Question

For the reaction, \[ \mathrm{Q + R \xrightleftharpoons[k_{-1}]{k_1} X \xrightarrow{k_2} P} \] Given: $k_1 = 2.5 \times 10^5$ L mol$^{-1}$ s$^{-1}$, $k_{-1} = 1.0 \times 10^4$ s$^{-1}$, and $k_2 = 10$ s$^{-1}$. Under steady-state approximation, the rate constant for the overall reaction in L mol$^{-1}$ s$^{-1}$ (rounded off to the nearest integer) is ________.

Hint:

In multi-step reactions, the steady-state approximation assumes the rate of formation and decomposition of intermediates are equal.

Answer 1

Correct Answer: 250

Step 1: Write the rate expression using steady-state approximation. At steady state for the intermediate $X$, \[ \frac{d[X]}{dt} = k_1[Q][R] - (k_{-1} + k_2)[X] = 0 \] \[ \Rightarrow [X] = \frac{k_1[Q][R]}{k_{-1} + k_2} \] Step 2: Rate of product formation. \[ r = k_2 [X] = \frac{k_1 k_2 [Q][R]}{k_{-1} + k_2} \] Hence, the effective rate constant: \[ k_{\text{eff}} = \frac{k_1 k_2}{k_{-1} + k_2} \] Step 3: Substitute the values. \[ k_{\text{eff}} = \frac{(2.5 \times 10^5)(10)}{1.0 \times 10^4 + 10} = \frac{2.5 \times 10^6}{10010} = 249.75 \] Step 4: Conclusion. Effective rate constant $k_{\text{eff}} \approx 246 \, \text{L mol}^{-1} \, \text{s}^{-1}$.