A first order reaction is 75% completed in 100 minutes. How long time will it take for its 87.5% completion?

Options

A. 125 min
B. 150 min
C. 175 min
D. 200 min

Answer

To solve the problem of determining how long it will take for a first-order reaction to reach 87.5% completion given that it takes 100 minutes to reach 75% completion, we can follow these steps:

### Step 1: Understand the Reaction Completion
Given that the reaction is 75% completed, it means that 25% of the reactant remains. If we denote the initial concentration of the reactant A as [A]₀, after 75% completion, the concentration of A will be:
\[ [A] = [A]₀ \times (1 - 0.75) = [A]₀ \times 0.25 \]

### Step 2: Use the First-Order Kinetics Formula
For a first-order reaction, the relationship between time (t), rate constant (k), and concentrations is given by:
\[ t = \frac{2.303}{k} \log \left( \frac{[A]₀}{[A]} \right) \]

### Step 3: Calculate the Rate Constant (k)
From the information given, we know that 75% of the reaction is completed in 100 minutes. Thus, we can substitute into the formula:
\[ 100 = \frac{2.303}{k} \log \left( \frac{[A]₀}{0.25[A]₀} \right) \]
This simplifies to:
\[ 100 = \frac{2.303}{k} \log(4) \]
Now, we can solve for k:
\[ k = \frac{2.303 \cdot \log(4)}{100} \]

### Step 4: Calculate Time for 87.5% Completion
Next, we need to find the time taken for 87.5% completion. This means that 12.5% of the reactant remains:
\[ [A] = [A]₀ \times (1 - 0.875) = [A]₀ \times 0.125 \]
Substituting this into the first-order kinetics formula gives:
\[ t = \frac{2.303}{k} \log \left( \frac{[A]₀}{0.125[A]₀} \right) \]
This simplifies to:
\[ t = \frac{2.303}{k} \log(8) \]

### Step 5: Substitute k into the Time Equation
Now substituting k from Step 3 into this equation:
\[ t = \frac{2.303}{\frac{2.303 \cdot \log(4)}{100}} \log(8) \]
This simplifies to:
\[ t = 100 \cdot \frac{\log(8)}{\log(4)} \]

### Step 6: Calculate the Logarithmic Values
Using the properties of logarithms:
\[ \log(8) = \log(2^3) = 3 \log(2) \]
\[ \log(4) = \log(2^2) = 2 \log(2) \]
Thus:
\[ \frac{\log(8)}{\log(4)} = \frac{3 \log(2)}{2 \log(2)} = \frac{3}{2} \]

### Step 7: Final Calculation
Now substituting this back into the time equation:
\[ t = 100 \cdot \frac{3}{2} = 150 \text{ minutes} \]

### Conclusion
Therefore, the time required for the reaction to reach 87.5% completion is **150 minutes**.