12.3 Rate Laws

Reaction Kinetics and Rate Laws

Rate laws give you a mathematical way to predict how fast a reaction proceeds based on the concentrations of its reactants. Understanding them is essential for controlling reaction speeds, and they also reveal clues about how reactions happen at the molecular level.

Components of Rate Laws

A rate law is an equation that connects the reaction rate to the concentrations of the reactants. The general form looks like this:

$Rate = k[A]^m[B]^n$

• $k$ is the rate constant, a value that depends on temperature and the specific reaction. A larger $k$ means a faster reaction at the same concentrations.
• $[A]$ and $[B]$ are the molar concentrations of reactants A and B.
• $m$ and $n$ are the reaction orders with respect to A and B. These are not taken from the balanced equation coefficients. They must be determined experimentally.

The overall reaction order is the sum of the individual orders ($m + n$). For example, if $m = 1$ and $n = 2$, the reaction is third-order overall.

Rate laws also help identify the rate-determining step in a multi-step reaction. The rate-determining step is the slowest elementary step, and the overall rate law should be consistent with it.

Calculation of Reaction Rates

To calculate a rate, plug the concentrations and rate constant directly into the rate law.

Example: For $A + B \rightarrow C$ with rate law $Rate = k[A]^2[B]$

Given: $[A]_0 = 0.1 \text{ M}$, $[B]_0 = 0.2 \text{ M}$, $k = 0.5 \text{ M}^{-2}\text{s}^{-1}$

1. Substitute into the rate law: $Rate = (0.5)(0.1)^2(0.2)$
2. Square the concentration of A: $(0.1)^2 = 0.01$
3. Multiply: $0.5 \times 0.01 \times 0.2 = 1 \times 10^{-3} \text{ M s}^{-1}$

Notice the units of $k$ change depending on the overall reaction order. For this third-order reaction, $k$ has units of $\text{M}^{-2}\text{s}^{-1}$.

Integrated Rate Laws

To predict how concentration changes over time, you use integrated rate laws. Each reaction order has its own form:

Half-life is the time it takes for the reactant concentration to drop to half its initial value. Notice that for a first-order reaction, the half-life is constant (it doesn't depend on concentration), while for zero- and second-order reactions, the half-life changes as concentration changes.

Determination of Reaction Orders

Reaction orders must be found experimentally. You can't just look at the balanced equation. There are two main methods:

Method of Initial Rates

This is the most common approach in intro chemistry. You run the reaction multiple times, changing one reactant's concentration while holding the others constant, and compare the initial rates.

1. Pick two trials where only one reactant's concentration changes.
2. Set up the ratio of rates: $\frac{Rate_2}{Rate_1} = \left(\frac{[A]_2}{[A]_1}\right)^m$
3. Solve for the exponent $m$.

Example: If you double $[A]$ and the rate quadruples, then $\frac{4}{1} = 2^m$, so $m = 2$ (second-order in A). If you double $[A]$ and the rate stays the same, $m = 0$ (zero-order in A).

1. Repeat for each reactant to find all the orders.
2. Once you know all the orders, substitute any trial's data back into the rate law and solve for $k$.

Graphical Method

Plot concentration data over time using the three linear forms from the integrated rate law table above. Whichever plot gives a straight line tells you the reaction order. The slope of that line gives you $k$.

Advanced Kinetics Concepts

• Zero-order reactions have a rate that doesn't depend on reactant concentration at all: $Rate = k$. These are less common but show up when a catalyst surface is saturated or an enzyme is at maximum capacity.
• Reaction mechanisms are sequences of elementary steps that describe what actually happens at the molecular level. The rate law you determine experimentally should match the rate law predicted by the mechanism's rate-determining step. If they don't match, the proposed mechanism is wrong.