Hydrolysis of cane sugar in presence of acid is a first order reaction with half life time of 4 hours what fraction of sucrose undergoes hydrolysis in 11 hrs ?

Options

A. `0.9345`
B. `0.6122`
C. `0.8511`
D. `0.412`

Answer

To solve the problem of determining the fraction of sucrose that undergoes hydrolysis in 11 hours, we will follow these steps:

### Step 1: Determine the rate constant (k)
For a first-order reaction, the half-life (\(t_{1/2}\)) is related to the rate constant (k) by the formula:

\[ k = \frac{0.693}{t_{1/2}} \]

Given that the half-life is 4 hours, we can substitute this value into the formula:

\[ k = \frac{0.693}{4} = 0.17325 \, \text{hours}^{-1} \]

### Step 2: Use the first-order kinetics equation
The first-order kinetics equation is given by:

\[ \ln\left(\frac{A_0}{A}\right) = kt \]

Where:
- \(A_0\) is the initial concentration of sucrose,
- \(A\) is the concentration of sucrose remaining after time \(t\),
- \(k\) is the rate constant,
- \(t\) is the time in hours.

### Step 3: Set up the equation for 11 hours
Assuming the initial concentration \(A_0 = 1\) (for simplicity), we can express the remaining concentration \(A\) as \(1 - X\), where \(X\) is the fraction of sucrose that has undergone hydrolysis.

Substituting the values into the equation:

\[ \ln\left(\frac{1}{1 - X}\right) = k \cdot t \]

For \(t = 11\) hours, we have:

\[ \ln\left(\frac{1}{1 - X}\right) = 0.17325 \cdot 11 \]

Calculating the right-hand side:

\[ 0.17325 \cdot 11 = 1.90575 \]

Thus, we have:

\[ \ln\left(\frac{1}{1 - X}\right) = 1.90575 \]

### Step 4: Solve for \(1 - X\)
To eliminate the natural logarithm, we exponentiate both sides:

\[ \frac{1}{1 - X} = e^{1.90575} \]

Calculating \(e^{1.90575}\):

\[ e^{1.90575} \approx 6.707 \]

Thus, we have:

\[ 1 - X = \frac{1}{6.707} \]

Calculating this gives:

\[ 1 - X \approx 0.149 \]

### Step 5: Solve for \(X\)
Now, we can find \(X\):

\[ X = 1 - 0.149 \approx 0.851 \]

### Conclusion
The fraction of sucrose that undergoes hydrolysis in 11 hours is approximately 0.851. Therefore, the correct answer is option C.

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