A rate law is a mathematical equation that shows exactly how the speed of a chemical reaction depends on the concentrations of its reactants. For a reaction involving reactants A and B, the general form is: rate = k[A]ˣ[B]ʸ, where k is the rate constant, the brackets represent concentrations, and the exponents x and y tell you how sensitive the rate is to each reactant. Those exponents are called the reaction orders, and here’s the critical point: they must be determined by experiment. You cannot simply read them off the balanced chemical equation.
The Parts of a Rate Law
Every rate law has three key components working together. The rate itself is how quickly reactant concentrations decrease (or product concentrations increase) over time. The rate constant k is a fixed number for a given reaction at a given temperature. Changing the concentration of a reactant changes the reaction rate, but it does not change k. The exponents (x and y) define the reaction order with respect to each reactant, and the sum of all the exponents gives you the overall reaction order.
If x = 1 and y = 2, for example, the reaction is first order in A, second order in B, and third order overall. That tells you the rate is directly proportional to A’s concentration but proportional to the square of B’s concentration. Double the amount of B, and the reaction rate quadruples.
Reaction Orders and What They Mean
The reaction order tells you the practical relationship between concentration and speed. The most common orders are zero, first, and second.
- Zero order (rate = k): The rate does not depend on the reactant’s concentration at all. No matter how much reactant you add, the reaction proceeds at the same speed. This occurs in some surface-catalyzed reactions where the surface is already saturated.
- First order (rate = k[A]): The rate is directly proportional to the concentration. Double the reactant, double the rate. First-order kinetics show up everywhere, including drug metabolism in the body, where a constant fraction of the drug is eliminated per unit time rather than a constant amount.
- Second order (rate = k[A]²): The rate depends on the square of the concentration. Double the reactant, and the rate increases fourfold.
Each order also has a distinctive half-life pattern. For first-order reactions, the half-life stays constant regardless of concentration, which is why radioactive decay (a first-order process) has a fixed half-life. For zero-order reactions, the half-life shrinks as concentration decreases. For second-order reactions, the half-life gets longer as the reactant is consumed.
Why Reaction Order Differs From the Balanced Equation
One of the most common mistakes in chemistry is assuming that the coefficients in a balanced equation are the same as the exponents in the rate law. This is only true for reactions that happen in a single elementary step. Most reactions proceed through multiple steps, and when they do, the rate law reflects the mechanism of the reaction rather than the overall stoichiometry.
Consider a reaction where two molecules of NO react with one molecule of oxygen to form NO₂. The balanced equation might suggest the rate depends on [NO]² and [O₂], but the actual rate law depends on what happens in each step of the mechanism. The rate-determining step, the slowest step in the sequence, acts as a bottleneck. Think of it like a funnel: the narrowest point controls how fast liquid flows through, regardless of how wide the rest of the funnel is. The rate law is built from the reactants involved in that slowest step, using the stoichiometric coefficients of that specific elementary step.
How Rate Laws Are Determined Experimentally
Since you cannot predict reaction orders from the balanced equation alone, chemists use experimental data to figure them out. The most common approach is the method of initial rates. You run the reaction several times, changing only one reactant’s concentration at a time, and measure the initial rate each time.
The logic is straightforward. Suppose you double the concentration of A while keeping B constant, and the rate doubles. That means the reaction is first order in A (exponent = 1). If doubling A causes the rate to quadruple, the reaction is second order in A (exponent = 2). If doubling A has no effect on the rate, it’s zero order in A.
For cases where the relationship isn’t obvious by inspection, you can solve for the exponent algebraically. Take the ratio of rates from two trials where only one concentration changed, then take the natural logarithm of both sides. The exponent equals ln(rate₁/rate₂) divided by ln([A]₁/[A]₂). Once you know all the exponents, you plug any single trial’s data back into the rate law to calculate k.
The Rate Constant and Temperature
The rate constant k is not truly constant across all conditions. It stays fixed at a specific temperature, but raise the temperature and k increases, often dramatically. This happens for two reasons: molecules collide more frequently at higher temperatures, and those collisions carry more kinetic energy, making them more likely to overcome the energy barrier needed for the reaction to occur.
The relationship between k and temperature follows the Arrhenius equation. In practical terms, the equation says that k increases exponentially as temperature rises. If you plot the natural logarithm of k against the inverse of temperature (in Kelvin), you get a straight line. The slope of that line reveals the activation energy of the reaction, which is the minimum energy molecules need to react when they collide. Catalysts also increase k, not by raising temperature, but by lowering that activation energy.
Units of the Rate Constant
The units of k depend on the overall reaction order, and this is a useful consistency check. The rate always has units of concentration per time (typically mol/L per second), so k’s units must compensate for whatever concentration terms appear in the rate law.
- Zero order: k has units of mol·L⁻¹·s⁻¹
- First order: k has units of s⁻¹
- Second order: k has units of L·mol⁻¹·s⁻¹
If you calculate k and the units don’t match what’s expected for that reaction order, something in your calculation is off. This makes unit analysis a quick way to verify your work.
Integrated Rate Laws and Concentration Over Time
The standard rate law (called the differential rate law) tells you the reaction speed at any instant. But if you want to know how much reactant remains after a certain amount of time, you need the integrated rate law. Each reaction order has its own version.
- Zero order: [A] = [A]₀ − kt. Concentration drops linearly with time.
- First order: ln[A] = ln[A]₀ − kt. Concentration decreases exponentially, meaning equal fractions are consumed in equal time intervals.
- Second order: 1/[A] = 1/[A]₀ + kt. Concentration drops steeply at first, then slows considerably.
These equations also help identify the reaction order from experimental data. If you plot concentration versus time and get a straight line, the reaction is zero order. If the natural log of concentration versus time is linear, it’s first order. If the inverse of concentration versus time is linear, it’s second order. The slope of whichever plot gives a straight line equals k (or negative k, depending on the form).