Rate: how fast concentration changes
Lesson 2.11.1 wrote the reaction rate as v = Δc/Δt. AP uses square brackets for concentration: [A] is the molar concentration of A (mol/L, or M).
A reaction rate反应速率 is the decrease in a reactant’s concentration, or the increase in a product’s concentration, per unit time, usually in M/s. A reactant’s concentration falls, so Δ[reactant] is negative; a minus sign in front keeps the rate positive:
Rate ratios equal coefficient ratios
For one reaction, the rates for different species are in the same ratio as their coefficients. In , if H₂ disappears at 0.090 M/s, N₂ disappears at 0.030 M/s and NH₃ appears at 0.060 M/s.
To get one number that doesn’t depend on which species you pick, AP divides each species’ rate by its coefficient. This is the rate of reaction反应速率（除以系数）. For aA + bB → cC + dD:
For the ammonia synthesis above, the rate of reaction is 0.030 M/s. When a question asks for the rate of a particular species, convert using that species’ coefficient; when it asks for the rate of reaction, divide by the coefficient.
Which reactant is consumed faster? Look at the coefficients. In X + 2Y → products, −Δ[Y]/Δt = 2 × (−Δ[X]/Δt), so Y is consumed faster. This has nothing to do with which reactant has the higher concentration or the higher order in the rate law.
Average and instantaneous rates
- The average rate平均速率 is the change in concentration over a time interval divided by that interval: the slope of the secant line between two points on a concentration–time curve.
- The instantaneous rate瞬时速率 is the rate at one moment: the slope of the tangent line at that point. The instantaneous rate at t = 0 is the initial rate初始速率, which U5.2 uses to find rate laws.
Most reactions are fastest at the start. As reactants are used up, there are fewer collisions per unit time, so the curve flattens and the rate decreases. Move the times below to compare average rates over different intervals, then switch to “Instantaneous rate” to see the tangent:
2N₂O₅(g) → 4NO₂(g) + O₂(g), [N₂O₅]₀ = 0.100 M (hypothetical data). Average rates are calculated from concentrations read to 3 significant figures.
How rates are measured
Any measurable property that changes as the reaction proceeds can be used to follow it:
| Method | When it works |
|---|---|
| Titrating samples taken at intervals | A reactant or product is an acid, base, or oxidizing agent (U4.6) |
| Pressure (at constant volume and temperature) | The number of moles of gas changes during the reaction |
| Color or absorbance | One species is colored, and absorbance is proportional to concentration (U3.13) |
| Mass | A gas escapes, so the system loses mass |
| Volume of gas collected | A gas forms and is collected in a gas buret or over water |
In , for example, the total moles of gas don’t change, so pressure can’t follow the reaction; but I₂ is violet, so absorbance can.
Factors that affect rate
The factors from 2.11.2 matter on the AP exam too: higher reactant concentration, higher temperature, and greater surface area of a solid all increase the rate, and a catalyst usually increases it greatly. U5.2 describes the effect of concentration quantitatively with rate laws, U5.5 explains the effects with the collision model, and U5.11 covers catalysts.
Example 1: Converting rates with coefficients
For , at one moment NO₂ is appearing at 4.0 × 10⁻⁵ M/s. Find the rate of disappearance of N₂O₅, the rate of appearance of O₂, and the rate of reaction.
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Find the coefficients. In one reaction, the rates of the species are in the same ratio as their coefficients. N₂O₅, NO₂, and O₂ have coefficients 2, 4, and 1.
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Rate of disappearance of N₂O₅. Multiply the NO₂ rate by the coefficient ratio 2/4 (all rates in M/s):
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Rate of appearance of O₂. Multiply by 1/4:
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Rate of reaction. Rate of reaction = any species’ rate ÷ its coefficient, and it doesn’t depend on which species you pick:
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Check. From the other two species: 2.0 × 10⁻⁵ ÷ 2 = 1.0 × 10⁻⁵ and 1.0 × 10⁻⁵ ÷ 1 = 1.0 × 10⁻⁵. All three agree.
Example 2: Average rates from data
Hypothetical data for the decomposition of N₂O₅: [N₂O₅] = 0.100 M at t = 0 s, 0.0607 M at 500 s, and 0.0368 M at 1000 s. Find the average rate of disappearance of N₂O₅ from 0 to 500 s and from 500 to 1000 s, and explain the difference.
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Write the definition. The reactant’s concentration falls, so Δ[N₂O₅] is negative; the minus sign makes the rate positive:
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0 to 500 s. In M/s:
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500 to 1000 s.
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Compare. The intervals are equally long, and the later one has the smaller average rate, about 0.6 times the first.
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Explain. As the reaction proceeds, [N₂O₅] decreases, so there are fewer N₂O₅ molecules per unit volume and fewer react per unit time; the rate decreases.
- Raising the temperature
A larger fraction of collisions has enough energy (U5.5).
- Using the same mass of marble as powder instead of chips with HCl
Greater surface area means more collisions per unit time.
- Adding water to dilute a reactant solution
The reactant concentration decreases.
- Adding a suitable catalyst
A catalyst provides a pathway with a lower activation energy (U5.11).
- Some time after the reaction starts (compared with the start)
Reactants are used up, so their concentrations fall.
- Compressing the container to raise the concentration of gaseous reactants
Higher concentration means more collisions per unit time.
- Lowering the temperature
A smaller fraction of collisions has enough energy.
- Adding argon, which doesn't react, at constant volume
The reactant concentrations don't change.
- Putting the same amounts of gaseous reactants in a larger container
A larger volume means lower concentrations.
- Using more concentrated HCl with zinc
[H⁺] is higher.
- Using a smaller volume of HCl at the same concentration with the same zinc
The rate depends on concentration, not on the volume of solution.
Common mistakes
Mistake 1: “Every species in a reaction has the same rate.” The rates for different species are in the ratio of their coefficients and are usually different numbers. Only after dividing each by its coefficient do you get a rate of reaction that is the same for every species.
Mistake 2: “A reactant’s rate is negative.” Δ[reactant] is negative, but the definition includes a minus sign, so the rate is always positive. “N₂O₅ disappears at 2.0 × 10⁻⁵ M/s” means its concentration falls by that much each second.
Mistake 3: “A reaction has one fixed rate.” The rate usually decreases as the reaction proceeds. An average rate describes only an interval, and the instantaneous rate is the tangent slope at one moment; average rates over different intervals are usually different.
Quick check
In 2NO + O₂ → 2NO₂, O₂ disappears at 0.010 M/s. At what rate does NO₂ appear?
The coefficients of O₂ and NO₂ are in a 1 : 2 ratio, so NO₂ appears twice as fast as O₂ disappears.
In 4NH₃ + 5O₂ → 4NO + 6H₂O, NH₃ disappears at 0.24 M/s. At what rate does H₂O appear?
0.24 × 6/4 = 0.36 M/s. Choice C flips the ratio (0.24 × 4/6); D is the rate for O₂.
In a reaction, [A] falls from 0.80 M to 0.50 M in 60 s. What is the average rate of disappearance of A over this interval?
−(0.50 M − 0.80 M) ÷ 60 s = 5.0 × 10⁻³ M/s. Rates are positive; B divides 0.50 by 60; C doesn't divide by time.
A reaction is fastest at the start and then slows down. Which explanation is best?
At constant temperature, k and the activation energy don't change; the rate falls because reactant concentrations fall.
H₂(g) + I₂(g) → 2HI(g) takes place in a sealed rigid container. Which method could NOT be used to follow its rate?
There are 2 mol of gas before and after, so the total pressure stays constant and can't show how far the reaction has gone. I₂'s color and absorbance change with [I₂].
On an [A]–t curve, what gives the instantaneous rate of disappearance of A at t = 100 s?
The instantaneous rate is the tangent slope; [A] is falling, so the slope is negative and the rate of disappearance is its negative. Option A is an average rate.
For X + 2Y → Z, which statement is correct?
−Δ[Y]/Δt = 2 × (−Δ[X]/Δt). The coefficients decide this, not the concentrations or the orders.
In 2MnO₄⁻ + 10Br⁻ + 16H⁺ → 2Mn²⁺ + 5Br₂ + 8H₂O, Br₂ appears at 2.5 × 10⁻³ M/s. At what rate does MnO₄⁻ disappear?
2.5 × 10⁻³ × 2/5 = 1.0 × 10⁻³ M/s. B flips the ratio; D is the rate of reaction (the Br₂ rate divided by 5).
H₂O₂ decomposes with a catalyst. The O₂ collected at t = 0, 20, 40, and 60 s is 0, 24, 40, and 50 mL. Which statement is best?
The three 20 s intervals give 24, 16, and 10 mL, so the rate is falling; the cause is that less H₂O₂ remains. Catalysts aren’t used up (U5.11).
Identical strips of magnesium react with two HCl samples, and 30 mL of H₂ is collected each time: trial 1 takes 50 s and trial 2 takes 20 s. Which statement is correct?
For the same amount of product, a shorter time means a higher average rate: 30 mL ÷ 20 s = 1.5 mL/s for trial 2 and 0.60 mL/s for trial 1.
AP-style free response
HI decomposes in a rigid container: . These hypothetical data were collected:
| t (s) | 0 | 100 | 200 | 300 |
|---|---|---|---|---|
| [HI] (M) | 0.500 | 0.400 | 0.333 | 0.286 |
(a) Calculate the average rate of disappearance of HI from 0 to 100 s.
(b) Calculate the average rate of appearance of I₂ and the rate of reaction over the same interval.
(c) Calculate the average rate of disappearance of HI from 200 to 300 s, compare it with (a), and explain the difference at the particle level.
(d) A student plans to follow the reaction by measuring the total pressure in the container. Evaluate the plan, and propose a method that would work.
Model answer and scoring
(a) −Δ[HI]/Δt = −(0.400 M − 0.500 M) ÷ 100 s = 1.00 × 10⁻³ M/s.
(b) I₂ has half the coefficient of HI: rate of appearance of I₂ = 5.0 × 10⁻⁴ M/s. Rate of reaction = rate of disappearance of HI ÷ 2 = 5.0 × 10⁻⁴ M/s.
(c) −(0.286 M − 0.333 M) ÷ 100 s = 4.7 × 10⁻⁴ M/s, smaller than in (a). [HI] is lower, so HI molecules collide less often per unit time, and the rate is lower.
(d) It won’t work: 2 mol of HI gas forms 1 mol of H₂ and 1 mol of I₂, so the total moles of gas, and therefore the total pressure at constant volume and temperature, stay the same. I₂ is colored, so the absorbance (color intensity) can be measured over time.
Scoring: (a) correct sign and units; (b) uses the 1 : 2 coefficient ratio; (c) correct value, explained by lower concentration and fewer collisions; (d) notes the unchanged moles of gas and proposes a sensible method such as color or absorbance.
Summary
- A reaction rate is the change in concentration per unit time: −Δ[ ]/Δt for reactants and Δ[ ]/Δt for products, in M/s.
- Rates for different species are in the ratio of their coefficients; dividing each by its coefficient gives the rate of reaction, which is the same for every species.
- The average rate is a secant slope and the instantaneous rate a tangent slope; rates usually decrease as reactants are used up.
- Any property that changes during the reaction can be used to follow it: concentration (titration), pressure, color or absorbance, mass, or gas volume.
AP tip: To convert rates, write “known rate × coefficient of the wanted species ÷ coefficient of the known species,” then check that the size makes sense: species with larger coefficients have larger rates. When a question distinguishes “the rate of a species” from “the rate of reaction,” check which one it asks for.