Rate of a reaction: average and instantaneous rate
The rate of a reaction is the change in concentration per unit time. Average rate covers an interval; instantaneous rate is the slope of the tangent at one moment.
What the rate of a reaction means
Some reactions are very fast. Silver chloride forms the moment silver nitrate meets sodium chloride solution. Some are very slow, like the rusting of iron. Others, like the inversion of cane sugar, go at a moderate speed. Chemical kinetics is the branch of chemistry that studies these speeds and how reactions happen. The rate of a reaction is the change in concentration of a reactant or a product in unit time. Square brackets mean molar concentration: [R] is the concentration of R in mol L⁻¹. A reactant is used up, so Δ[R] is negative. We put a minus sign before it so that the rate is always positive.
Average rate vs instantaneous rate
The rate usually falls as the reaction goes on, because the reactant concentration falls. So the curve of [R] against t gets flatter with time.
How to draw the concentration-time graph
- Draw two axes. Label the x-axis Time (s) and the y-axis [R] (mol L⁻¹).
- Draw a curve that starts high on the y-axis at [R]₀ and falls, getting flatter as time goes on.
- For the average rate, mark two points at t₁ and t₂ and join them with a straight line. Label Δ[R] (vertical side) and Δt (horizontal side).
- For the instantaneous rate, draw a tangent that touches the curve at one time t. Label it "slope = d[R]/dt".
- You may draw the product curve [P] on the same axes: it starts at zero and rises, getting flatter.
Rate when the coefficients are not 1
In 2HI(g) → H₂(g) + I₂(g), two moles of HI are used for every mole of H₂ formed. So HI disappears twice as fast as H₂ appears. To get one rate for the whole reaction, divide each rate term by its coefficient: Rate = −½ d[HI]/dt = d[H₂]/dt = d[I₂]/dt. In the same way, for 5Br⁻ + BrO₃⁻ + 6H⁺ → 3Br₂ + 3H₂O: Rate = −(1/5) d[Br⁻]/dt = −d[BrO₃⁻]/dt = −(1/6) d[H⁺]/dt = (1/3) d[Br₂]/dt = (1/3) d[H₂O]/dt.
Worked example: average rate of 2HI → H₂ + I₂
In the reaction 2HI(g) → H₂(g) + I₂(g), the concentration of HI falls from 0.80 mol L⁻¹ to 0.62 mol L⁻¹ in 15 minutes. Find (a) the average rate of reaction and (b) the rate of formation of H₂, in mol L⁻¹ min⁻¹ and in mol L⁻¹ s⁻¹.
- Given: [HI]₁ = 0.80 mol L⁻¹, [HI]₂ = 0.62 mol L⁻¹, Δt = 15 min.
- Δ[HI] = 0.62 − 0.80 = −0.18 mol L⁻¹.
- Formula: Rate = −½ Δ[HI]/Δt.
- Rate = −½ × (−0.18 mol L⁻¹) ÷ 15 min = 0.09 ÷ 15 = 6.0 × 10⁻³ mol L⁻¹ min⁻¹.
- Rate of formation of H₂ = Δ[H₂]/Δt = rate of reaction = 6.0 × 10⁻³ mol L⁻¹ min⁻¹.
- In seconds: 6.0 × 10⁻³ ÷ 60 = 1.0 × 10⁻⁴ mol L⁻¹ s⁻¹.
(a) and (b) both 6.0 × 10⁻³ mol L⁻¹ min⁻¹ = 1.0 × 10⁻⁴ mol L⁻¹ s⁻¹
Worked example: linking the rates of different species
For 2N₂O₅(g) → 4NO₂(g) + O₂(g), N₂O₅ disappears at 1.6 × 10⁻⁴ mol L⁻¹ s⁻¹. Find the rate of reaction and the rates of formation of NO₂ and O₂.
- Rate = −½ d[N₂O₅]/dt = ¼ d[NO₂]/dt = d[O₂]/dt.
- Rate of reaction = ½ × 1.6 × 10⁻⁴ = 8.0 × 10⁻⁵ mol L⁻¹ s⁻¹.
- d[NO₂]/dt = 4 × 8.0 × 10⁻⁵ = 3.2 × 10⁻⁴ mol L⁻¹ s⁻¹.
- d[O₂]/dt = 1 × 8.0 × 10⁻⁵ = 8.0 × 10⁻⁵ mol L⁻¹ s⁻¹.
Rate of reaction 8.0 × 10⁻⁵; NO₂ forms at 3.2 × 10⁻⁴; O₂ forms at 8.0 × 10⁻⁵ (all in mol L⁻¹ s⁻¹)
Milk turns to curd faster on a hot day in Chennai than in a cold Shimla winter, and a refrigerator keeps food fresh longer. Both are about controlling the rate of chemical reactions.
2 marks: “Define average rate and instantaneous rate of a reaction. How is the instantaneous rate found from a graph?” Average rate = change in concentration ÷ time interval, −Δ[R]/Δt. Instantaneous rate = rate at one moment, −d[R]/dt. Draw a tangent to the [R]-t curve at that time; the rate is minus its slope.
2 marks: “For N₂ + 3H₂ → 2NH₃, NH₃ forms at 2.0 × 10⁻⁴ mol L⁻¹ s⁻¹. Find the rates of disappearance of N₂ and H₂.” Rate = ½ d[NH₃]/dt = 1.0 × 10⁻⁴. So −d[N₂]/dt = 1.0 × 10⁻⁴ mol L⁻¹ s⁻¹ and −d[H₂]/dt = 3 × 1.0 × 10⁻⁴ = 3.0 × 10⁻⁴ mol L⁻¹ s⁻¹.
1 marks: “Why is a minus sign put before Δ[R] in the rate expression?” [R] decreases, so Δ[R] is negative; the minus sign makes the rate a positive quantity.
Reactants get a minus, products a plus, and every term is divided by its own coefficient.