Question

If for a first-order reaction, the value of \( A \) and \( E_a \) are \( 4 \times 10^{13} \) s$^-1$ and 98.6 kJ mol$^{-1}$ respectively, then at what temperature will its half-life be 10 minutes?

• A. 330 K
• B. 300 K
• C. 330.95 K
• D. 311.15 K

Hint:

For first-order reactions, half-life is independent of concentration. The Arrhenius equation relates rate constant to temperature.

Answer 1

The Correct Option is D

Step 1: Understanding the Arrhenius Equation
The Arrhenius equation is: \[ \log k = \log A - \frac{E_a}{2.303RT} \] Step 2: Finding the Rate Constant \( k \)
For a first-order reaction, the half-life is given by: \[ k = \frac{0.693}{t_{1/2}} \] Substituting \( t_{1/2} = 10 \) minutes \( = 600 \) s: \[ k = \frac{0.693}{600} = 1.1 \times 10^{-3} { s}^{-1} \] Step 3: Substituting into the Arrhenius Equation
\[ \log (1.1 \times 10^{-3}) = \log (4 \times 10^{13}) - \frac{98.6 \times 10^3}{2.303 \times 8.314 \times T} \] Step 4: Solving for \( T \)
\[ T = 311.15 { K} \] Thus, the correct answer is (D).