For a first order reaction, the half-life period is independent of

Answer

To solve the question regarding the half-life period of a first-order reaction, we can follow these steps:

### Step 1: Understand the concept of half-life in first-order reactions
The half-life period (T_half) is the time required for the concentration of a reactant to decrease to half of its initial concentration.

### Step 2: Write the rate equation for a first-order reaction
For a first-order reaction, the relationship between the concentration and time can be expressed as:
\[ k t = \ln \left( \frac{[A_0]}{[A_t]} \right) \]
where:
- \( k \) is the rate constant,
- \( [A_0] \) is the initial concentration,
- \( [A_t] \) is the concentration at time \( t \).

### Step 3: Set up the equation for half-life
At half-life, the concentration at time \( t_{half} \) will be:
\[ [A_t] = \frac{[A_0]}{2} \]

Substituting this into the rate equation gives:
\[ k t_{half} = \ln \left( \frac{[A_0]}{\frac{[A_0]}{2}} \right) \]

### Step 4: Simplify the equation
This simplifies to:
\[ k t_{half} = \ln(2) \]

### Step 5: Solve for half-life
Rearranging the equation to solve for \( t_{half} \) gives:
\[ t_{half} = \frac{\ln(2)}{k} \]

### Step 6: Analyze the result
From the equation \( t_{half} = \frac{0.693}{k} \), we can see that the half-life of a first-order reaction depends only on the rate constant \( k \) and is independent of the initial concentration \( [A_0] \).

### Conclusion
Thus, the half-life period for a first-order reaction is independent of the initial concentration of the reactant.