To determine the rate law for the reaction [tex]\( A + B \rightarrow C \)[/tex], we need to carefully analyze the influence of the concentrations of [tex]\( A \)[/tex] and [tex]\( B \)[/tex] on the initial rate of the reaction, using the experimental data provided.
Here is the experimental data again for reference:
[tex]\[
\begin{tabular}{c|c|c|c}
\text{Experiment Number} & [A] & [B] & \text{Initial Rate} \\
\hline
1 & 0.451 & 0.855 & 1.13 \\
\hline
2 & 0.451 & 1.77 & 1.13 \\
\hline
3 & 1.35 & 0.885 & 10.17 \\
\hline
\end{tabular}
\][/tex]
### Step-by-Step Solution:
#### Step 1: Determine the order with respect to [tex]\( B \)[/tex]
- Compare experiments 1 and 2, where the concentration of [tex]\( A \)[/tex] is kept constant while the concentration of [tex]\( B \)[/tex] is varied.
- In experiments 1 and 2:
- [tex]\([A]_1 = [A]_2 = 0.451\)[/tex]
- [tex]\([B]_1 = 0.855\)[/tex], [tex]\([B]_2 = 1.77\)[/tex]
- Initial rates: [tex]\( \text{Rate}_1 = 1.13 \)[/tex], [tex]\( \text{Rate}_2 = 1.13 \)[/tex]
Since the initial rates do not change when the concentration of [tex]\( B \)[/tex] is changed from 0.855 to 1.77, we can conclude that the reaction is zero-order with respect to [tex]\( B \)[/tex].
Rate ratio for [tex]\( B \)[/tex]:
[tex]\[
\frac{\text{Rate}_2}{\text{Rate}_1} = \frac{1.13}{1.13} = 1
\][/tex]
Concentration ratio for [tex]\( B \)[/tex]:
[tex]\[
\left( \frac{[B]_2}{[B]_1} \right)^0 = 1 \quad \text{(since any number to the power of 0 is 1)}
\][/tex]
#### Step 2: Determine the order with respect to [tex]\( A \)[/tex]
- Compare experiments 1 and 3, where the concentration of [tex]\( B \)[/tex] is kept constant while the concentration of [tex]\( A \)[/tex] is varied.
- In experiments 1 and 3:
- [tex]\([B]_1 \approx [B]_3 = 0.855 \approx 0.885\)[/tex] (approximately the same)
- [tex]\([A]_1 = 0.451\)[/tex], [tex]\([A]_3 = 1.35\)[/tex]
- Initial rates: [tex]\( \text{Rate}_1 = 1.13 \)[/tex], [tex]\( \text{Rate}_3 = 10.17 \)[/tex]
Rate ratio for [tex]\( A \)[/tex]:
[tex]\[
\frac{\text{Rate}_3}{\text{Rate}_1} = \frac{10.17}{1.13} = 9.0
\][/tex]
Concentration ratio for [tex]\( A \)[/tex]:
[tex]\[
\left( \frac{[A]_3}{[A]_1} \right)^2 = \left( \frac{1.35}{0.451} \right)^2 \approx 8.9601
\][/tex]
Since the rate ratio is approximately the square of the concentration ratio, we conclude that the reaction is second-order with respect to [tex]\( A \)[/tex].
Putting these observations together, the rate law for the given reaction is:
[tex]\[ \text{rate} = k [A]^2 [B]^0 \][/tex]
Since [tex]\([B]^0 = 1\)[/tex], the rate law simplifies to:
[tex]\[ \text{rate} = k [A]^2 \][/tex]
Thus, the correct rate law for the reaction is:
[tex]\[ \boxed{k[A]^2} \][/tex]
