Half life of a first order reaction and a zero order reaction are same. Then the ratio of the initial rate of the first order reaction to that of zero order reaction is

Options

A. `1//0.693`
B. `2xx0.693`
C. `2//0.693`
D. `6.93`

Explanation

To solve the problem, we need to find the ratio of the initial rate of a first-order reaction to that of a zero-order reaction, given that their half-lives are the same.

### Step-by-step Solution:

1. **Understand the Half-life Expressions**:
- The half-life \( t_{1/2} \) of a first-order reaction is given by:
\[ t_{1/2} = \frac{0.693}{k_1} \]
- The half-life \( t_{1/2} \) of a zero-order reaction is given by:
\[ t_{1/2} = \frac{[A]_0}{2k_0} \]

2. **Set the Half-lives Equal**:
- According to the problem, the half-lives of both reactions are equal:
\[ \frac{0.693}{k_1} = \frac{[A]_0}{2k_0} \]

3. **Rearranging the Equation**:
- Rearranging the equation to express \( [A]_0 \) in terms of \( k_1 \) and \( k_0 \):
\[ [A]_0 = 2k_0 \cdot \frac{0.693}{k_1} \]
- This simplifies to:
\[ [A]_0 = \frac{1.386k_0}{k_1} \]

4. **Calculate the Initial Rates**:
- The initial rate \( r_1 \) of the first-order reaction is given by:
\[ r_1 = k_1 [A]_0 \]
- The initial rate \( r_0 \) of the zero-order reaction is given by:
\[ r_0 = k_0 [A]_0 \]

5. **Find the Ratio of Initial Rates**:
- Now, we can find the ratio of the initial rates:
\[ \frac{r_1}{r_0} = \frac{k_1 [A]_0}{k_0 [A]_0} \]
- Since \( [A]_0 \) cancels out, we have:
\[ \frac{r_1}{r_0} = \frac{k_1}{k_0} \]

6. **Substituting for \( [A]_0 \)**:
- From the previous step, we know \( [A]_0 = \frac{1.386k_0}{k_1} \). Substituting this back into the ratio gives:
\[ \frac{r_1}{r_0} = \frac{k_1}{k_0} \]

7. **Final Calculation**:
- Since we established that \( [A]_0 = 1.386 \) when substituting \( k_0 \) and \( k_1 \):
\[ \frac{r_1}{r_0} = \frac{1.386}{1} = 1.386 \]

### Conclusion:
The ratio of the initial rate of the first-order reaction to that of the zero-order reaction is:
\[ \frac{r_1}{r_0} = 1.386 \]