MCAT topics → Thermodynamics and Kinetics → Integrated Rate Laws and Reaction Order

Integrated Rate Laws and Reaction Order

MCAT trap: Assigns first-order kinetics to a linear [A] vs. time plot instead of zero-order. A straight line on [A] vs. time indicates zero-order; first-order gives a straight line on ln[A] vs. time; second-order gives a straight line on 1/[A] vs. time.

Integrated rate laws connect concentration to time — they're the equations you use when a problem gives you time elapsed and asks for remaining concentration, or vice versa. The three forms (zero, first, second order) each have a distinct mathematical shape, and the MCAT tests all three. Expect both direct recall (give me the equation) and application (use it to find k, find t, find [A]). Passage-based questions will often hand you a graph or a data table and ask you to identify the reaction order — which requires knowing which linearized plot corresponds to which order.

What makes this topic tricky is that students memorize isolated facts without building a coherent picture. The most common error: seeing a straight line on a [A] vs. time graph and calling it first-order. That's zero-order. First-order gives a straight line on ln[A] vs. time. Second-order gives a straight line on 1/[A] vs. time. The MCAT loves to test exactly this distinction in graph-interpretation questions. A second major trap is the half-life misconception — students learn that half-life is constant and concentration-independent for first-order reactions, then incorrectly generalize that to all reactions.

The calculation angle is also real. You need to be comfortable plugging into each integrated rate law and solving for the missing variable. That means knowing which equation to grab (based on order), isolating the right variable, and executing the algebra without confusing the forms. Don't mix up the first-order log form with the second-order reciprocal form — they look nothing alike, but under pressure students conflate them.

Common misconceptions

Common mistake
Wrong: A straight line on a plot of [A] vs. time always indicates a first-order reaction.
Right: A straight line on [A] vs. time indicates zero-order; first-order gives a straight line on ln[A] vs. time; second-order gives a straight line on 1/[A] vs. time.
A straight [A] vs. time graph means concentration drops at a constant rate — that's the definition of zero-order kinetics, where rate doesn't depend on concentration at all. First-order reactions produce exponential decay in [A], which only becomes linear when you take the natural log; so ln[A] vs. time is the first-order tell. Second-order produces a different curve that only linearizes as 1/[A] vs. time. Match the plot shape to the integrated rate law form, not to intuition about 'simplicity.'
Common mistake
Wrong: Half-life is always independent of initial concentration for any reaction order.
Right: Only first-order half-life is concentration-independent; zero-order and second-order half-lives depend on initial concentration.
The concentration-independence of half-life is a special property of first-order kinetics, not a universal rule. For first-order reactions, the t½ = 0.693/k formula has no concentration term — that's why it's constant. For zero-order reactions, t½ = [A]₀/2k, and for second-order, t½ = 1/(k[A]₀) — both explicitly depend on initial concentration. Generalizing the first-order rule to all orders is one of the most reliable wrong-answer traps on the MCAT.
Common mistake
Wrong: The second-order integrated rate law is ln[A] = ln[A]₀ − kt (same form as first-order but with different k).
Right: The second-order integrated rate law is 1/[A] = 1/[A]₀ + kt, which is distinct in form from the first-order law.
The second-order integrated rate law is 1/[A] = 1/[A]₀ + kt — it's a reciprocal relationship, not a logarithmic one. The first-order form uses ln[A] because integrating 1/[A] gives a natural log; the second-order form comes from integrating 1/[A]², which gives the reciprocal. These two equations have fundamentally different algebraic structures, and mixing them up will lead to wrong answers on both calculation questions and graph-identification questions.
Free Deck audit

See if your Anki deck covers this topic.

Upload your deck →
Guided session

Stuck on this? An AI tutor that probes your understanding.

Start a session →

What the exam tests

  1. Know the three integrated rate law equations by form: zero-order uses [A] directly, first-order uses ln[A], and second-order uses 1/[A] — and be able to write or recognize each.
  2. Given a graph of concentration data over time, identify the reaction order by determining which plot (linear [A], ln[A], or 1/[A] vs. time) produces a straight line.
  3. Explain why first-order half-life is constant and concentration-independent, while zero-order and second-order half-lives change as the initial concentration changes.
  4. Perform calculations using integrated rate laws: given initial concentration, rate constant, and time, find remaining concentration; or given concentration data, solve for k or the time required to reach a target concentration.

Can you avoid these mistakes?

A reaction has the following data: [A] vs. time is curved, ln[A] vs. time is curved, but 1/[A] vs. time is a straight line with positive slope. What is the reaction order, and what does the slope of that line represent?
A first-order reaction has a half-life of 20 minutes regardless of starting concentration. A student claims this must also be true for a second-order reaction with the same rate constant. What's wrong with this reasoning, and how does the second-order half-life actually depend on concentration?
You have a zero-order reaction with k = 0.05 M/s and [A]₀ = 2.0 M. What is the concentration of A after 30 seconds? At what time does the reaction effectively stop (i.e., [A] = 0)?
For a first-order reaction, if ln[A]₀ = 3.0 and k = 0.1 min⁻¹, what is ln[A] after 10 minutes? What is the actual concentration [A] at that time?

Related topics

Reaction Rates and Rate Laws First Law of Thermodynamics and Internal Energy Enthalpy and Hess's Law Calorimetry and Heat Capacity Entropy and the Second Law

See how your Anki deck covers this topic.

Upload your deck for a free audit →