If the time required to reach any fixed fractional conversion varies inversely with the initial concentration of reactants, what is the order of the reaction?

Introduction / Context:
Time-to-fractional-conversion criteria are practical diagnostics for reaction order. Recognizing these signatures is valuable in lab-scale kinetics.

Given Data / Assumptions:

• Initial concentration C0 varies between runs.
• Measured time to a fixed conversion X is inversely proportional to C0.

Concept / Approach:
For a second-order reaction with rate r = k C^2, the integrated form yields t at fixed conversion proportional to 1/C0. For a first-order reaction, the time to a fixed fraction is independent of C0.

Step-by-Step Solution:
1) Second order integrated law: 1/C - 1/C0 = k t.2) At a fixed conversion X, C = C0*(1 - X).3) Substitute to find t = (1/(k*C0)) * (X/(1 - X)). Thus t ∝ 1/C0.4) Therefore, inverse dependence means second order.

Verification / Alternative check:
First order gives t = (1/k) * ln(1/(1 - X)), independent of C0—contrary to the observation.

Why Other Options Are Wrong:
Zero/first/third orders have different C0 dependencies; only second order yields the observed inverse proportionality.

Common Pitfalls:
Confusing half-life patterns (first-order half-life independent of C0) with second-order behavior (half-life ∝ 1/C0).

Final Answer:
Second order.