Integrated Rate Law Equations

Why This Matters

Integrated rate laws let you predict exactly how much reactant remains at any point during a reaction. They connect reaction order to a characteristic equation, graph shape, half-life behavior, and rate constant units. These concepts appear frequently on the AP exam, both in multiple-choice questions asking you to identify reaction order from data and in FRQs where you calculate concentrations, determine half-lives, or analyze graphs.

Each reaction order produces a unique mathematical fingerprint. Zero-order reactions proceed at a constant rate regardless of concentration. First-order reactions slow down proportionally as reactant depletes. Second-order reactions slow down even more dramatically. Don't just memorize the equations: understand why each order behaves differently and how to identify them from graphs, half-life patterns, and unit analysis.

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The Core Equations: Mathematical Forms by Order

Each reaction order has a distinct integrated rate law relating concentration to time. The mathematical form determines everything else: graph shape, half-life behavior, and rate constant units.

Zero-Order Integrated Rate Law

$[A] = [A]_0 - kt$

Concentration decreases linearly with time, making this the simplest form. The rate is constant and independent of concentration; the reaction proceeds at a steady pace until the reactant is completely gone.

Zero-order kinetics commonly show up in surface-catalyzed reactions where the catalyst surface is saturated. Once every active site is occupied, adding more reactant molecules doesn't speed anything up.

First-Order Integrated Rate Law

$\ln[A] = \ln[A]_0 - kt$

This natural log form is what you'll use for graphing and most calculations. There's also an equivalent exponential decay form:

$[A] = [A]_0 e^{-kt}$

The concentration decreases by the same fraction in each equal time interval. This is the most frequently tested order because it applies to radioactive decay, many decomposition reactions, and biological processes.

Second-Order Integrated Rate Law

$\frac{1}{[A]} = \frac{1}{[A]_0} + kt$

The reciprocal form creates a positive slope when graphed against time. The rate depends on concentration squared (or on the product of two reactant concentrations in bimolecular steps), so the reaction slows dramatically as concentration drops because far fewer productive collisions occur.

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Half-Life Behavior: The Concentration Connection

Half-life equations reveal how reaction order affects the time needed to consume half the reactant. The relationship between half-life and initial concentration is one of the most useful diagnostic tools you have.

Zero-Order Half-Life

$t_{1/2} = \frac{[A]_0}{2k}$

Half-life is directly proportional to initial concentration. As the reaction progresses and less reactant remains, each successive half-life gets shorter. Students often mix this up on exams, so pay attention: for zero-order, successive half-lives decrease.

First-Order Half-Life

$t_{1/2} = \frac{0.693}{k}$

This is the famous constant half-life equation. There's no concentration term at all, so whether you start with 1 M or 0.001 M, the half-life is identical. If you see successive half-lives that are equal in experimental data, the reaction is first-order.

Second-Order Half-Life

$t_{1/2} = \frac{1}{k[A]_0}$

Half-life is inversely proportional to initial concentration. Each successive half-life is longer than the last because the concentration keeps dropping. A useful calculation shortcut: doubling the initial concentration cuts the half-life in half.

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Graphical Analysis: Finding the Straight Line

The graphical method is the most reliable way to determine reaction order from experimental data. Whichever plot yields a straight line reveals the order.

Zero-Order Graph

Plot $[A]$ vs. time. A linear graph with a negative slope confirms zero-order kinetics. The slope equals $-k$, so you can read the rate constant directly. The y-intercept gives $[A]_0$.

First-Order Graph

Plot $\ln[A]$ vs. time. Linearity confirms first-order behavior. The slope equals $-k$ (the line slopes downward). This is the most commonly tested graphing scenario on AP exams.

Second-Order Graph

Plot $\frac{1}{[A]}$ vs. time. A straight line with a positive slope indicates second-order. The slope equals $+k$ (positive, unlike the other two orders). The y-intercept equals $\frac{1}{[A]_0}$, so take the reciprocal to find the actual initial concentration.

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Rate Constant Units: The Dimensional Analysis Check

The rate law must always yield units of $\text{M/s}$ for the reaction rate, so the units of $k$ change depending on how many concentration terms appear. Working backward from units is a quick way to verify reaction order.

Zero-Order Rate Constant Units

Units: $\text{M/s}$ (equivalently, $\text{mol·L}^{-1}\text{·s}^{-1}$)

Since rate $= k$ with no concentration terms, $k$ itself must carry the full rate units of concentration per time.

First-Order Rate Constant Units

Units: $\text{s}^{-1}$

Dimensional check: rate $= k[A]$, so $k = \frac{\text{rate}}{[A]} = \frac{\text{M/s}}{\text{M}} = \text{s}^{-1}$. Notice these are inverse-time units only, with no concentration component. This is consistent with the half-life equation $t_{1/2} = 0.693/k$, which only works dimensionally if $k$ has units of inverse time.

Second-Order Rate Constant Units

Units: $\text{M}^{-1}\text{s}^{-1}$

Dimensional check: rate $= k[A]^2$, so $k = \frac{\text{rate}}{[A]^2} = \frac{\text{M/s}}{\text{M}^2} = \text{M}^{-1}\text{s}^{-1}$. If a problem gives you $k$ with these units, you immediately know it's second-order without needing any other information.

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Determining Reaction Order: Experimental Methods

Real kinetics problems require you to identify reaction order from data. The AP exam tests several approaches.

Method of Initial Rates

Compare rates at different starting concentrations while holding everything else constant. If doubling $[A]$ doubles the rate, the reaction is first-order in A. If doubling $[A]$ quadruples the rate, it's second-order. If doubling $[A]$ has no effect on rate, it's zero-order.

On FRQs, you'll typically set up ratios of rate laws from two trials to solve for the order algebraically.

Integrated Rate Law Method

Plot all three graphs ($[A]$ vs. time, $\ln[A]$ vs. time, $1/[A]$ vs. time) and identify which one is linear. This is the most definitive method because it uses all data points across the entire reaction, not just initial values. If you're given a table of concentration vs. time data, this is usually the expected approach.

Half-Life Analysis

Observe how $t_{1/2}$ changes across different initial concentrations or successive intervals:

• Constant half-life → first-order
• Decreasing half-life (successive half-lives get shorter) → zero-order
• Increasing half-life (successive half-lives get longer) → second-order

This is the quickest diagnostic when half-life data is provided directly in the problem.

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Quick Reference Table

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Self-Check Questions

1. If successive half-lives in an experiment are 20 s, 40 s, and 80 s, what is the reaction order, and how do you know?

2. A reaction has a rate constant of $0.025 \text{ M}^{-1}\text{s}^{-1}$. What is the reaction order, and what integrated rate law equation would you use?

3. Compare and contrast how you would graphically determine whether a reaction is first-order versus second-order. What would you plot, and what would you look for?

4. For a zero-order reaction with $[A]_0 = 0.80 \text{ M}$ and $k = 0.020 \text{ M/s}$, calculate the half-life. How would this half-life change if you started with $[A]_0 = 0.40 \text{ M}$?

5. An FRQ provides concentration-time data and asks you to determine reaction order. Describe the systematic approach you would use, including what calculations or graphs you would create.