Rate law inference:\nFor the reaction X → Y, tripling [X] increases the rate ninefold. What is the order of the reaction with respect to X?

Introduction / Context:
Determining reaction order from how the rate scales with concentration is a common kinetic skill. If the rate follows rate = k[X]^n, changes in [X] map to predictable changes in rate, allowing n to be inferred from a single experiment when the change is clear.

Given Data / Assumptions:

• Single-reactant rate law: rate ∝ [X]^n.
• When [X] → 3[X], rate → 9 × rate.
• Temperature and k are constant.

Concept / Approach:
Compare ratios: (rate_new / rate_old) = ( [X]_new / [X]_old )^n. Substitute 9 = 3^n and solve for n.

Step-by-Step Solution:
Given: 9 = 3^n.Take logs or recognize powers: 3^2 = 9.Therefore, n = 2 (second order in X).

Verification / Alternative check:
A double-increase test: doubling [X] in a second-order law quadruples the rate; consistent with tripling causing a ninefold increase.

Why Other Options Are Wrong:
Zero order: rate independent of [X]; first order: triples, not ninefold; third order: 3^3 = 27; fractional 1.5 gives 3^1.5 ≈ 5.2, not 9.

Common Pitfalls:
Confusing order with molecularity, or reading 9 as 3×3 accidentally.

Final Answer:
2