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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.

Using the experimental data provided, what is the rate law for this reaction involving reactants A, B, and C?

Learn how to determine the rate law for a reaction using experimental data. This example involves calculating the reaction orders for reactants A, B, and C by analyzing how the rate changes with respect to changes in concentrations. Suitable for high school and college-level chemistry students.

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