To determine the rate law for this reaction, we need to analyze how the rate changes with respect to changes in the concentrations of the reactants $[A]$, $[B]$, and $[C]$. The rate law generally takes the form:

[ \text{Rate} = k[A]^x[B]^y[C]^z ]

Where:

• $k$ is the rate constant.
• $x$, $y$, and $z$ are the orders of reaction with respect to $A$, $B$, and $C$, respectively.

Step-by-Step Analysis:

1. Comparing Experiments 1 and 2:

   • $[A]$ increases from 0.100 to 0.275 (factor of 2.75), while $[B]$ and $[C]$ remain constant.
   • The rate increases from 4.62 to 34.96.
   • This change is primarily due to $[A]$, so we calculate the effect on the rate:

   $\frac{34.96}{4.62} = 7.57$

   • We know that $(2.75)^x = 7.57$, solving for $x$:

   $x \approx 2$

2. Comparing Experiments 2 and 3:

   • $[A]$ increases from 0.275 to 0.335, and $[B]$ increases from 0.220 to 0.315, while $[C]$ remains constant.
   • The rate increases from 34.96 to 51.88.
   • To isolate the effect of $[B]$, we need to account for both $[A]$ and $[B]$.
   • We previously determined that $x = 2$, so now we focus on $[B]$:

   $\frac{51.88}{34.96} = 1.48$

   • $[B]$ changes from 0.220 to 0.315 (a factor of 1.43). Using the known relation $(1.43)^y = 1.48$, solving for $y$:

   $y \approx 1$

3. Comparing Experiments 3 and 4:

   • $[A]$ and $[B]$ both change, and $[C]$ decreases from 0.315 to 0.250.
   • To determine the effect of $[C]$, calculate the ratio of the rates and isolate $[C]$'s contribution:

   $\frac{18.20}{51.88} = 0.35$

   • $[C]$ changes from 0.315 to 0.250, a factor of approximately 0.79. Using $(0.79)^z = 0.35$, solving for $z$:

   $z \approx 2$

Conclusion:

The rate law is:

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

The correct answer is:

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

Do you need any further clarification on the calculations or the rate law concept?

Here are 5 related questions:

1. How do you calculate the reaction order for a specific reactant?
2. Why is it important to keep the concentrations of other reactants constant when determining the reaction order of one reactant?
3. How does the rate constant $k$ vary with temperature?
4. What are the units of the rate constant $k$ for different orders of reactions?
5. Can the order of a reaction be fractional or zero, and what does that imply?

Tip: When determining the reaction order, always look for pairs of experiments where the concentration of only one reactant changes while others are held constant.