Integrated Rate Law Equations

Why This Matters

Integrated rate laws are the mathematical backbone of chemical kinetics—they let you predict exactly how much reactant remains at any point in time. On the AP exam, you're being tested on your ability to connect reaction order to its characteristic equation, graph shape, half-life behavior, and rate constant units. These concepts show up repeatedly in multiple-choice questions asking you to identify reaction order from data, and in FRQs where you'll need to calculate concentrations, determine half-lives, or analyze experimental graphs.

The key insight is that each reaction order produces a unique mathematical fingerprint. Zero-order reactions march along at a constant pace 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 that relates 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 mathematical form
• Rate is constant and independent of concentration; the reaction proceeds at a steady pace until reactant is completely depleted
• Common in surface-catalyzed reactions where the catalyst surface is saturated, so adding more reactant doesn't speed things up

First-Order Integrated Rate Law

• $\ln[A] = \ln[A]_0 - kt$—the natural log form is what you'll use for graphing and calculations
• Exponential decay form: $[A] = [A]_0 e^{-kt}$—concentration decreases by the same fraction in each time interval
• 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
• Rate depends on concentration squared (or the product of two reactant concentrations in bimolecular collisions)
• Reaction slows dramatically as concentration drops because fewer molecular 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 a powerful diagnostic tool.

Zero-Order Half-Life

• $t_{1/2} = \frac{[A]_0}{2k}$—half-life is directly proportional to initial concentration
• Half-life decreases as the reaction progresses because less reactant means less time to halve what remains
• Exam trap: Students often forget that successive half-lives get shorter, not longer, for zero-order reactions

First-Order Half-Life

• $t_{1/2} = \frac{0.693}{k}$—the famous constant half-life equation with no concentration dependence
• Half-life is independent of $[A]_0$—whether you start with 1 M or 0.001 M, the half-life is identical
• Key identifier: If successive half-lives 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
• Half-life increases as concentration decreases; each successive half-life is longer than the last
• Doubling the initial concentration cuts the half-life in half—a useful relationship for calculations

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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 reaction order.

Zero-Order Graph

• Plot $[A]$ vs. time—a linear graph with negative slope confirms zero-order kinetics
• Slope equals $-k$—the rate constant can be read directly from the graph's slope
• Y-intercept equals $[A]_0$—provides the initial concentration if unknown

First-Order Graph

• Plot $\ln[A]$ vs. time—linearity confirms first-order behavior
• Slope equals $-k$—note the negative sign; the line slopes downward
• Most common graphing question on AP exams because first-order reactions are so prevalent

Second-Order Graph

• Plot $\frac{1}{[A]}$ vs. time—a straight line with positive slope indicates second-order
• Slope equals $+k$—unlike the other orders, this slope is positive
• Y-intercept equals $\frac{1}{[A]_0}$—remember to take the reciprocal to find actual initial concentration

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Rate Constant Units: The Dimensional Analysis Check

Rate constant units vary by order because the rate law equation must always yield units of M/s for the reaction rate. Working backward from units is a quick way to verify reaction order.

Zero-Order Rate Constant Units

• Units: $\text{M/s}$ or $\text{mol·L}^{-1}\text{·s}^{-1}$—rate equals $k$ directly, so $k$ has rate units
• Dimensional check: rate = $k$, so $k$ must have units of concentration per time
• Memory trick: Zero concentration terms in the rate law means $k$ carries all the units

First-Order Rate Constant Units

• Units: $\text{s}^{-1}$—inverse time only, no concentration units
• Dimensional check: rate = $k[A]$, so $k$ = rate/$[A]$ = (M/s)/M = $\text{s}^{-1}$
• Appears in half-life equation $t_{1/2} = 0.693/k$, which only works if $k$ has units of inverse time

Second-Order Rate Constant Units

• Units: $\text{M}^{-1}\text{s}^{-1}$—inverse molarity times inverse seconds
• Dimensional check: rate = $k[A]^2$, so $k$ = rate/$[A]^2$ = (M/s)/M² = $\text{M}^{-1}\text{s}^{-1}$
• Quick identifier: If a problem gives $k$ with these units, you immediately know it's second-order

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Determining Reaction Order: Experimental Methods

Real kinetics problems require you to identify reaction order from data. Multiple methods exist, and the AP exam tests all of them.

Method of Initial Rates

• Compare rates at different starting concentrations—if doubling $[A]$ doubles the rate, it's first-order in A
• Systematic approach: Hold all concentrations constant except one, then observe how rate changes
• FRQ favorite: You'll often need to set up ratios of rate laws to solve for reaction orders

Integrated Rate Law Method

• Plot all three graphs ($[A]$, $\ln[A]$, $1/[A]$ vs. time) and identify which is linear
• Most definitive method because it uses all data points, not just initial values
• Exam tip: If given a table of concentration vs. time data, this method is usually expected

Half-Life Analysis

• Observe how $t_{1/2}$ changes with different initial concentrations
• Constant half-life = first-order; decreasing half-life = zero-order; increasing half-life = second-order
• Quick 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.