For first order reaction `t_(0.75)` is 138.6 sec. Its specific rate constant is (in`s^(-1)`)

Options

A. `10^(-2)`
B. `10^(-4)`
C. `10^(-5)`
D. `10^(-6)`

Answer

To find the specific rate constant (k) for a first-order reaction given that \( t_{0.75} = 138.6 \) seconds, we can follow these steps:

### Step 1: Understand the relationship between \( t_{0.75} \) and half-life
For a first-order reaction, the time taken to reach 75% completion (\( t_{0.75} \)) is related to the half-life (\( t_{1/2} \)). Specifically, \( t_{0.75} \) can be expressed in terms of the half-life as:
\[ t_{0.75} = 2 \cdot t_{1/2} \]

### Step 2: Calculate the half-life
Given that \( t_{0.75} = 138.6 \) seconds, we can find the half-life:
\[ t_{1/2} = \frac{t_{0.75}}{2} = \frac{138.6 \, \text{s}}{2} = 69.3 \, \text{s} \]

### Step 3: Use the half-life formula for first-order reactions
The half-life for a first-order reaction is given by the formula:
\[ t_{1/2} = \frac{0.693}{k} \]
Where \( k \) is the specific rate constant.

### Step 4: Rearrange the formula to solve for \( k \)
Rearranging the formula to solve for \( k \):
\[ k = \frac{0.693}{t_{1/2}} \]

### Step 5: Substitute the value of \( t_{1/2} \)
Substituting \( t_{1/2} = 69.3 \, \text{s} \) into the equation:
\[ k = \frac{0.693}{69.3} \]

### Step 6: Calculate \( k \)
Now, performing the calculation:
\[ k = \frac{0.693}{69.3} \approx 0.01 \, \text{s}^{-1} = 10^{-2} \, \text{s}^{-1} \]

### Conclusion
Thus, the specific rate constant \( k \) is:
\[ \boxed{10^{-2} \, \text{s}^{-1}} \]

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