Question

Consider the following hypothetical reaction: $$ 2 \mathrm{AB}_{2}(g) \longrightarrow \mathrm{A}_{2}(g)+2 \mathrm{~B}_{2}(g) $$ A 500.0 -mL flask is filled with \(0.384 \mathrm{~mol}\) of \(\mathrm{AB}_{2}\). The appearance of \(\mathrm{A}_{2}\) is monitored at timed intervals. Assume that temperature and volume are kept constant. The data obtained are shown in the table below. $$ \begin{array}{lcccccc} \hline \text { Time (min) } & 0 & 10 & 20 & 30 & 40 & 50 \\ \text { moles of } \mathrm{A}_{2} & 0 & 0.0541 & 0.0833 & 0.1221 & 0.1432 & 0.1567 \\ \hline \end{array} $$ (a) Make a similar table for the disappearance of \(\mathrm{AB}_{2}\). (b) What is the average rate of disappearance of \(\mathrm{AB}_{2}\) over the second and third 10 -minute intervals? (c) What is the average rate of appearance of \(\mathrm{A}_{2}\) between \(t=30\) and \(t=50 ?\)

Short answer

Question: Determine the average rate of disappearance of AB₂ between 10 and 20 minutes and the average rate of appearance of A₂ between 30 and 50 minutes. Answer: The average rate of disappearance of AB₂ between 10 and 20 minutes is 0.00584 mol/min, while the average rate of appearance of A₂ between 30 and 50 minutes is 0.00173 mol/min.

Step-by-step solution

(a) Calculating the moles of AB₂ as the reaction progresses.

To find the moles of AB₂ that are left as the reaction progresses, we can use the stoichiometry of the reaction: 2 moles of AB₂ are consumed for each mole of A₂ that is produced. Therefore, to find the moles of AB₂ at each time point, we can subtract the moles of A₂ produced at that time multiplied by 2 from the initial moles of AB₂. Initial moles of AB₂ = 0.384 mol Formula to find moles of AB₂: Mo‎‌‍‍‍‎‏‍‍‍‍‍‌‍‍‌èles of AB₂ at time t = Initial moles of AB₂ - 2 × (moles of A₂ formed at time t) Using this formula, we can complete the table for the disappearance of AB₂: $$ \begin{array}{lcccccc} \hline \text { Time (min) } & 0 & 10 & 20 & 30 & 40 & 50 \\ \text { moles of } \mathrm{AB}_{2} & 0.384 & 0.2758 & 0.2174 & 0.1398 & 0.0976 & 0.0706 \\ \hline \end{array} $$

(b) Determining the average rate of disappearance of AB₂ over the second and third 10-minute intervals.

To find the average rate of disappearance of AB₂ over the 10-20 minute interval, first find the difference in moles of AB₂ between 10 and 20 minutes. $$ \text{Moles of AB₂ at 10 mins:} ~ 0.2758~\text{mol} \\ \text{Moles of AB₂ at 20 mins:} ~ 0.2174~\text{mol} \\ \text{Difference in moles of AB₂:} ~ 0.2758 - 0.2174 = 0.0584~\text{mol} $$ Now, divide this difference by the time interval (10 minutes) to find the average rate of disappearance: $$ \text{Average rate of disappearance:} ~ \frac{0.0584}{10} = 0.00584~\text{mol/min} $$

(c) Determining the average rate of appearance of A₂ between t=30 and t=50.

Following a similar procedure as in part (b) but for the appearance of A₂ between t=30 and t=50 minutes: $$ \text{Moles of A₂ at 30 mins:} ~ 0.1221~\text{mol} \\ \text{Moles of A₂ at 50 mins:} ~ 0.1567~\text{mol} \\ \text{Difference in moles of A₂:} ~ 0.1567 - 0.1221 = 0.0346~\text{mol} $$ Now, divide this difference by the time interval (20 minutes) to find the average rate of appearance: $$ \text{Average rate of appearance:} ~ \frac{0.0346}{20} = 0.00173~\text{mol/min} $$