Last updated: 2026-07-01

What a rate law describes

A rate law connects the rate of a reaction to the concentrations of its reactants. For a reaction that depends on a single reactant A, it takes the form

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

where k is the rate constant and n is the reaction order in A. The order is an experimental quantity: it cannot be read from the balanced equation and must be measured. The method of initial rates is the most direct way to find it.

The method of initial rates

The idea is to run the reaction twice with different starting concentrations of A and record the initial rate each time — the rate at the very start, before products accumulate. Writing the rate law for both experiments and dividing one by the other makes the rate constant cancel:

$$\frac{r_2}{r_1} = \left(\frac{[A]_2}{[A]_1}\right)^{\,n}$$

Taking logarithms isolates the order:

$$n = \frac{\ln\left(r_2 / r_1\right)}{\ln\left([A]_2 / [A]_1\right)}$$

Once n is known, the rate constant follows by substituting it back into either experiment:

$$k = \frac{r_1}{[A]_1^{\,n}}$$

Symbol	Quantity	Unit
[A]₁, [A]₂	Concentrations of A in the two experiments	mol/L
r₁, r₂	Initial rates in the two experiments	mol/(L·s)
n	Reaction order in A	dimensionless
k	Rate constant	order-dependent

Worked example

A reaction is studied at two concentrations of A:

• Experiment 1: [A]₁ = 0.10 mol/L, r₁ = 0.0020 mol/(L·s)
• Experiment 2: [A]₂ = 0.20 mol/L, r₂ = 0.0080 mol/(L·s)

1. Concentration ratio: [A]₂/[A]₁ = 0.20 / 0.10 = 2.
2. Rate ratio: r₂/r₁ = 0.0080 / 0.0020 = 4.
3. Order: n = ln(4) / ln(2) = 1.386 / 0.693 = 2.
4. Rate constant: k = r₁ / [A]₁ⁿ = 0.0020 / (0.10)² = 0.20 L/(mol·s).

Doubling the concentration multiplied the rate by four, the signature of a second-order reaction. The rate constant comes out to 0.20 L/(mol·s).

Reading the order

The shortcut for whole-number orders is to watch what a change in concentration does to the rate.

Concentration change	Rate change	Order
Double [A]	Unchanged	0
Double [A]	Doubles	1
Double [A]	Quadruples	2

A zero-order reaction proceeds at a fixed rate regardless of how much A is present — common when a surface or catalyst is saturated. A first-order reaction tracks the concentration directly, and a second-order reaction responds to the square of it.

Units of the rate constant

Because the rate is always in mol/(L·s) but the concentration is raised to different powers, the units of k change with the order:

Order	Units of k
0	mol/(L·s)
1	s⁻¹
2	L/(mol·s)

This calculator reports k as a plain number; attach the units that match the order you found when you record the result.

When the order is not a whole number

A fractional order — say 0.5 or 1.5 — is a real and common result. It usually signals that the observed reaction is the sum of several elementary steps rather than a single collision, so the overall order need not be an integer. If a measurement lands close to but not exactly on a whole number, the difference is often just experimental scatter; repeating the experiment or adding more concentration points sharpens the estimate.

Related calculations

The order and rate constant found here feed directly into the integrated rate laws, which describe how concentration falls over time for zero-, first-, and second-order reactions. The average-reaction-rate tool helps turn raw concentration-versus-time data into the initial rates this method needs.

Frequently Asked Questions (FAQ)

What is the method of initial rates?

The method of initial rates finds a rate law by comparing the initial reaction rate at different starting concentrations. For a single reactant A with rate = k·[A]ⁿ, two experiments are enough: the ratio of the rates equals the ratio of the concentrations raised to the power n, so the order follows from a single logarithm. Measuring the rate at the very start avoids complications from product build-up or the reverse reaction.

How do I find the reaction order from two experiments?

Divide the second rate by the first, divide the second concentration by the first, then take n = ln(r₂/r₁) ÷ ln([A]₂/[A]₁). If doubling the concentration leaves the rate unchanged the order is 0; if it doubles the rate the order is 1; if it quadruples the rate the order is 2. Small departures from a whole number usually come from measurement error.

How do I find the rate constant once I know the order?

Put the order back into rate = k·[A]ⁿ using either experiment and solve for k = rate ÷ [A]ⁿ. Both experiments should give the same k within experimental error, which is a useful check that the order is correct.

What are the units of the rate constant?

The units of k depend on the overall order so that rate = k·[A]ⁿ stays consistent in mol/(L·s). A first-order constant has units of s⁻¹, a second-order constant L/(mol·s), and a zero-order constant mol/(L·s). Because the units shift with the order, the rate constant here is reported as a plain number; attach the order-appropriate units when you record it.