Key Concepts of Reaction Rate Laws

Why This Matters

Reaction rate laws are the mathematical backbone of chemical kinetics—they tell you how fast reactions happen and why that speed changes under different conditions. On the AP exam, you're being tested on your ability to connect experimental data to rate expressions, predict how concentration changes over time, and explain the molecular-level reasons behind reaction speeds. These concepts show up everywhere from enzyme kinetics in biochemistry to industrial process optimization.

Don't just memorize formulas here. You need to understand what each rate law form reveals about the reaction mechanism, how to extract reaction orders from experimental data, and why temperature has such a dramatic effect on reaction speed. Master the connections between rate laws, integrated rate laws, and half-life expressions, and you'll be ready for any kinetics question the exam throws at you.

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Foundational Concepts: What Rate Laws Describe

Before diving into specific reaction orders, you need a solid grasp of what rate laws actually measure and how they're structured. Rate laws quantify the relationship between reactant concentrations and the speed at which products form.

Reaction Rate

• Change in concentration per unit time—typically expressed as $-\frac{\Delta[A]}{\Delta t}$ for reactants or $+\frac{\Delta[B]}{\Delta t}$ for products
• Instantaneous vs. average rates—instantaneous rates (the slope of the concentration-time curve at a point) are more useful for kinetic analysis
• Units are always $M/s$ or $mol \cdot L^{-1} \cdot s^{-1}$—this consistency helps you check your work on calculations

Rate Law Equation

• $Rate = k[A]^m[B]^n$—where $k$ is the rate constant and exponents $m$ and $n$ are reaction orders determined experimentally
• Cannot be predicted from stoichiometry—the rate law reflects the mechanism, not the balanced equation
• Each reaction has a unique rate law—you must derive it from experimental data, typically using the method of initial rates

Order of Reaction

• Exponents in the rate law—zero, first, second, or even fractional orders indicate how concentration affects rate
• Overall order = sum of individual orders—for $Rate = k[A]^2[B]^1$, the overall order is 3
• Fractional orders suggest complex mechanisms—they often indicate multi-step reactions with intermediates

Rate Constant

• The proportionality factor $k$—specific to each reaction and temperature, with units that depend on overall reaction order
• Higher $k$ means faster reaction—it reflects the intrinsic speed of the reaction under given conditions
• Temperature-dependent—$k$ increases exponentially with temperature according to the Arrhenius equation

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Reaction Orders: How Concentration Affects Rate

Different reactions respond to concentration changes in distinct ways. The reaction order determines the mathematical relationship between concentration and rate, and it dictates the shape of concentration-time graphs.

Zero-Order Reactions

• Rate is independent of concentration—the rate law is simply $Rate = k$, meaning the reaction proceeds at constant speed
• Integrated rate law: $[A] = [A]_0 - kt$—concentration decreases linearly with time
• Common when a catalyst is saturated—enzyme reactions at high substrate concentrations or surface-catalyzed reactions often show zero-order behavior

First-Order Reactions

• Rate is directly proportional to one reactant—doubling $[A]$ doubles the rate, with $Rate = k[A]$
• Integrated rate law: $\ln[A] = \ln[A]_0 - kt$—a plot of $\ln[A]$ vs. time gives a straight line with slope $-k$
• Constant half-life: $t_{1/2} = \frac{0.693}{k}$—radioactive decay and many decomposition reactions follow first-order kinetics

Second-Order Reactions

• Rate depends on concentration squared—either $Rate = k[A]^2$ or $Rate = k[A][B]$ for two different reactants
• Integrated rate law: $\frac{1}{[A]} = \frac{1}{[A]_0} + kt$—a plot of $\frac{1}{[A]}$ vs. time is linear with slope $k$
• Half-life depends on initial concentration: $t_{1/2} = \frac{1}{k[A]_0}$—later half-lives get progressively longer as concentration drops

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Time-Dependent Analysis: Integrated Rate Laws and Half-Life

While differential rate laws describe instantaneous rates, integrated rate laws let you calculate concentrations at any point during the reaction—essential for predicting reaction progress.

Integrated Rate Laws

• Connect concentration to time directly—derived by integrating the differential rate law
• Each order has a characteristic linear plot—zero-order: $[A]$ vs. $t$; first-order: $\ln[A]$ vs. $t$; second-order: $\frac{1}{[A]}$ vs. $t$
• The linear plot determines reaction order—whichever transformation gives a straight line reveals the order

Half-Life

• Time for $[A]$ to drop to $\frac{1}{2}[A]_0$—a convenient measure for comparing reaction speeds
• Order-dependent formulas—zero-order: $t_{1/2} = \frac{[A]_0}{2k}$; first-order: $t_{1/2} = \frac{0.693}{k}$; second-order: $t_{1/2} = \frac{1}{k[A]_0}$
• Diagnostic for reaction order—if half-life stays constant, it's first-order; if it increases with decreasing concentration, it's second-order

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Experimental Determination: Finding the Rate Law

You can't predict rate laws from balanced equations—they must be determined experimentally. The method of initial rates is your primary tool for this.

Method of Initial Rates

• Measure rate at the very start of reaction—before significant concentration changes occur, using varying initial concentrations
• Compare trials systematically—if doubling $[A]$ doubles the rate, the order with respect to $A$ is 1; if it quadruples the rate, the order is 2
• Calculate $k$ after determining orders—substitute known values into the rate law and solve for the rate constant

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Temperature and Energy: Why Reactions Speed Up

Temperature dramatically affects reaction rates because it changes both collision frequency and the fraction of molecules with sufficient energy. The Arrhenius equation and collision theory explain this connection quantitatively.

Collision Theory

• Reactions require effective collisions—molecules must collide with sufficient energy (activation energy) and proper orientation
• Higher temperature increases collision energy—more molecules exceed the activation energy threshold
• Concentration increases collision frequency—more particles per volume means more collisions per second

Activation Energy

• $E_a$ is the energy barrier for reaction—the minimum energy required to break bonds and form the transition state
• Higher $E_a$ means slower reaction—fewer molecules have enough energy to overcome the barrier
• Determines temperature sensitivity—reactions with high $E_a$ speed up more dramatically with temperature increases

Arrhenius Equation

• $k = Ae^{-E_a/RT}$—relates rate constant to temperature, where $A$ is the pre-exponential factor and $R = 8.314 \, J/(mol \cdot K)$
• Linear form: $\ln k = -\frac{E_a}{R} \cdot \frac{1}{T} + \ln A$—plot $\ln k$ vs. $\frac{1}{T}$ to get slope $= -\frac{E_a}{R}$
• Two-point form: $\ln\frac{k_2}{k_1} = \frac{E_a}{R}\left(\frac{1}{T_1} - \frac{1}{T_2}\right)$—use this when given rate constants at two temperatures

Temperature Dependence of Reaction Rates

• Rule of thumb: rate doubles every 10°C—a rough approximation for many reactions near room temperature
• Exponential relationship—small temperature changes cause large rate changes due to the exponential term in the Arrhenius equation
• Critical for industrial and biological processes—temperature control is essential for reaction optimization

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Catalysis: Speeding Up Reactions Without Changing $E_a$ of Products

Catalysts provide an alternative reaction pathway with lower activation energy. They're not consumed in the reaction and don't change the equilibrium position—just how fast equilibrium is reached.

Catalysts and Their Effects on Reaction Rates

• Lower the activation energy—provide an alternative mechanism with a smaller energy barrier
• Homogeneous catalysts—same phase as reactants (acids catalyzing ester hydrolysis in solution)
• Heterogeneous catalysts—different phase, typically solid surfaces (catalytic converters in cars, Haber process iron catalyst)

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Quick Reference Table

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Self-Check Questions

1. Given that a reaction's half-life decreases as the reaction progresses, what is the reaction order, and how would you confirm this graphically?

2. Compare and contrast how you would use the method of initial rates versus integrated rate law plots to determine a reaction's order with respect to a single reactant.

3. Two reactions have the same rate constant at 300 K, but Reaction A has $E_a = 50 \, kJ/mol$ and Reaction B has $E_a = 100 \, kJ/mol$. Which reaction's rate will increase more when temperature is raised to 350 K? Explain using the Arrhenius equation.

4. A catalyst increases a reaction's rate by a factor of 1000 at constant temperature. What does this tell you about the change in activation energy? (Hint: use the Arrhenius equation)

5. For a second-order reaction with $Rate = k[A][B]$, how would you design an experiment to determine the order with respect to each reactant independently?