Question

The following data were collected in two studies of the

reaction

2A + B → C + D

Where

In experiment 1, [B]0 = 5.0 M. In experiment 2, [B]0 =10.0 M.

a. Why is [B] much greater than [A]?

b. Give the rate law and value for kfor this reaction.

c. Which of the following mechanisms could be correct for this reaction? Justify your choice.

i. A+B ↔ E (fast equilibrium)

E + B ↔ C + D (slow)

ii. A + B ↔ E (fast equilibrium)

E + A ↔ C + D (slow)

iii. A + A ↔ E (slow)

E + B ↔ C + D (fast)

Short answer

1. Pseudo-order rate-law equation
2. Rate law = k[A]²[B]

K = 0.050 L² mol^-2 s^-1

3. Only ii is possible

Step-by-step solution

Describing the reason

[B]>>[A], so [B] can be considered constant over the experiments. This gives us a pseudo-order rate-law equation.

Determining the rate law and k for the reaction

In each case the half-life doubles as time increases (in experiment 1 the first half-life is 40 s. the second half-life is 80 s ; in experiment 2, the first half-life is 20 s, the second half-life is the 40s.) This occurs for a second-order reaction, so the reaction is second order in [A]. Between expt. 1 and expt. 2, we double [B] and the reaction rate doubles, thus it is first order in [B]. The overall rate-law equation is rate = k[A]²[B]

Using,

$$\begin{gathered} t_{1/2}=\frac{1}{k\left[A\right]} \\ Weget, \\ k=\frac{1}{\left(40\right)\left(10\times {10}^{-2}\right)}=0.25Lmo{l}^{-1}{s}^{-1} \end{gathered}$$

But this is actually k’ where Rate = k’[A]² and k’ = k[B]

$K=\frac{k\text{'}}{\left[B\right]}=\frac{0.25}{5.0}=0.050L^{2}mo{l}^{-2}{s}^{-1}$

Determining the correct mechanism for the reaction

1. This mechanism gives the wrong stoichiometry, so it can’t be correct.

$$\begin{gathered} Rate=k\left[E\right]\left[A\right] \\ {k}_1\left[A\right]\left[B\right]=k_1\left[E\right] \\ \left[E\right]=\frac{k_1\left[A\right]\left[B\right]}{k_{-1}} \\ Rate=\frac{k{k}_1}{k_{-1}}{\left[A\right]}^2\left[B\right] \end{gathered}$$

This mechanism gives the correct stoichiometry and gives the correct rate law when it is derived from the mechanism. This is a possible mechanism for this reaction.

$Rate=k{\left[A\right]}^2$

This mechanism gives the wrong derived rate law, so it can’t be correct. Only mechanism ii is possible.