In Collision Theory and Reaction Rate you saw why reactions speed up when particles collide more often and harder. Now we put a number on "how fast." Concentrations are in moles per liter (molarity), so a rate is a change in molarity per unit of time, usually $\text{M/s}$.
🔑 Key idea
For $a\mathrm{A} + b\mathrm{B} \to c\mathrm{C} + d\mathrm{D}$, the rate of reaction is
$$\text{rate} = -\frac{1}{a}\frac{\Delta[\mathrm{A}]}{\Delta t} = -\frac{1}{b}\frac{\Delta[\mathrm{B}]}{\Delta t}$$ $$= \frac{1}{c}\frac{\Delta[\mathrm{C}]}{\Delta t} = \frac{1}{d}\frac{\Delta[\mathrm{D}]}{\Delta t}$$
It is always positive, and every substance gives the same value.
💡 Why it works
Reactant concentrations fall, so $\Delta[\mathrm{A}]$ is negative. The minus sign turns it into a positive rate. The coefficients matter because the equation fixes the ratio in which things are used and made. In $\mathrm{N_2 + 3H_2 \to 2NH_3}$, hydrogen disappears three times as fast as nitrogen, because three $\mathrm{H_2}$ react for every $\mathrm{N_2}$. Dividing each change by its coefficient removes that difference, so the reaction has one rate, no matter which substance you watch.
A rate measured between two times is an average rate. Rates usually slow down as reactants are used up, so the rate at one moment, the instantaneous rate, is the slope of the tangent to the concentration curve at that time (with a minus sign for a reactant). The initial rate is the instantaneous rate at $t = 0$, before products build up.
Drag the point along the curve. The tangent's slope is the instantaneous rate; it is steepest at the start and flattens as the reactant runs down.
Worked examples
✏️ Worked example: One rate for the whole reaction
In $\mathrm{2N_2O_5 \to 4NO_2 + O_2}$, $[\mathrm{N_2O_5}]$ falls from $0.0200\ \text{M}$ to $0.0170\ \text{M}$ in $100\ \text{s}$. Find the rate at which $\mathrm{N_2O_5}$ is used, the rate of reaction, and the rates at which $\mathrm{NO_2}$ and $\mathrm{O_2}$ form.
$\Delta[\mathrm{N_2O_5}] = 0.0170 - 0.0200 = -0.0030\ \text{M}$, so $\mathrm{N_2O_5}$ is used at $\dfrac{0.0030\ \text{M}}{100\ \text{s}} = 3.0 \times 10^{-5}\ \text{M/s}$.
Rate of reaction: divide by its coefficient, 2: $\tfrac{1}{2}(3.0 \times 10^{-5}) = 1.5 \times 10^{-5}\ \text{M/s}$.
$\mathrm{NO_2}$ has coefficient 4: it forms at $4(1.5 \times 10^{-5}) = 6.0 \times 10^{-5}\ \text{M/s}$. $\mathrm{O_2}$ has coefficient 1: $1.5 \times 10^{-5}\ \text{M/s}$.
In $\mathrm{N_2 + 3H_2 \to 2NH_3}$, ammonia forms at $0.060\ \text{M/s}$. How fast is $\mathrm{H_2}$ used?
Rate of reaction $= \tfrac{1}{2}(0.060) = 0.030\ \text{M/s}$.
$\mathrm{H_2}$ has coefficient 3: $3(0.030) = 0.090\ \text{M/s}$. Its concentration falls, so $\Delta[\mathrm{H_2}]/\Delta t = -0.090\ \text{M/s}$, but the rate is reported as a positive number.
Answer: $0.090\ \text{M/s}$.
⚠️ Common mistake
Reporting a reactant's rate as a negative number, or comparing rates of different substances without the coefficients. The rate is positive by definition; the minus sign in the formula is there to make it so. Quick check: after dividing each substance's rate by its coefficient, all of them must agree.
Rates are measured by following anything that tracks one concentration: the color of a solution in a spectrometer, the pressure of a gas, the conductivity of ions or the volume of gas collected.
Practice
Pass with 3 right answers, the last 2 in a row. Each right answer earns 2 XP.