If `50%` of the rectant is converted into a product in a first order reaction in 25 minutes, how much of it would react in 100 minutes?

Options

A. `93.75%`
B. `87.5%`
C. `75%`
D. `100%`

Explanation

To solve the problem step by step, we will follow the principles of first-order kinetics.

### Step 1: Understand the given data
We know that in a first-order reaction, the half-life (T½) is constant and is given as 25 minutes. This means that in 25 minutes, 50% of the reactant is converted into the product.

### Step 2: Calculate the rate constant (K)
The formula for the half-life of a first-order reaction is:
\[ T_{1/2} = \frac{0.693}{K} \]
Given \( T_{1/2} = 25 \) minutes, we can rearrange the formula to find K:
\[ K = \frac{0.693}{25} \]
Calculating this gives:
\[ K = 0.02772 \, \text{min}^{-1} \]

### Step 3: Determine the total time for reaction
We need to find out how much of the reactant will be converted into the product in 100 minutes.

### Step 4: Use the first-order kinetics equation
The first-order reaction can be described by the equation:
\[ \ln\left(\frac{A}{A - X}\right) = Kt \]
Where:
- \( A \) = initial concentration
- \( X \) = amount reacted
- \( t \) = time

### Step 5: Calculate the amount reacted after 100 minutes
We can express the equation in terms of the logarithm:
\[ \ln\left(\frac{A}{A - X}\right) = K \cdot t \]
Substituting \( K \) and \( t \):
\[ \ln\left(\frac{A}{A - X}\right) = 0.02772 \cdot 100 \]
Calculating the right side gives:
\[ \ln\left(\frac{A}{A - X}\right) = 2.772 \]

### Step 6: Exponentiate to solve for the ratio
Taking the exponential of both sides:
\[ \frac{A}{A - X} = e^{2.772} \]
Calculating \( e^{2.772} \) gives approximately 16.

### Step 7: Set up the equation for the initial concentration
Let’s assume the initial concentration \( A = 100 \):
\[ \frac{100}{100 - X} = 16 \]

### Step 8: Solve for X
Cross-multiplying gives:
\[ 100 = 16(100 - X) \]
Expanding this:
\[ 100 = 1600 - 16X \]
Rearranging to find \( X \):
\[ 16X = 1600 - 100 \]
\[ 16X = 1500 \]
\[ X = \frac{1500}{16} = 93.75 \]

### Step 9: Conclusion
Thus, after 100 minutes, 93.75% of the reactant has been converted into the product.

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