In the Wilhelmey equation of a first order reaction `c_(t) = c_(0)e^(-kt)`. If the initial concentration `c_(0)` is increased `m` times, then

Options

A. The value of `k` will increase `m` times
B. The value of `k` will decrease `m` times
C. The value of `k` will remain unchanged
D. None of these

Explanation

To solve the problem, we need to analyze the effect of increasing the initial concentration \( c_0 \) on the rate constant \( k \) in the context of a first-order reaction described by the equation:

\[ c_t = c_0 e^{-kt} \]

### Step-by-Step Solution:

1. **Understand the First-Order Reaction**:
The given equation \( c_t = c_0 e^{-kt} \) describes the concentration of a reactant at time \( t \) for a first-order reaction. Here, \( c_0 \) is the initial concentration, \( c_t \) is the concentration at time \( t \), and \( k \) is the rate constant.

**Hint**: Recall that in a first-order reaction, the rate of reaction depends linearly on the concentration of one reactant.

2. **Effect of Increasing Initial Concentration**:
The problem states that the initial concentration \( c_0 \) is increased \( m \) times, which means the new initial concentration becomes \( mc_0 \).

**Hint**: Consider how the rate of reaction is defined for first-order kinetics.

3. **Rate of Reaction**:
For a first-order reaction, the rate \( r \) is given by:
\[ r = k [A] \]
where \( [A] \) is the concentration of the reactant. If we increase \( c_0 \) to \( mc_0 \), the rate becomes:
\[ r' = k (mc_0) \]
However, the rate constant \( k \) itself is a characteristic of the reaction and does not change with concentration.

**Hint**: Remember that the rate constant \( k \) is independent of the concentration for first-order reactions.

4. **Conclusion**:
Since the rate constant \( k \) is independent of the initial concentration in a first-order reaction, increasing \( c_0 \) to \( mc_0 \) does not affect the value of \( k \). Therefore, the value of \( k \) remains constant.

**Final Answer**: The value of \( k \) will remain unchanged.

### Summary of the Answer:
When the initial concentration \( c_0 \) is increased \( m \) times in a first-order reaction, the rate constant \( k \) remains constant.