In a first order reaction, 20% reaction is completed in 24 minutes.The percentage of reactat remaining after 48 minutes is

Options

A. 60
B. 64
C. 81
D. 80

Answer

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

### Step 1: Understand the Reaction
In a first-order reaction, the rate of reaction is directly proportional to the concentration of the reactant. We are given that 20% of the reactant has reacted in 24 minutes.

### Step 2: Determine Remaining Concentration
If 20% of the reactant has reacted, then 80% of the reactant remains. We can express this as:
- Initial concentration (A₀) = 100%
- Remaining concentration (A) = 100% - 20% = 80%

### Step 3: Use the First-Order Rate Equation
The first-order rate equation can be expressed as:
\[ k = \frac{2.303}{t} \log \left( \frac{A₀}{A} \right) \]
Where:
- \( k \) = rate constant
- \( t \) = time
- \( A₀ \) = initial concentration
- \( A \) = concentration at time \( t \)

### Step 4: Calculate the Rate Constant (k)
Substituting the values we have:
- \( t = 24 \) minutes
- \( A₀ = 100 \)
- \( A = 80 \)

So, we can write:
\[ k = \frac{2.303}{24} \log \left( \frac{100}{80} \right) \]
\[ k = \frac{2.303}{24} \log \left( 1.25 \right) \]

Using the logarithm:
\[ \log(1.25) \approx 0.09691 \]

Now substituting this value:
\[ k = \frac{2.303}{24} \times 0.09691 \]
\[ k \approx \frac{0.223}{24} \approx 0.00929 \, \text{min}^{-1} \]

### Step 5: Find Remaining Concentration After 48 Minutes
Now we need to find the remaining concentration after 48 minutes. We will use the same first-order equation:
\[ k = \frac{2.303}{t} \log \left( \frac{A₀}{A} \right) \]

Here, \( t = 48 \) minutes and we need to find \( A \):
\[ 0.00929 = \frac{2.303}{48} \log \left( \frac{100}{A} \right) \]

Rearranging gives:
\[ \log \left( \frac{100}{A} \right) = 0.00929 \times 48 / 2.303 \]
\[ \log \left( \frac{100}{A} \right) \approx 0.193 \]

### Step 6: Solve for A
Now we can solve for \( A \):
\[ \frac{100}{A} = 10^{0.193} \]
Calculating \( 10^{0.193} \approx 1.57 \):
\[ A \approx \frac{100}{1.57} \approx 63.69 \]

### Step 7: Calculate Percentage Remaining
The percentage of the reactant remaining is:
\[ \text{Percentage remaining} = A \approx 63.69\% \]

### Conclusion
The percentage of the reactant remaining after 48 minutes is approximately **64%**.

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