14.2 Rate Laws
PRE-HEALTH PREP
Rate Laws
Reactant Orders and the Rate Constant Explained
General Chemistry 14.2
Lesson Overview
Rate laws express how the speed of a chemical reaction depends on reactant concentrations through the relationship Rate = k[A]x[B]y, where k is the rate constant and the exponents represent reactant orders. Reactant orders—most commonly zero, first, or second—must be determined from experimental data by systematically varying one concentration at a time and observing the effect on reaction rate. The rate constant k is calculated by substituting known values into the rate law equation, and its units depend on the overall reaction order, providing a predictive tool for any set of concentrations.
Table of Contents
What Is a Rate Law in Chemistry?
Think about what "laws" do in science. The law of gravity lets you predict that a dropped object will fall. A rate law works the same way: it allows you to predict the rate of a chemical reaction based on the concentrations of the reactants involved. Understanding rate laws is a foundational skill in chemical kinetics.
A typical rate law equation takes this form:
Each piece of this expression carries specific meaning:
- k — the rate constant, a proportionality constant specific to the reaction at a given temperature. Note: this is a lowercase k, not a capital K (which represents the equilibrium constant in the next chapter).
- [A] and [B] — the molar concentrations of the reactants. Typically only reactants appear in a rate law, though exceptions exist.
- x and y — the reactant orders with respect to each reactant. The exponent x is the order with respect to A, and y is the order with respect to B.
- Overall reaction order — the sum of all individual reactant orders. For this general rate law, the overall order equals x + y.
Important Distinction: k vs. K
The rate constant uses a lowercase k. In the next chapter, you will encounter a capital K that represents the equilibrium constant—something entirely different. Keep these symbols straight — mixing them up is a common source of lost points on exams.
3 Types of Reactant Orders
Reactant orders in a rate law are most commonly integers — particularly 0, 1, and 2 — and they sometimes correspond to how many molecules of a given reactant participate in the slow step of the reaction mechanism. However, the orders do not necessarily match the stoichiometric coefficients in the balanced equation. Sometimes they line up; sometimes they do not. The only reliable way to determine reactant orders is from experimental data.
Common Mistake: Assuming Orders Match Coefficients
You cannot simply look at the coefficients in a balanced equation and assume they are the reactant orders. While this occasionally works out, it is not a safe assumption. Reactant orders must be determined experimentally.
Are Reactant Orders Always Integers?
In a typical general chemistry course, you will almost exclusively encounter integer orders (0, 1, or 2). However, fractional and decimal orders are entirely possible in real kinetics experiments. So the statement "reactant orders are always integers" is false — a popular true/false trap.
Zero Order Reactants
Consider a simple reaction A → B with the rate law:
Since anything raised to the zero power equals one, the concentration of A simply drops out of the expression. The reaction rate equals k regardless of how much A is present. Whether you use 1 M or 10 M of reactant A, the rate remains the same.
Hallmark of Zero Order
The concentration of a zero-order reactant has no effect on the reaction rate. Change it however you like — double it, triple it, cut it in half — and the rate stays the same.
First Order Reactants
Now consider a reaction M → N with the rate law:
Since anything to the first power is itself, the exponent of 1 is typically omitted. The reaction rate is now directly proportional to the concentration of M. If you double the concentration of M, the rate doubles. Triple it, and the rate triples. Increase it by a factor of 10, and the rate increases tenfold.
Hallmark of First Order
Whatever you do to the concentration, the rate does the exact same thing. Double the concentration → double the rate. This direct proportionality is the signature of a first-order reactant.
Second Order Reactants
For a reaction X → Y with the rate law:
The reaction rate is now proportional to the square of the concentration of X, which means this reactant has a much larger impact on the rate. If you double the concentration of X, the rate increases by a factor of 22 = 4. If you triple it, the rate jumps by a factor of 32 = 9. A tenfold increase in concentration produces a hundredfold increase in rate.
Hallmark of Second Order
Whatever you do to the concentration, the rate does the same thing squared. Double the concentration → rate quadruples (22 = 4). Triple the concentration → rate increases ninefold (32 = 9).
Comparing All 3 Reactant Orders
| Order | Exponent | Effect of Doubling [Reactant] | Rate Depends On |
|---|---|---|---|
| Zero | 0 | No change in rate | Independent of concentration |
| First | 1 | Rate doubles (×2) | Proportional to [reactant] |
| Second | 2 | Rate quadruples (×4) | Proportional to [reactant]2 |
What About Third Order?
In principle, third-order reactants exist — doubling the concentration would increase the rate by 23 = 8. In practice, they are extremely rare, and you are unlikely to encounter them in a general chemistry course.
Determining Rate Laws from Experimental Data
The "Change One Variable at a Time" Principle
Before diving into the math, consider this analogy. Imagine you turn the key on your vehicle and nothing happens — no engine turnover, no sound at all. The problem could be a dead battery or a dead starter. If you replace both parts simultaneously and the vehicle starts, you have fixed the issue, but you have no idea which part was actually faulty (and which one you could have returned for a refund).
The smarter approach is to change only one variable at a time. Replace the battery first. If the vehicle starts, the battery was the problem. If not, swap the starter instead. By isolating each variable, you identify exactly what caused the change.
Applying This to Kinetics Experiments
The same logic applies when determining rate laws from experimental data. We run multiple experimental trials, but between any two trials we compare, only one reactant concentration should change. That way, if the reaction rate changes, we know exactly which reactant is responsible.
How to Choose the Right Trial Pairs
When comparing experimental trials to determine the order with respect to a particular reactant:
- Find two trials where only that reactant's concentration differs.
- All other reactant concentrations must remain constant between the two trials.
- Compare what happened to the concentration and what happened to the reaction rate.
If both reactant concentrations change between two trials, any change in rate cannot be attributed to either one individually — and the comparison becomes useless for determining reactant orders.
Worked Example: Finding the Rate Law for NO + Cl2 → NOCl
Rate laws must be determined from experimental data.
2NO(g) + Cl2(g) ⇌ 2NOCl(g)
The following data were obtained for the reaction above:
Consider the reaction between nitrogen monoxide and chlorine gas to form nitrosyl chloride. Three experimental trials were conducted at the same temperature, with different initial concentrations. This step-by-step example shows how to determine the rate law equation from data.
| Trial | [NO] (M) | [Cl2] (M) | Rate (M/s) |
|---|---|---|---|
| 1 | 0.10 | 0.10 | 0.18 |
| 2 | 0.10 | 0.20 | 0.36 |
| 3 | 0.20 | 0.20 | 1.44 |
The expected form of the rate law equation is:
Our job is to determine the reactant orders x and y, and then calculate the rate constant k.
Finding the Reactant Order with Respect to NO
We need two trials where only the NO concentration changes. Comparing Trial 1 to Trial 2 does not work because both [NO] and [Cl2] differ. Comparing Trial 1 to Trial 3 fails for the same reason. However, Trials 2 and 3 are ideal: [Cl2] stays at 0.20 M while [NO] changes.
- Concentration change: [NO] goes from 0.10 → 0.20 M (doubles, ×2)
- Rate change: 0.36 → 1.44 M/s (1.44 ÷ 0.36 = 4, a ×4 increase)
The concentration doubled, and the rate quadrupled. What power connects these? Since 2x = 4, we get x = 2. NO is second order.
The Logic
The concentration of NO was multiplied by 2. The rate was multiplied by 4. Ask: 2 to what power equals 4? Answer: 22 = 4, so the reactant order with respect to NO is 2.
Finding the Reactant Order with Respect to Cl2
Now we need two trials where only the Cl2 concentration changes. Trials 1 and 2 fit perfectly: [NO] stays at 0.10 M while [Cl2] changes.
- Concentration change: [Cl2] goes from 0.10 → 0.20 M (doubles, ×2)
- Rate change: 0.18 → 0.36 M/s (0.36 ÷ 0.18 = 2, a ×2 increase)
The concentration doubled, and the rate also doubled. Since 2y = 2, we get y = 1. Cl2 is first order.
Overall Reaction Order
With NO second order (2) and Cl2 first order (1), the overall reaction order is 2 + 1 = 3 (third order overall).
How to Calculate the Rate Constant k
Now that the reactant orders are known, the rate law equation takes the form:
Rearranging to solve for the rate constant k:
Plug in values from any trial — the rate constant k should come out the same regardless of which trial you choose (it is, after all, a constant). Using Trial 1 for its simple numbers:
Tip: Pick the Easiest Numbers
When calculating the rate constant k from experimental data, you can use any trial. Choose whichever has the friendliest numbers. In a real lab setting, you might calculate k from each trial and confirm that the values are consistent.
Rate Constant k Units for 4 Reaction Orders
The units of the rate constant k depend on the overall reaction order, because different numbers of concentration terms appear in the denominator of the rate law equation. For our example, the rate has units of M/s and the denominator has units of M2 × M = M3, giving k units of M−2 s−1.
Here is a quick-reference table linking rate constant units to overall reaction order:
| Overall Reaction Order | Units of Rate Constant k |
|---|---|
| 0 (zero order) | M · s−1 |
| 1 (first order) | s−1 |
| 2 (second order) | M−1 · s−1 |
| 3 (third order) | M−2 · s−1 |
Rate Constant Units as a Shortcut
If a problem gives you a rate constant with units of s−1, you immediately know the reaction is first order overall. If the units are M−1 s−1, it is second order. This shortcut works in both directions: if you know the overall reaction order, you know the units of k, and vice versa.
Writing the Complete Rate Law
Putting everything together, the complete rate law equation for this reaction is:
This is the power of a rate law in chemical kinetics. Once established, you no longer need to run the reaction in the lab to know how fast it will go. Want to know the reaction rate when [NO] = 2.0 M and [Cl2] = 3.0 M? Just plug those values in and calculate. The rate law lets you predict the reaction rate for any set of concentrations, just as the law of gravity lets you predict that a dropped object will fall.
Making a Rate Prediction
Suppose [NO] = 2.0 M and [Cl2] = 3.0 M:
Rate = 180 × (2.0)2 × (3.0) = 180 × 4.0 × 3.0 = 2160 M/s
No beakers required — the rate law does the work for you.
Coefficients vs. Reactant Orders: A Critical Distinction
In this particular reaction (NO + Cl2 → NOCl), the reactant orders happen to match the stoichiometric coefficients. This is a special case, not the general rule. Do not assume coefficients equal reactant orders — they must always be determined from experimental data. We will encounter one specific exception later in this chapter where you can read the orders directly from the equation.
Key Takeaways
- A rate law equation expresses the reaction rate as a function of reactant concentrations and allows you to predict rates without running the experiment: Rate = k [A]x[B]y.
- The rate constant (lowercase k) is a proportionality constant that remains fixed at a given temperature.
- Reactant orders are most commonly 0, 1, or 2, but can be fractions — they must be determined from experimental data, not from balanced equation coefficients.
- The overall reaction order is the sum of all individual reactant orders.
- To determine reactant orders experimentally, compare two trials where only one reactant concentration changes, then observe how the reaction rate responds.
- A zero-order reactant does not affect the rate; a first-order reactant changes the rate proportionally; a second-order reactant changes the rate by the square of the concentration change.
- The units of rate constant k depend on the overall reaction order: M · s−1 (zero), s−1 (first), M−1 s−1 (second), M−2 s−1 (third).
- Once the rate law is established, it serves as a predictive tool — plug in any concentrations to calculate the expected reaction rate.