A first order reaction is 20% complete in one hour. At the end of 3 hrs the extent of the reaction is:

Options

A. `60% `
B. `52.2% `
C. `63% `
D. `44.4% `

Answer

To solve the problem step by step, we will use the first-order reaction kinetics formula and the information provided in the question.

### Step 1: Understand the first-order reaction formula
The rate constant \( k \) for a first-order reaction can be calculated using the formula:
\[ k = \frac{2.303}{t} \log \left( \frac{A_0}{A_t} \right) \]
where:
- \( A_0 \) is the initial concentration,
- \( A_t \) is the concentration at time \( t \),
- \( t \) is the time in hours.

### Step 2: Calculate the rate constant \( k \) for the first hour
Given that the reaction is 20% complete in 1 hour, we can express this as:
- \( A_t = A_0 - 0.2 A_0 = 0.8 A_0 \)

Now substituting into the formula:
\[ k = \frac{2.303}{1} \log \left( \frac{A_0}{0.8 A_0} \right) \]
This simplifies to:
\[ k = 2.303 \log \left( \frac{1}{0.8} \right) = 2.303 \log (1.25) \]

Calculating \( \log (1.25) \):
\[ \log (1.25) \approx 0.09691 \]
Thus,
\[ k \approx 2.303 \times 0.09691 \approx 0.223 \, \text{hr}^{-1} \]

### Step 3: Calculate the extent of the reaction after 3 hours
Now we need to find out how much of the reaction is complete after 3 hours. Using the same first-order reaction formula:
\[ k = \frac{2.303}{3} \log \left( \frac{A_0}{A_t} \right) \]
Let \( x \) be the fraction of the reaction that is complete after 3 hours. Then:
- \( A_t = A_0 - x A_0 = (1 - x) A_0 \)

Substituting into the formula:
\[ 0.223 = \frac{2.303}{3} \log \left( \frac{A_0}{(1 - x) A_0} \right) \]
This simplifies to:
\[ 0.223 = \frac{2.303}{3} \log \left( \frac{1}{1 - x} \right) \]

### Step 4: Solve for \( x \)
Rearranging gives:
\[ \log \left( \frac{1}{1 - x} \right) = \frac{3 \times 0.223}{2.303} \]
Calculating the right-hand side:
\[ \frac{3 \times 0.223}{2.303} \approx 0.290 \]

Now, taking the antilogarithm:
\[ \frac{1}{1 - x} = 10^{0.290} \approx 1.95 \]
Thus:
\[ 1 - x = \frac{1}{1.95} \approx 0.513 \]
So:
\[ x \approx 1 - 0.513 \approx 0.487 \text{ or } 48.7\% \]

### Step 5: Calculate the extent of the reaction
To find the extent of the reaction, we multiply by 100:
\[ \text{Extent of reaction} \approx 48.7\% \text{ after 3 hours.} \]

### Final Answer
At the end of 3 hours, the extent of the reaction is approximately **48.7%**.

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