75% of a first order process is completed in 30 min .The tim e required for 93.75% A completion of same process(in hr)?

Options

A. 1
B. 120
C. 2
D. 0.25

Explanation

To solve the problem step by step, we will use the first-order reaction kinetics formula:

### Step 1: Understand the given information
We know that 75% of the reaction is completed in 30 minutes. This means that 25% of the reactant remains.

### Step 2: Use the first-order reaction formula
The formula for the time taken for a first-order reaction is given by:
\[ t = \frac{2.303}{k} \log \left( \frac{A}{A - x} \right) \]
Where:
- \( t \) = time taken
- \( k \) = rate constant
- \( A \) = initial concentration of the reactant
- \( x \) = amount reacted

### Step 3: Calculate the rate constant \( k \)
For the first scenario (75% completion):
- \( A = 100 \) (initial amount)
- \( x = 75 \) (amount reacted)
- Remaining amount \( A - x = 25 \)

Substituting these values into the formula:
\[ 30 = \frac{2.303}{k} \log \left( \frac{100}{25} \right) \]
\[ 30 = \frac{2.303}{k} \log(4) \]
Since \( \log(4) \approx 0.6020 \):
\[ 30 = \frac{2.303}{k} \times 0.6020 \]
Now, rearranging to find \( k \):
\[ k = \frac{2.303 \times 0.6020}{30} \]
Calculating \( k \):
\[ k \approx 0.046 \, \text{min}^{-1} \approx 0.05 \, \text{min}^{-1} \]

### Step 4: Calculate time for 93.75% completion
Now, for the second scenario (93.75% completion):
- Amount reacted \( x = 93.75 \)
- Remaining amount \( A - x = 100 - 93.75 = 6.25 \)

Substituting these values into the formula:
\[ t = \frac{2.303}{0.05} \log \left( \frac{100}{6.25} \right) \]
Calculating \( \log(16) \) (since \( \frac{100}{6.25} = 16 \)):
\[ \log(16) \approx 1.2041 \]
Now substituting back into the time formula:
\[ t = \frac{2.303}{0.05} \times 1.2041 \]
Calculating \( t \):
\[ t \approx \frac{2.303 \times 1.2041}{0.05} \approx 55.5 \, \text{minutes} \]

### Step 5: Convert time to hours
To convert minutes to hours:
\[ t \approx \frac{55.5}{60} \approx 0.925 \, \text{hours} \]
This is approximately 0.93 hours, which is roughly 55.5 minutes.

### Final Answer:
The time required for 93.75% completion of the same process is approximately **55.5 minutes** or **0.93 hours**.