First-Order Reactions

10 min read Updated Mar 26, 2026

First-order reactions are the most important reaction order for the MCAT. Radioactive decay, many drug metabolism pathways, and numerous biological processes follow first-order kinetics. If you learn only one reaction order thoroughly, make it this one.

Rate Law

Integrated Rate Law

Graphical Analysis

[A] vs. time: Curved (exponential decay). NOT linear.
ln[A] vs. time: Straight line with slope = -k and y-intercept = ln[A]₀.

Graph of ln[H2O2] versus time in hours showing a straight line with negative slope, demonstrating first-order kinetics for hydrogen peroxide decomposition, with data points at 0, 6, 12, 18, and 24 hours — A plot of ln[H₂O₂] vs. time for the decomposition of hydrogen peroxide. The straight line confirms first-order kinetics. The slope equals -k. *Credit: OpenStax Chemistry 2e, CC BY 4.0*

If you plot the data and ln[A] vs. t gives a straight line, you have confirmed first-order kinetics. This is the diagnostic test for first-order reactions.

Half-Life: The Star of First-Order Kinetics

The half-life is the time required for the concentration to drop to half its initial value. For first-order reactions:

Graph of reactant concentration in molarity versus time in seconds for a first-order reaction, showing an exponential decay curve starting at 0.10 M with red brackets marking four successive half-life intervals of equal length approximately 100 seconds each, as concentration drops from 0.10 to 0.05 to 0.025 to 0.0125 to 0.00625 M — First-order decay with constant half-life. Each red bracket ( $t_{1/2}$ ) spans the same time interval (~100 s), even as the concentration drops. After one half-life: 0.10 → 0.05 M. After two: 0.05 → 0.025 M. After three: 0.025 → 0.0125 M. The half-life never changes - this is the defining feature of first-order kinetics. *Credit: Lumen Learning / OpenStax Introductory Chemistry, CC BY 4.0*

Calculating with Half-Lives

After n half-lives, the fraction of the original sample remaining is ( $\frac{1}{2}$ )^n:

Half-Lives Elapsed	Fraction Remaining	Percent Remaining
0	1	100%
1	$\frac{1}{2}$	50%
2	$\frac{1}{4}$	25%
3	$\frac{1}{8}$	12.5%
4	$\frac{1}{16}$	6.25%
5	$\frac{1}{32}$	3.125%

Radioactive Decay

Radioactive decay is the classic example of a first-order process. The rate of decay depends only on the amount of radioactive isotope present, not on temperature, pressure, or chemical environment.

Pharmacokinetics Connection

Most drugs are eliminated from the body via first-order kinetics. The half-life of a drug tells physicians how often to dose: if the half-life is 6 hours, giving a dose every 6 hours maintains a relatively stable blood concentration. This is why “take every 4-6 hours” appears on medication labels.

A radioactive isotope has a half-life of 8 hours. If you start with 120 mg, how much remains after 24 hours?

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15 mg. 24 hours / 8 hours per half-life = 3 half-lives. After 3 half-lives: 120 → 60 → 30 → 15 mg. Or use (

\frac{1}{2}

)³ x 120 = (

\frac{1}{8}

)(120) = 15 mg.

A first-order reaction has k = 0.0231 s⁻¹. What is the half-life?

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30 seconds. t_(

\frac{1}{2}

) = 0.693 / k = 0.693 / 0.0231 = 30 s. Note that 0.693 / 0.0231 can be estimated as 0.7 / 0.023 ≈ 30.