Kinetics Formulas

Why This Matters

Kinetics is where chemistry gets dynamic—you're moving beyond what reactions can happen (thermodynamics) to how fast they actually occur. The AP exam loves testing your ability to connect mathematical formulas to physical meaning: why does a first-order half-life stay constant while a second-order half-life depends on concentration? How does the Arrhenius equation explain why reactions speed up when heated? These aren't just equations to memorize—they're tools for predicting and explaining reaction behavior.

You're being tested on your ability to interpret graphical data, derive rate laws from mechanisms, and connect temperature to reaction rates through activation energy. The formulas in this guide appear in multiple-choice calculations, but they really shine in FRQs where you'll need to explain why a plot is linear, calculate rate constants from experimental data, or predict how changing conditions affects reaction speed. Don't just memorize the math—know what concept each formula illustrates and when to apply it.

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Rate Laws and Reaction Orders

The rate law tells you how concentration affects reaction speed. The key insight: reaction order must be determined experimentally for overall reactions—you can't just read it from coefficients (unless it's an elementary step).

Rate Law Equation

$\text{rate} = k[A]^m[B]^n$

• Exponents (m, n) represent reaction orders—these must be determined experimentally and indicate how sensitive the rate is to each reactant's concentration
• Rate constant (k) is temperature-dependent and specific to each reaction; its units change based on overall reaction order
• Overall order = m + n—this determines the units of k and which integrated rate law applies

Rate-Determining Step

$\text{overall rate} = \text{rate of slowest step}$

• The slowest elementary step controls the overall reaction rate—like the narrowest part of a funnel limiting flow
• Rate laws can be derived directly from elementary steps—stoichiometric coefficients do equal rate law exponents for elementary reactions
• Intermediates in the rate law indicate a more complex mechanism requiring the steady-state or equilibrium approximation

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Integrated Rate Laws

Integrated rate laws connect concentration to time, allowing you to predict concentrations at any point and determine reaction order from graphical data. The key: each order produces a different linear plot.

Zero-Order Integrated Rate Law

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

• Plot $[A]$ vs. $t$ gives a straight line with slope $= -k$; concentration decreases at a constant rate regardless of how much reactant remains
• Units of k: $\text{M} \cdot \text{s}^{-1}$—rate is literally constant, common for surface-catalyzed reactions where the catalyst is saturated
• Half-life: $t_{1/2} = \frac{[A]_0}{2k}$—directly proportional to initial concentration, so each successive half-life is shorter

First-Order Integrated Rate Law

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

• Plot $\ln[A]$ vs. $t$ gives a straight line with slope $= -k$; concentration decreases exponentially
• Units of k: $\text{s}^{-1}$—the rate constant has no concentration dependence built in
• Radioactive decay follows first-order kinetics—this is the most commonly tested order because half-life behavior is so distinctive

Second-Order Integrated Rate Law

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

• Plot $\frac{1}{[A]}$ vs. $t$ gives a straight line with slope $= +k$; note the positive slope, unlike zero and first order
• Units of k: $\text{M}^{-1} \cdot \text{s}^{-1}$—rate depends on concentration squared (or product of two concentrations)
• Half-life: $t_{1/2} = \frac{1}{k[A]_0}$—inversely proportional to initial concentration, so each successive half-life doubles

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Half-Life Relationships

Half-life connects rate constants to practical timescales. The first-order formula is by far the most tested because of its elegant simplicity and real-world applications.

First-Order Half-Life

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

• Half-life is independent of initial concentration—whether you start with 1 mol or 1000 mol, it takes the same time to lose half
• Directly relates to the rate constant—a larger k means faster decay and shorter half-life
• Essential for radioactive decay calculations—carbon-14 dating, nuclear medicine, and decay chain problems all use this relationship

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Temperature and Activation Energy

The Arrhenius equation is the bridge between kinetics and thermodynamics—it explains why temperature affects rate through the lens of activation energy and molecular collisions.

Arrhenius Equation

$k = Ae^{-E_a/RT}$

• $E_a$ (activation energy) is the minimum energy required for reaction; higher $E_a$ means slower reaction at any given temperature
• $A$ (pre-exponential factor) represents collision frequency and proper orientation—the "best-case scenario" rate constant if every collision succeeded
• Exponential dependence on $\frac{1}{T}$ means small temperature increases cause large rate increases; doubling temperature doesn't double the rate—it can increase it by orders of magnitude

Two-Point Arrhenius Equation

$\ln\left(\frac{k_2}{k_1}\right) = \frac{E_a}{R}\left(\frac{1}{T_1} - \frac{1}{T_2}\right)$

• Compares rate constants at two temperatures—allows you to calculate $E_a$ from experimental data without knowing $A$
• Rearranged form of Arrhenius equation—derived by taking $\ln(k_2) - \ln(k_1)$ and eliminating the $A$ term
• Watch your algebra: if $T_2 > T_1$, then $\frac{1}{T_1} > \frac{1}{T_2}$, making the right side positive when $k_2 > k_1$

Collision Theory Equation

$k = pZe^{-E_a/RT}$

• $Z$ represents collision frequency—how often molecules collide, increasing with concentration and temperature
• $p$ is the steric factor—the fraction of collisions with correct orientation; most collisions fail because molecules aren't aligned properly
• Combines with Arrhenius: $A = pZ$, so the pre-exponential factor has physical meaning rooted in collision geometry

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Catalysis and Energy Barriers

Catalysts change how fast equilibrium is reached without changing where equilibrium lies. Understanding their effect on activation energy is crucial for both kinetics and connecting to thermodynamic vs. kinetic control.

Catalyst Effect on Activation Energy

$E_a(\text{catalyzed}) < E_a(\text{uncatalyzed})$

• Catalysts provide an alternative reaction pathway with a lower energy barrier—they don't change $\Delta G$ or $\Delta H$, only the path between reactants and products
• Both forward and reverse rates increase equally—equilibrium position is unchanged, but it's reached faster
• Catalysts are regenerated—they participate in the mechanism but aren't consumed overall, appearing in intermediate steps but not the net equation

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Quick Reference Table

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Self-Check Questions

1. Which two integrated rate laws produce plots with negative slopes, and which one has a positive slope? What does this tell you about how to identify reaction order graphically?

2. A reaction's half-life doubles when the initial concentration is cut in half. What is the reaction order, and which integrated rate law applies?

3. Compare and contrast the Arrhenius equation and collision theory: what physical meaning does the pre-exponential factor $A$ have in terms of molecular behavior?

4. If a catalyst is added to a reversible reaction at equilibrium, what happens to the forward rate, reverse rate, and equilibrium constant? Explain using the relationship $K_{eq} = \frac{k_f}{k_r}$.

5. An FRQ gives you rate constant values at 300 K and 350 K. Write out the equation you would use to calculate activation energy and identify what each variable represents.