Question

The first order rate constant for the decomposition of \( \text{N}_2\text{O}_5 \) is \( 6.93 \times 10^{-4}\, \text{sec}^{-1} \). Its half-life period is

• A. \( 1000\,s \)
• B. \( 100\,s \)
• C. \( 10\,s \)
• D. \( 1\,s \)
• E. \( 10000\,s \)

Hint:

For first-order reactions, half-life is independent of concentration: \[ t_{1/2} = \frac{0.693}{k} \]

Answer 1

The Correct Option is A

Step 1: Recall half-life formula for first order reaction.
\[ t_{1/2} = \frac{0.693}{k} \]

Step 2: Substitute the given rate constant.
\[ k = 6.93 \times 10^{-4}\, \text{sec}^{-1} \]

Step 3: Write the expression.
\[ t_{1/2} = \frac{0.693}{6.93 \times 10^{-4}} \]

Step 4: Simplify the numbers.
\[ \frac{0.693}{6.93} = 0.1 \] So, \[ t_{1/2} = \frac{0.1}{10^{-4}} = 0.1 \times 10^4 \]

Step 5: Calculate final value.
\[ t_{1/2} = 1000\,s \]

Step 6: Check unit.
Since \( k \) is in \( \text{sec}^{-1} \), half-life is in seconds.

Step 7: Final conclusion.
\[ \boxed{1000\,s} \] Therefore, correct option is \[ \boxed{(1)\ 1000\,s} \]