75 % of first order reaction is complete in 30 minutes. What is the time required for 93.75 % of the reaction (in minutes) ?

Options

A. 45
B. 120
C. 90
D. 60

Answer

To solve the problem, we need to determine the time required for 93.75% completion of a first-order reaction, given that 75% of the reaction is complete in 30 minutes.

### Step-by-Step Solution:

1. **Understanding the Reaction Completion:**
- Initially, we have 100% of the reactant.
- If 75% of the reaction is complete, then 25% of the reactant remains.
- Therefore, after 30 minutes, the concentration of the reactant left (A) is 25%.

2. **Using the First-Order Reaction Formula:**
- The formula for a first-order reaction is:
\[ k = \frac{2.303}{t} \log \left( \frac{[A_0]}{[A]} \right) \]
- Here, \([A_0]\) is the initial concentration (100%), and \([A]\) is the concentration at time \(t\) (25%).
- Substituting the values into the formula for the time \(t = 30\) minutes:
\[ k = \frac{2.303}{30} \log \left( \frac{100}{25} \right) \]
- Simplifying the logarithm:
\[ \log \left( \frac{100}{25} \right) = \log(4) \]
- Therefore, we have:
\[ k = \frac{2.303}{30} \log(4) \]

3. **Finding Time for 93.75% Completion:**
- For 93.75% completion, the remaining reactant is:
\[ 100\% - 93.75\% = 6.25\% \]
- Now, we will use the same first-order reaction formula to find the time \(t\) for this completion:
\[ t = \frac{2.303}{k} \log \left( \frac{[A_0]}{[A]} \right) \]
- Substituting the values:
\[ t = \frac{2.303}{k} \log \left( \frac{100}{6.25} \right) \]
- Simplifying the logarithm:
\[ \log \left( \frac{100}{6.25} \right) = \log(16) = \log(4^2) = 2 \log(4) \]
- Therefore, we can write:
\[ t = \frac{2.303}{k} \cdot 2 \log(4) \]

4. **Substituting \(k\) from the First Calculation:**
- From the earlier calculation, we know:
\[ k = \frac{2.303}{30} \log(4) \]
- Substituting this value of \(k\) into the equation for \(t\):
\[ t = \frac{2.303}{\frac{2.303}{30} \log(4)} \cdot 2 \log(4) \]
- The \(\log(4)\) cancels out:
\[ t = 30 \cdot 2 = 60 \text{ minutes} \]

### Final Answer:
The time required for 93.75% completion of the reaction is **60 minutes**.