Sucrose hydrolyses in acid solution into glucose and fructose following first order rate law with a half-life of 3.33 h at `25^(@)C`. After 9h , the fraction of sucrose remaining if f. The value of `log_(10)((1)/(f))` is _________`xx 10^(-2)`. (Rounded off to the nearest integer )
[Assume , `ln 10 = 2.303, ln 2= 0.693`]

Answer

To solve the problem, we need to determine the fraction of sucrose remaining after 9 hours of hydrolysis and then calculate \( \log_{10} \left( \frac{1}{f} \right) \).

### Step-by-Step Solution:

1. **Identify the half-life and rate constant**:
The half-life (\( t_{1/2} \)) of the reaction is given as 3.33 hours. For a first-order reaction, the rate constant (\( k \)) can be calculated using the formula:
\[ k = \frac{0.693}{t_{1/2}} \]
Substituting the half-life value:
\[ k = \frac{0.693}{3.33} \approx 0.208 \, \text{h}^{-1} \]

2. **Use the first-order kinetics equation**:
The relationship between time, rate constant, and concentrations in a first-order reaction is given by:
\[ t = \frac{2.303}{k} \log \left( \frac{[A_0]}{[A]} \right) \]
Here, \( [A_0] \) is the initial concentration and \( [A] \) is the concentration at time \( t \). In this case, we can set \( [A_0] = 1 \) (initial concentration of sucrose) and \( [A] = \frac{1}{f} \) (final concentration).

3. **Substituting values into the equation**:
We know the time \( t = 9 \) hours, so we can substitute the values into the equation:
\[ 9 = \frac{2.303}{0.208} \log \left( \frac{1}{f} \right) \]

4. **Rearranging the equation**:
Rearranging gives:
\[ \log \left( \frac{1}{f} \right) = 9 \times \frac{0.208}{2.303} \]

5. **Calculating the right-hand side**:
First, calculate \( \frac{0.208}{2.303} \):
\[ \frac{0.208}{2.303} \approx 0.0904 \]
Now, multiply by 9:
\[ \log \left( \frac{1}{f} \right) \approx 9 \times 0.0904 \approx 0.8136 \]

6. **Final calculation**:
To express this in the form \( xx \times 10^{-2} \):
\[ 0.8136 = 81.36 \times 10^{-2} \]
Rounding off to the nearest integer gives:
\[ \log_{10} \left( \frac{1}{f} \right) \approx 81 \]

### Final Answer:
The value of \( \log_{10} \left( \frac{1}{f} \right) \) is **81**.