Chemistry · Unit 5 Kinetics · 14 min read · Updated 2026-05-11
Concentration changes over time — AP Chemistry
AP Chemistry · Unit 5 Kinetics · 14 min read
1. Integrated Rate Laws by Reaction Order★★☆☆☆⏱ 4 min
An integrated rate law expresses reactant concentration as an explicit function of time, derived by integrating the differential rate law for a given reaction order. For each common order, the integrated rate law can be rearranged to the linear form $y = mx + b$ to enable experimental identification of reaction order. Standard notation: $[A]_0$ = initial concentration of reactant A at $t=0$, $[A]_t$ = concentration at time $t$, $k$ = rate constant, $t_{1/2}$ = half-life.
For a zero-order reaction with differential rate law $\text{rate} = -\frac{d[A]}{dt} = k$, the integrated form is:
[A]_t = -kt + [A]_0
This is linear when plotting $[A]$ (y-axis) vs $t$ (x-axis), with slope equal to $-k$ and intercept equal to $[A]_0$.
For a first-order reaction with differential rate law $\text{rate} = -\frac{d[A]}{dt} = k[A]$, the integrated form is:
\ln[A]_t = -kt + \ln[A]_0
This is linear when plotting $\ln[A]$ (y-axis) vs $t$ (x-axis), with slope equal to $-k$ and intercept equal to $\ln[A]_0$.
For a second-order reaction in a single reactant with differential rate law $\text{rate} = -\frac{d[A]}{dt} = k[A]^2$, the integrated form is:
\frac{1}{[A]_t} = kt + \frac{1}{[A]_0}
This is linear when plotting $1/[A]$ (y-axis) vs $t$ (x-axis), with slope equal to $+k$ and intercept equal to $1/[A]_0$.
Exam tip: When asked to determine reaction order from concentration-time data, always confirm which transformed concentration gives a straight line; constant half-life is a useful shortcut only for first-order reactions.
2. Half-Life of Reactions★★☆☆☆⏱ 3 min
Each reaction order has a unique half-life relationship derived directly from its integrated rate law, which can be used to quickly identify order and calculate time to reach a given concentration.
For first-order reactions, substituting $[A]_t = [A]_0/2$ into the integrated rate law gives:
t_{1/2} = \frac{\ln 2}{k} \approx \frac{0.693}{k}
The key unique property of first-order half-life is that it is *independent of initial concentration*, so it remains constant throughout the entire reaction.
For zero-order reactions, the half-life relationship is:
t_{1/2} = \frac{[A]_0}{2k}
Half-life depends directly on initial concentration, so it increases as the reaction proceeds and $[A]$ decreases.
For second-order reactions, the half-life relationship is:
t_{1/2} = \frac{1}{k[A]_0}
Half-life depends inversely on initial concentration, so it also increases as the reaction proceeds and $[A]$ decreases.
Exam tip: When calculating time to reach a given percentage of initial concentration for first-order reactions, convert the fraction to powers of 1/2 to avoid logarithm calculation errors.
3. Graphical Determination of Reaction Order★★★☆☆⏱ 3 min
A common AP Chemistry exam task is to determine reaction order from experimental data using graphical methods, based on the linear form of each integrated rate law. The core rule is: whichever transformation of concentration gives a straight line when plotted against time confirms the matching reaction order.
AP exam questions may ask you to identify order from provided graphs, draw the correct transformed graph from raw data, or calculate the rate constant from the slope of the correct linear graph.
Exam tip: Remember that for second-order reactions, the slope of $1/[A]$ vs $t$ is positive and equal to $k$, while zero and first-order plots have negative slopes equal to $-k$. Sign errors for $k$ are extremely common in graph questions.
4. AP-Style Practice Check★★★☆☆⏱ 4 min
Common Pitfalls
Why: Students often memorize the simple first-order half-life formula and forget that other orders have concentration-dependent half-life formulas.
Why: The integrated rate law for zero and first order gives slope = $-k$, but students often copy the slope value without adjusting for sign.
Why: Only first-order reactions have constant half-life independent of initial concentration; other orders have changing half-life as concentration drops.
Why: Confusion between the three linear forms of integrated rate laws, especially between first and second order.
Why: The standard $1/[A]_t = kt + 1/[A]_0$ formula is only derived for second order in a single reactant or two reactants with equal initial concentrations.