A reaction has rate r = k [A]^m [B]^n. If doubling [A] increases the formation rate of product C by a factor of 2.82 and tripling [B] increases the rate by a factor of 9, what is the overall reaction order (m + n)?

Difficulty: Medium

Correct Answer: 7/2

Explanation:


Introduction / Context:
Determining reaction order from how the rate responds to concentration changes is a fundamental kinetics skill. Here, separate experiments vary [A] and [B] to infer exponents m and n and then the overall order m + n.


Given Data / Assumptions:

  • Rate law: r = k [A]^m [B]^n.
  • When [A] → 2[A], rate multiplies by 2.82.
  • When [B] → 3[B], rate multiplies by 9.
  • Other concentration is held constant in each test.


Concept / Approach:
Use the proportionality of rate to each concentration raised to its order. For A: 2^m = 2.82. For B: 3^n = 9. Solve m and n, then sum them.


Step-by-Step Solution:
1) For A: 2^m = 2.82 ⇒ m = log(2.82)/log(2) ≈ 1.496 ≈ 1.5, i.e., 3/2.2) For B: 3^n = 9 ⇒ n = log(9)/log(3) = 2.3) Overall order = m + n ≈ 1.5 + 2 = 3.5 = 7/2.


Verification / Alternative check:
Plug m = 3/2 and n = 2 back: 2^(3/2) = 2.828 (≈ 2.82) and 3^2 = 9. The predicted factors match the observations.



Why Other Options Are Wrong:
7/4 and 5/4 are too small; 5/2 is 2.5 (insufficient). Option 3 ignores the non-integer order for A; accurate calculation gives 3.5.



Common Pitfalls:
Rounding m prematurely to 1 or 2; not isolating the change of one reactant at a time; forgetting to sum m and n for overall order.



Final Answer:
7/2.

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