What is the rate of a reaction?
Rate means how fast. For a reaction R → P, rate is the change in concentration in one unit of time. Concentration is written in square brackets, like [R], in mol L⁻¹.
The reactant is used up, so Δ[R] is negative. We put a minus sign so that the rate is always positive:
Rate = −Δ[R]/Δt = +Δ[P]/Δt
The unit is mol L⁻¹ s⁻¹. For gases we may use atm s⁻¹.
Average rate and instantaneous rate
Average rate is the rate over a time gap. On the [R]–t graph, join two points with a straight line (a secant). Minus its slope is the average rate.
Instantaneous rate is the rate at one exact moment. Draw a tangent (a line that just touches the curve) at that time. Minus its slope is the instantaneous rate: r = −d[R]/dt.
The curve gets flatter with time, so the instantaneous rate keeps falling as the reactant is used up.
Rate with more than one substance (stoichiometry)
When the numbers in the balanced equation are not 1, divide each change by its number. For 2HI → H₂ + I₂:
Rate = −½ Δ[HI]/Δt = Δ[H₂]/Δt = Δ[I₂]/Δt
So HI disappears twice as fast as H₂ appears, but the reaction has one rate.
Factors that change the rate
- Concentration (or pressure for gases): more particles in the same space → more collisions.
- Temperature: particles move faster and more of them have enough energy. Often a 10 °C rise nearly doubles the rate.
- Catalyst: gives an easier path, so the reaction is faster. It is not used up.
- Nature of reactants and surface area: ionic reactions in water are very fast; powder reacts faster than a lump.
- Light: some reactions (like photosynthesis) need light.
Rate law and rate constant
For aA + bB → products, experiments give the rate law:
Rate = k [A]^x [B]^y
x and y come from experiment. They may or may not equal a and b. k is the rate constant: the rate when every concentration is 1 mol L⁻¹. k changes with temperature, but not with concentration.
Units of k = (mol L⁻¹)^(1−n) s⁻¹, where n is the order. Zero order: mol L⁻¹ s⁻¹. First order: s⁻¹. Second order: L mol⁻¹ s⁻¹.
Order of a reaction
Order = the sum of the powers in the rate law, x + y. It can be 0, 1, 2, 3 or even a fraction. It is found only by experiment.
Test: double [A] and keep others fixed. Rate same → order 0 in A; rate ×2 → order 1; rate ×4 → order 2.
Pseudo first order: when one reactant is in huge excess (like water in the hydrolysis of an ester, or of sugar), its concentration hardly changes. A second-order reaction then behaves like first order.
Molecularity and elementary steps
An elementary reaction happens in one single step. Many reactions are complex: they happen in several steps (the mechanism).
Molecularity is the number of particles that must collide at the same time in one elementary step: unimolecular (1), bimolecular (2), termolecular (3). It is always a whole number and never zero. More than 3 is almost impossible because so many particles rarely meet at once.
For a complex reaction, the slowest step (the rate-determining step) controls the overall rate.
| Order | Molecularity |
|---|---|
| From experiment | From the mechanism (theory) |
| Can be 0 or a fraction | Whole number 1, 2, 3; never 0 |
| For the overall reaction | Only for each elementary step |
Try it at home
Take two glasses of water, one cold and one warm. Drop half an effervescent tablet (or a spoon of baking soda plus lemon juice) in each at the same time. Count the seconds until the fizzing stops. The warm glass wins: temperature raises the rate. Now try a crushed tablet against a whole one: more surface, faster rate. In the 3D, predict first, then move the slider to check.
Key formulas and definitions
- Rate = −Δ[R]/Δt = +Δ[P]/Δt
- Instantaneous rate r = −d[R]/dt
- For aA + bB → cC + dD: rate = −(1/a)Δ[A]/Δt = −(1/b)Δ[B]/Δt = (1/c)Δ[C]/Δt
- Rate law: rate = k[A]^x[B]^y, order = x + y
- Unit of k = (mol L⁻¹)^(1−n) s⁻¹
Worked examples
1. In R → P, [R] falls from 0.80 mol/L to 0.50 mol/L in 20 s. Find the average rate.
Δ[R] = 0.50 − 0.80 = −0.30 mol/L. Average rate = −Δ[R]/Δt = 0.30/20 = 0.015 mol L⁻¹ s⁻¹.
2. For 2N₂O₅ → 4NO₂ + O₂, N₂O₅ disappears at 0.02 mol L⁻¹ s⁻¹. Find the rate of formation of NO₂ and O₂, and the rate of reaction.
Rate of reaction = ½ × 0.02 = 0.01 mol L⁻¹ s⁻¹. NO₂ forms at 4 × 0.01 = 0.04 mol L⁻¹ s⁻¹. O₂ forms at 1 × 0.01 = 0.01 mol L⁻¹ s⁻¹.
3. Rate = k[A][B]². What is the order with respect to A, to B, and overall?
Order in A = 1, order in B = 2, overall order = 1 + 2 = 3.
4. Find the unit of k for a second-order reaction (concentration in mol/L, time in s).
Unit = (mol L⁻¹)^(1−2) s⁻¹ = (mol L⁻¹)⁻¹ s⁻¹ = L mol⁻¹ s⁻¹.
5. Rate = k[A]². If [A] is made 3 times, how does the rate change?
New rate = k(3[A])² = 9k[A]². The rate becomes 9 times.
6. Data for A + B → products: [A]=0.1,[B]=0.1 → rate 2×10⁻³; [A]=0.2,[B]=0.1 → 4×10⁻³; [A]=0.1,[B]=0.2 → 8×10⁻³ (mol L⁻¹ s⁻¹). Find the rate law and k.
Doubling A (rows 1→2): rate ×2, so order in A = 1. Doubling B (rows 1→3): rate ×4, so order in B = 2. Rate = k[A][B]². k = 2×10⁻³ / (0.1 × 0.1²) = 2×10⁻³ / 10⁻³ = 2 L² mol⁻² s⁻¹.
7. The mechanism of 2NO + O₂ → 2NO₂ is: (1) 2NO ⇌ N₂O₂ (fast), (2) N₂O₂ + O₂ → 2NO₂ (slow). What is the molecularity of each step?
Step 1 has 2 NO molecules colliding: bimolecular. Step 2 has N₂O₂ and O₂: also bimolecular. The slow step 2 decides the rate.
Common mistakes
- Forgetting the minus sign for a reactant: rate is always positive, so use −Δ[R]/Δt.
- Taking the order from the balanced equation. Order comes only from experiment.
- Mixing up order and molecularity: order can be 0 or ½, molecularity cannot.
- Forgetting to divide by the coefficient (like ½ for 2HI) when writing the rate of reaction.