For a first order reaction `t_(0.75)` is `1386 s`. Therefore, the specific rate constant is

Options

A. `10^(-1) s^(-1)`
B. `10^(-3) s^(-1)`
C. `10^(-2) s^(-1)`
D. `10^(-4) s^(-1)`

Explanation

To find the specific rate constant (k) for a first-order reaction given that \( t_{0.75} = 1386 \, s \), we will follow these steps:

### Step 1: Understand the relationship between \( t_{0.75} \) and \( t_{0.5} \)
For a first-order reaction, there is a known relationship between the time taken for 75% completion of the reaction (\( t_{0.75} \)) and the half-life (\( t_{0.5} \)):
\[ t_{0.75} = 2 \times t_{0.5} \]

### Step 2: Calculate \( t_{0.5} \)
Using the relationship from Step 1, we can express \( t_{0.5} \) as:
\[ t_{0.5} = \frac{t_{0.75}}{2} \]
Substituting the given value:
\[ t_{0.5} = \frac{1386 \, s}{2} = 693 \, s \]

### Step 3: Use the formula for half-life of a first-order reaction
The half-life (\( t_{0.5} \)) for a first-order reaction is related to the specific rate constant (k) by the formula:
\[ t_{0.5} = \frac{0.693}{k} \]
We can rearrange this to solve for k:
\[ k = \frac{0.693}{t_{0.5}} \]

### Step 4: Substitute \( t_{0.5} \) into the equation for k
Now, substituting the value of \( t_{0.5} \):
\[ k = \frac{0.693}{693 \, s} \]

### Step 5: Calculate k
Performing the calculation:
\[ k = 0.001 \, s^{-1} = 10^{-3} \, s^{-1} \]

### Final Answer
The specific rate constant \( k \) is:
\[ k = 10^{-3} \, s^{-1} \]

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