`C_(o)` =initial concentration of the reactant `C_(t)` = concentration of the reactant at time t, k=rate constant of the reaction. Then the equation applicable for a first order reaction is

Options

A. `C_(t) = C_(o)e^(-kt)`
B. `C_(t) = C_(o)e^(kt)`
C. `C_(o) = C_(t)e^(-kt)`
D. `(C_(0)/C_(t)) = 1`

Explanation

To derive the equation applicable for a first-order reaction using the given terms, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Terms**:
- Let \( C_0 \) be the initial concentration of the reactant.
- Let \( C_t \) be the concentration of the reactant at time \( t \).
- Let \( k \) be the rate constant of the reaction.

2. **Using the Integrated Rate Law for First-Order Reactions**:
- The integrated rate law for a first-order reaction is given by the equation:
\[ C_t = C_0 e^{-kt} \]
- This equation states that the concentration of the reactant at time \( t \) is equal to the initial concentration multiplied by the exponential of the negative rate constant times time.

3. **Rearranging the Equation**:
- To derive this equation, we can start from the definition of a first-order reaction, which can be expressed in terms of the natural logarithm:
\[ \ln\left(\frac{C_t}{C_0}\right) = -kt \]
- By exponentiating both sides, we can eliminate the logarithm:
\[ \frac{C_t}{C_0} = e^{-kt} \]
- Multiplying both sides by \( C_0 \) gives us:
\[ C_t = C_0 e^{-kt} \]

4. **Final Equation**:
- Thus, the equation applicable for a first-order reaction is:
\[ C_t = C_0 e^{-kt} \]