For an elementary reaction , `X(g)toY(g)+Z(g)`
the half life period is 10 min. In what period of time would the concentration of X be reduced to 10% of original concentration?

Options

A. `20 min`
B. `33 min`
C. `15 min`
D. `25 min`

Explanation

To solve the problem step by step, we will follow these calculations:

### Step 1: Understand the Reaction and Given Data
The reaction is given as:
\[ X(g) \rightarrow Y(g) + Z(g) \]
We know that this is an elementary reaction, which means the order of the reaction is equal to its molecularity. Here, the molecularity is 1, so the order of the reaction is also 1.

### Step 2: Use the Half-Life Formula for First Order Reactions
The half-life (\( t_{1/2} \)) for a first-order reaction is given by the formula:
\[ t_{1/2} = \frac{0.693}{k} \]
Where \( k \) is the rate constant.

Given that the half-life is 10 minutes, we can rearrange this formula to find \( k \):
\[ k = \frac{0.693}{t_{1/2}} = \frac{0.693}{10 \text{ min}} \]

### Step 3: Calculate the Rate Constant \( k \)
Calculating \( k \):
\[ k = \frac{0.693}{10} = 0.0693 \text{ min}^{-1} \]

### Step 4: Determine the Time to Reduce Concentration to 10%
We need to find the time (\( t \)) required for the concentration of \( X \) to reduce to 10% of its original concentration. This means that if the initial concentration is 100%, the remaining concentration will be 10%.

Using the first-order reaction formula:
\[ t = \frac{2.303}{k} \log \left( \frac{[A]_0}{[A]} \right) \]
Where:
- \([A]_0\) is the initial concentration (100%)
- \([A]\) is the remaining concentration (10%)

Substituting the values:
\[ t = \frac{2.303}{0.0693} \log \left( \frac{100}{10} \right) \]

### Step 5: Calculate the Logarithm
Calculating the logarithm:
\[ \log \left( \frac{100}{10} \right) = \log(10) = 1 \]

### Step 6: Substitute and Calculate Time \( t \)
Now substituting back into the equation:
\[ t = \frac{2.303}{0.0693} \times 1 \]

Calculating \( t \):
\[ t \approx \frac{2.303}{0.0693} \approx 33.23 \text{ minutes} \]

### Final Answer
Thus, the time required for the concentration of \( X \) to be reduced to 10% of its original concentration is approximately:
\[ \boxed{33.23 \text{ minutes}} \]

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