Integrated rate laws: concentration versus time

From rate laws to concentration versus time

The rate law rate = k[A]^m tells you the speed right now. But chemists often need the concentration later — how much reactant remains after an hour, or how long until 90% is gone. Integrating the differential rate law over time answers exactly that.

Each order integrates to a different relationship:

• Zeroth order: $[A]_t = -kt + [A]_0$
• First order: $[A]_t = [A]_0e^{-kt}$, or equivalently $\ln[A]_t = -kt + \ln[A]_0$
• Second order: $\dfrac{1}{[A]_t} = kt + \dfrac{1}{[A]_0}$

All three assume constant temperature (so k does not change) and a single reactant dominating the rate expression.

Using the first-order law

The first-order form is the workhorse of kinetics. Because the fraction remaining after time t is $e^{-kt}$, the math is clean. Example: a first-order decomposition has k = 0.0500 s⁻¹ and starts at [A]₀ = 0.500 M. After 30.0 s:

$$[A]_{30} = 0.500 \times e^{-(0.0500)(30.0)} = 0.500 \times e^{-1.50} = 0.500 \times 0.223 = 0.112\ \text{M}$$

Notice that after one "time constant" (t = 1/k = 20 s), about 37% ($e^{-1}$) remains — a useful mental benchmark. The same equation governs radioactive decay, which is why half-life and first-order kinetics are inseparable.

Second and zeroth order in practice

Second-order reactions fall off faster at first, then linger: 1/[A] grows linearly with time. Example: for 2A → products with k = 0.80 M⁻¹s⁻¹ and [A]₀ = 0.40 M, the time to reach [A] = 0.10 M is

$$t = \frac{1/k}{ } \Rightarrow t = \frac{\frac{1}{0.10} - \frac{1}{0.40}}{0.80} = \frac{10 - 2.5}{0.80} = 9.4\ \text{s}$$

Zeroth order is the simplest: concentration drops at a constant rate, like a taxi meter, until the reactant runs out (the law stops applying at [A] = 0 — negative concentrations are unphysical).

Straight-line tests for order

Because each integrated law rearranges into y = mx + b form, the right plot is a straight line — and the wrong ones curve:

• Zeroth order: [A] vs t is linear, slope = −k
• First order: ln[A] vs t is linear, slope = −k
• Second order: 1/[A] vs t is linear, slope = +k

In the lab, you plot all three and see which is straightest. The slope hands you k directly, and the linearity confirms the order — two results from one graph.

Variables and units

Common mistakes

Real-world applications

• Pharmacokinetics: first-order integrated laws model how drug concentration falls between doses.

• Environmental cleanup: predicting how long a pollutant persists in groundwater.

• Materials aging: modeling the fading of dyes and the degradation of polymers.

• Nuclear chemistry: every radioactive decay calculation is a first-order integrated rate law.