Second-Order Reactions

10 min read Updated Mar 26, 2026

A second-order reaction is one where the rate depends on the square of one reactant’s concentration, or on the product of two reactants’ concentrations. Physically, this often corresponds to a reaction that requires two molecules to collide - a bimolecular event.

Rate Law

Second-order reactions can take two forms:

When the rate law is k[A][B] with first-order dependence on each of two reactants, it suggests a collision between one molecule of A and one molecule of B as the rate-determining step.

Integrated Rate Law

For a reaction that is second order with respect to a single reactant:

Graphical Analysis

[A] vs. time: Curved (not linear). The curve decreases steeply at first, then levels off.
ln[A] vs. time: Also curved (NOT linear). This rules out first-order.
1/[A] vs. time: Straight line with slope = +k and y-intercept = 1/[A]₀.

If 1/[A] vs. t gives a straight line, the reaction is second order. Note that the slope is positive (unlike zero-order and first-order, where the slope is negative).

Two side-by-side graphs for the dimerization of butadiene C4H6: left panel shows ln[C4H6] versus time in seconds with a curved line confirming the reaction is NOT first order, right panel shows 1/[C4H6] versus time with a straight line confirming second-order kinetics — Determining reaction order for the dimerization of C₄H₆. Left: ln[C₄H₆] vs. time is curved, ruling out first-order. Right: 1/[C₄H₆] vs. time is linear with a positive slope equal to k, confirming second-order kinetics. *Credit: OpenStax Chemistry 2e, CC BY 4.0*

Half-Life

This makes intuitive sense with the dance partner analogy: as the party empties out, it takes longer and longer to find someone to dance with.

Master Comparison Table

Property	Zero Order	First Order	Second Order
Rate Law	rate = k	rate = k[A]	rate = k[A]²
Integrated Law	[A] = [A]₀ - kt	ln[A] = ln[A]₀ - kt	1/[A] = 1/[A]₀ + kt
Linear Plot	[A] vs. t	ln[A] vs. t	1/[A] vs. t
Slope of Linear Plot	-k	-k	+k
Half-Life	[A]₀ / 2k	0.693 / k	1 / (k[A]₀)
Half-Life Depends on [A]₀?	Yes (shorter each time)	No (constant!)	Yes (longer each time)
Units of k	M·s⁻¹	s⁻¹	M⁻¹·s⁻¹

A reaction has a half-life that doubles each time. What is the reaction order?

Click to reveal answer

Second order. For second-order reactions, t_(

\frac{1}{2}

) = 1/(k[A]₀). As [A]₀ decreases by half each half-life, the next half-life is twice as long. Only second-order kinetics show increasing half-lives.

You plot three graphs: [A] vs. t, ln[A] vs. t, and 1/[A] vs. t. The 1/[A] vs. t plot is the only one that gives a straight line. What is the reaction order, and what does the slope equal?

Click to reveal answer

Second order. The slope equals +k (positive). For second-order reactions, 1/[A] = 1/[A]₀ + kt is linear with a positive slope equal to the rate constant k.