How fast does a reactant disappear?
The rate of a chemical reaction usually depends on the concentration of its reactants. A rate law such as rate = k[A]ⁿ links the two, where k is the rate constant and n the order of the reaction. Integrating that equation over time gives the integrated rate law, which tells you how much reactant is left after any time, how long it takes to reach a given concentration and what the half-life is.
The three common orders
| Order | Integrated law | Half-life | Straight-line plot |
|---|---|---|---|
| 0 | [A] = [A]₀ − kt | [A]₀ ÷ 2k | [A] against t |
| 1 | [A] = [A]₀ × e^(−kt) | ln 2 ÷ k | ln[A] against t |
| 2 | 1/[A] = 1/[A]₀ + kt | 1 ÷ (k[A]₀) | 1/[A] against t |
The time to reach a target concentration comes from solving each law for t, for example t = ln([A]₀ ÷ [A]) ÷ k at first order.
Example
A first-order reaction with k = 0.01 s⁻¹ and an initial concentration of 1 mol/L:
- After 100 s: [A] = e^(−1) = 0.3679 mol/L, so 36.8% remains
- Half-life: 0.6931 ÷ 0.01 = 69.31 s
- Time to fall to 0.1 mol/L: ln 10 ÷ 0.01 = 230.3 s
- Rate at 100 s: 0.01 × 0.3679 = 0.00368 mol/(L·s)
Where each order appears
- Zero order: reactions limited by a saturated catalyst or enzyme, such as alcohol elimination by the liver at high blood levels, and some surface reactions.
- First order: radioactive decay, many decompositions such as N₂O₅, and the elimination of most drugs.
- Second order: reactions where two molecules of the same species collide, such as the dimerization of NO₂ or of butadiene.
Half-lives that change
At first order, every half-life is the same. At zero order, each successive half-life is half as long as the one before, because the reactant is consumed at a constant rate. At second order, each one is twice as long, because the rate falls quickly as the reactant runs out.
Temperature
The rate constant k depends strongly on temperature, often doubling for every 10 °C rise near room temperature. The Arrhenius equation, k = A × e^(−Ea/RT), describes that dependence and lets you convert k from one temperature to another before using this calculator.