In a reversible reaction, two substances are in equilibrium. If the concentration of each one is reduced to half, the equilibrium constant will be

Options

A. reduced to half of its original value
B. doubled
C. same
D. reduced to one fourth its original value.

Explanation

To solve the problem, we need to understand the concept of equilibrium constant (K) in a reversible reaction. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Equilibrium Constant
The equilibrium constant (K) for a reversible reaction is defined as the ratio of the concentrations of the products to the concentrations of the reactants, each raised to the power of their coefficients in the balanced equation. It is given by the formula:

\[ K = \frac{[C]^c [D]^d}{[A]^a [B]^b} \]

where \( [A] \) and \( [B] \) are the concentrations of the reactants, and \( [C] \) and \( [D] \) are the concentrations of the products.

### Step 2: Analyze the Effect of Concentration Change
According to the problem, the concentrations of both substances (let's assume they are reactants) are reduced to half. This means if the original concentrations were \( [A] \) and \( [B] \), the new concentrations will be:

\[ [A]_{new} = \frac{[A]}{2} \]
\[ [B]_{new} = \frac{[B]}{2} \]

### Step 3: Substitute the New Concentrations into the Equilibrium Expression
Now, we substitute these new concentrations into the equilibrium expression:

\[ K_{new} = \frac{[C]^c [D]^d}{\left(\frac{[A]}{2}\right)^a \left(\frac{[B]}{2}\right)^b} \]

### Step 4: Simplify the New Equilibrium Expression
When we simplify the expression, we can factor out the \( \frac{1}{2} \):

\[ K_{new} = \frac{[C]^c [D]^d}{\frac{[A]^a}{2^a} \cdot \frac{[B]^b}{2^b}} \]

This can be rewritten as:

\[ K_{new} = K \cdot 2^{(a+b)} \]

### Step 5: Conclusion About the Equilibrium Constant
However, the equilibrium constant \( K \) is only dependent on temperature and does not change with the concentration of the reactants or products. Therefore, even though we have halved the concentrations, the equilibrium constant remains unchanged.

### Final Answer
The equilibrium constant will remain the same.