At equilibrium, the concentrations of reactants and products are related by the equilibrium constant \(K_c\).
We must first compare the reaction quotient \(Q_c\) with \(K_c\) to determine the direction of the reaction.
Step 2: Key Formula or Approach:
1. \(Q_c = \frac{[C]^2}{[A][B]}\).
2. Equilibrium expression: \(K_c = \frac{[C]_{eq}^2}{[A]_{eq}[B]_{eq}}\).
Step 3: Detailed Explanation:
1. Determine Direction:
Initial: \([A]=1, [B]=1, [C]=1\).
\(Q_c = \frac{1^2}{1 \times 1} = 1\).
Since \(Q_c<K_c\) (\(1<100\)), the reaction will proceed in the forward direction.

2. Equilibrium Concentration Setup:
Let \(y\) be the change in concentration of A and B.
At equilibrium: \([A] = 1 - y\), \([B] = 1 - y\), \([C] = 1 + 2y\).
\[ 100 = \frac{(1 + 2y)^2}{(1 - y)^2} \]
Taking square root on both sides:
\[ 10 = \frac{1 + 2y}{1 - y} \Rightarrow 10 - 10y = 1 + 2y \Rightarrow 12y = 9 \Rightarrow y = 0.75 \]

3. Final Equilibrium Concentration of C:
\[ [C]_{eq} = 1 + 2y = 1 + 2(0.75) = 2.5 \text{ M} \]
\[ [C]_{eq} = 25 \times 10^{-1} \text{ M} \]
So, \(x = 25\).
Step 4: Final Answer:
The value of \(x\) is 25.