Half life of a first order reaction in 10 min. What `%` of reaction will be completed in 100 min ? `

Options

A. 0.25
B. 0.5
C. 0.999
D. 0.75

Explanation

To solve the problem of determining the percentage of a first-order reaction that will be completed in 100 minutes, given that its half-life is 10 minutes, we can follow these steps:

### Step 1: Understand the relationship between half-life and rate constant.
For a first-order reaction, the rate constant \( k \) can be calculated using the formula:
\[ k = \frac{0.693}{t_{1/2}} \]
where \( t_{1/2} \) is the half-life of the reaction.

### Step 2: Calculate the rate constant \( k \).
Given that the half-life \( t_{1/2} \) is 10 minutes:
\[ k = \frac{0.693}{10 \text{ min}} = 0.0693 \text{ min}^{-1} \]

### Step 3: Use the first-order reaction formula to find the amount reacted after 100 minutes.
The formula for a first-order reaction is:
\[ \ln \left( \frac{A}{A - x} \right) = kt \]
where:
- \( A \) is the initial amount (100%),
- \( x \) is the amount reacted,
- \( t \) is the time (100 minutes).

### Step 4: Substitute the known values into the equation.
Substituting \( A = 100 \), \( k = 0.0693 \text{ min}^{-1} \), and \( t = 100 \text{ min} \):
\[ \ln \left( \frac{100}{100 - x} \right) = 0.0693 \times 100 \]
\[ \ln \left( \frac{100}{100 - x} \right) = 6.93 \]

### Step 5: Solve for \( x \).
To solve for \( x \), we exponentiate both sides:
\[ \frac{100}{100 - x} = e^{6.93} \]
Calculating \( e^{6.93} \):
\[ e^{6.93} \approx 1020.43 \]
Thus:
\[ 100 = 1020.43(100 - x) \]
Expanding and rearranging gives:
\[ 100 = 102043 - 1020.43x \]
\[ 1020.43x = 102043 - 100 \]
\[ 1020.43x = 102043 - 100 = 101943 \]
\[ x = \frac{101943}{1020.43} \approx 99.9 \]

### Step 6: Calculate the percentage of the reaction completed.
The percentage of the reaction completed is given by:
\[ \text{Percentage completed} = \frac{x}{A} \times 100 = \frac{99.9}{100} \times 100 = 99.9\% \]

### Final Answer:
Thus, the percentage of the reaction that will be completed in 100 minutes is **99.9%**.

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