The problem requires determining the reaction order with respect to $[A]$ and $[B]$ using the rate law:

[ \text{Rate} = k[A]^{\text{order}_A}[B]^{\text{order}_B} ]

We'll analyze the given data to find the reaction orders with respect to $[A]$ and $[B]$.

Step 1: Data Table Recap

Trial	$[A]$ (mol/L)	$[B]$ (mol/L)	Rate (mol/(L·min))
1	0.771	0.682	0.0893
2	1.54	0.682	0.357
3	0.771	1.36	0.357
4	0.774	0.291	0.0164

Step 2: Determine Order of Reaction with Respect to $[A]$

To isolate the effect of $[A]$, we compare trials where $[B]$ remains constant. Comparing Trial 1 and Trial 2:

• Trial 1: $[A]_1 = 0.771 \, \text{mol/L}, \, \text{Rate}_1 = 0.0893 \, \text{mol/(L·min)}$
• Trial 2: $[A]_2 = 1.54 \, \text{mol/L}, \, \text{Rate}_2 = 0.357 \, \text{mol/(L·min)}$
• $[B]$ remains constant at 0.682 mol/L.

Now, calculate the ratio of the rates and the concentrations:

[ \frac{\text{Rate}_2}{\text{Rate}_1} = \frac{k[A]_2^{\text{order}_A}[B]^{\text{order}_B}}{k[A]_1^{\text{order}_A}[B]^{\text{order}_B}} = \left(\frac{[A]_2}{[A]_1}\right)^{\text{order}_A} ]

$\frac{0.357}{0.0893} = \left(\frac{1.54}{0.771}\right)^{\text{order}_A}$

$4 = 2^{\text{order}_A}$

This gives us $\text{order}_A = 2$.

Step 3: Determine Order of Reaction with Respect to $[B]$

Now, compare Trial 1 and Trial 3, where $[A]$ remains constant:

• Trial 1: $[B]_1 = 0.682 \, \text{mol/L}, \, \text{Rate}_1 = 0.0893 \, \text{mol/(L·min)}$
• Trial 3: $[B]_3 = 1.36 \, \text{mol/L}, \, \text{Rate}_3 = 0.357 \, \text{mol/(L·min)}$
• $[A]$ remains constant at 0.771 mol/L.

Again, calculate the ratio of the rates and concentrations:

[ \frac{\text{Rate}_3}{\text{Rate}_1} = \frac{k[A]^{\text{order}_A}[B]_3^{\text{order}_B}}{k[A]^{\text{order}_A}[B]_1^{\text{order}_B}} = \left(\frac{[B]_3}{[B]_1}\right)^{\text{order}_B} ]

$\frac{0.357}{0.0893} = \left(\frac{1.36}{0.682}\right)^{\text{order}_B}$

$4 = 2^{\text{order}_B}$

This gives us $\text{order}_B = 2$.

Step 4: Conclusion

• The reaction order with respect to $[A]$ is $2$.
• The reaction order with respect to $[B]$ is $2$.
• Therefore, the overall rate law is:

[ \text{Rate} = k[A]^2[B]^2 ]

Would you like more details on this or have any questions?

Here are 5 related questions you might find helpful:

1. How do you calculate the rate constant $k$ given reaction data?
2. What happens to the reaction rate if only $[B]$ is doubled?
3. How does temperature affect the rate constant $k$?
4. What is the difference between zero-order and first-order reactions?
5. How would the rate law change if the reaction order with respect to $A$ was 1 instead of 2?

Tip: When determining reaction orders experimentally, always look for trials where only one reactant concentration changes while the others remain constant.