Which one is correct for first order reaction.

Options

A. `t_(75%)/t_(50%)`=1.5
B. `t_(75%)/t_(50%)`=3
C. `t_(99.9%)/t_(50%)`=10
D. `t_(87.5%)/t_(50%)`=3

Answer

To determine which statement is correct for a first-order reaction, we will analyze the relationships between the time taken for certain percentages of the reactant to be converted into products.

### Step-by-step Solution:

1. **Understanding First-Order Reactions**:
The rate of a first-order reaction is directly proportional to the concentration of one reactant. The integrated rate law for a first-order reaction can be expressed as:
\[ k t = \ln \left( \frac{[A]_0}{[A]} \right) \]
where \( k \) is the rate constant, \( t \) is time, \([A]_0\) is the initial concentration, and \([A]\) is the concentration at time \( t \).

2. **Calculating Time for 50% Dissociation**:
For 50% dissociation:
\[ k t_{50\%} = \ln \left( \frac{100}{50} \right) = \ln(2) \]
Thus, we have:
\[ t_{50\%} = \frac{\ln(2)}{k} \]

3. **Calculating Time for 75% Dissociation**:
For 75% dissociation:
\[ k t_{75\%} = \ln \left( \frac{100}{25} \right) = \ln(4) = 2 \ln(2) \]
Therefore:
\[ t_{75\%} = \frac{2 \ln(2)}{k} \]

4. **Finding the Ratio of Time for 75% to 50%**:
Now, we can find the ratio:
\[ \frac{t_{75\%}}{t_{50\%}} = \frac{\frac{2 \ln(2)}{k}}{\frac{\ln(2)}{k}} = 2 \]

5. **Calculating Time for 87.5% Dissociation**:
For 87.5% dissociation:
\[ k t_{87.5\%} = \ln \left( \frac{100}{12.5} \right) = \ln(8) = 3 \ln(2) \]
Thus:
\[ t_{87.5\%} = \frac{3 \ln(2)}{k} \]

6. **Finding the Ratio of Time for 87.5% to 50%**:
The ratio is:
\[ \frac{t_{87.5\%}}{t_{50\%}} = \frac{\frac{3 \ln(2)}{k}}{\frac{\ln(2)}{k}} = 3 \]

7. **Calculating Time for 99.9% Dissociation**:
For 99.9% dissociation:
\[ k t_{99.9\%} = \ln \left( \frac{100}{0.1} \right) = \ln(1000) = 3 \ln(10) \]
Thus:
\[ t_{99.9\%} = \frac{3 \ln(10)}{k} \]

8. **Finding the Ratio of Time for 99.9% to 50%**:
The ratio is:
\[ \frac{t_{99.9\%}}{t_{50\%}} = \frac{\frac{3 \ln(10)}{k}}{\frac{\ln(2)}{k}} = \frac{3 \ln(10)}{\ln(2)} \approx 10 \]

### Conclusion:
From the calculations, we have the following relationships:
- \( t_{75\%} = 2 \times t_{50\%} \)
- \( t_{87.5\%} = 3 \times t_{50\%} \)
- \( t_{99.9\%} \approx 10 \times t_{50\%} \)

Thus, the correct statement for a first-order reaction is:
- **Option D**: \( t_{87.5\%} = 3 \times t_{50\%} \)
- **Option C**: \( t_{99.9\%} \approx 10 \times t_{50\%} \)