Question

Half-life (\( t_{1/2} \)) of a first order reaction is 1386 s. The value of rate constant is:

• A. \(0.5 \times 10^{-4} \, \text{s}^{-1}\)
• B. \(5.0 \times 10^{-4} \, \text{s}^{-1}\)
• C. \(0.5 \times 10^{-5} \, \text{s}^{-1}\)
• D. \(0.5 \times 10^{-3} \, \text{s}^{-1}\)

Hint:

For first order reactions: \[ t_{1/2} = \frac{0.693}{k} \] If half-life is known, rate constant can be found directly without concentration data.

Answer 1

The Correct Option is A

Concept: For a
first order reaction, the half-life is independent of initial concentration and is given by: \[ t_{1/2} = \frac{0.693}{k} \] where \( k \) is the rate constant.
Step 1: Use the half-life formula. Given: \[ t_{1/2} = 1386 \, \text{s} \] \[ k = \frac{0.693}{t_{1/2}} = \frac{0.693}{1386} \]
Step 2: Simplify the value. Note that: \[ 1386 = 0.693 \times 2000 \] Hence, \[ k = \frac{0.693}{0.693 \times 2000} = \frac{1}{2000} \]
Step 3: Convert to scientific notation. \[ k = \frac{1}{2000} = 5 \times 10^{-4} \div 10 = 0.5 \times 10^{-4} \, \text{s}^{-1} \]
Step 4: Match with options. \[ \therefore \text{Correct answer = (A)} \]