ChemBench

Rate Law Calculator

How fast reactions proceed depends on the order — each has its own integrated rate law and half-life formula.

[A]₀ (mol/L)

Rate constant k

Time (s)

Reaction order

Final concentration [A]

0.368 mol/L

Half-life: 13.863 s
Reaction order: 1

What you entered

[A]₀: 1 mol/Lk: 0.05Time: 20 sOrder: 1

1
[A] = [A]₀ × e^(−kt) (first order)
1 × e^(−0.05 × 20)= 0.3679 M
2
t½ = ln(2) ÷ k
0.6931 ÷ 0.05= 13.8629 s

Result

Final [A]: 0.368 mol/L

For a first-order reaction with k = 0.05, after 20 s the concentration drops from 1 to 0.368 mol/L. Half-life = 13.863 s.

Concentration remaining and half-life by reaction order, starting from 1.00 M

Every row starts at [A]0 = 1.00 M and applies the integrated rate law for its order: [A] = [A]0 - kt for zero order, [A] = [A]0 x e^(-kt) for first order, and 1/[A] = 1/[A]0 + kt for second order. The units of k differ by order, which is why each is written out in the table.

Order	Rate constant k	[A] after 10 s (M)	[A] after 60 s (M)	Half-life (s)
Zero	0.005 M/s	0.950	0.700	100.0
Zero	0.010 M/s	0.900	0.400	50.0
Zero	0.050 M/s	0.500	0.000	10.0
First	0.005 1/s	0.951	0.741	138.6
First	0.010 1/s	0.905	0.549	69.3
First	0.050 1/s	0.607	0.050	13.9
First	0.100 1/s	0.368	0.002	6.9
Second	0.010 1/(M s)	0.909	0.625	100.0
Second	0.050 1/(M s)	0.667	0.250	20.0
Second	0.100 1/(M s)	0.500	0.143	10.0

Compare the three orders at the same k = 0.05 and the difference is stark. The zero-order row hits 0.000 M at 60 s because a zero-order reaction consumes reactant at a fixed rate and runs out completely at t = [A]0 / k, which is 20 s here; the calculator floors the concentration at zero rather than going negative. First order never reaches zero, only halving every 13.9 s no matter what the concentration is. Second order fades slowest at long times because the rate collapses as the reactant thins out. Only the first-order half-life is independent of starting concentration: the zero-order and second-order half-lives shown here apply to a 1.00 M start and change if you start elsewhere. Real rate constants are also temperature dependent, so use the Arrhenius equation before applying a k measured at one temperature to another.

Reaction order changes everything

A zeroth-order reaction consumes reactant at a constant rate regardless of concentration. A first-order reaction (like radioactive decay) has an exponential decay with a constant half-life. A second-order reaction slows down dramatically as concentration drops — each order produces a distinctly different concentration-vs-time curve.

Half-life depends on order

Only first-order reactions have a concentration-independent half-life (t½ = ln2/k) — for zeroth order, half-life depends on initial concentration (t½ = [A]₀/2k), and for second order it depends inversely on concentration (t½ = 1/k[A]₀). This is why radioactive half-lives are constant but chemical half-lives often aren't.

Frequently asked questions

A first-order reaction has k = 0.05 s⁻¹ and starts at 1.0 M — what is the concentration after 30 seconds?

[A] = [A]₀ × e^(-kt) = 1.0 × e^(-0.05×30) = 1.0 × e^(-1.5) = 0.223 M.

How do I determine the reaction order from experimental data?

Plot concentration vs time (zeroth order gives a straight line), ln[A] vs time (first order gives a straight line), or 1/[A] vs time (second order gives a straight line). Whichever plot is linear reveals the order.

How is the rate law calculator different from the radioactive decay calculator?

Radioactive decay is always first-order. The rate law calculator handles zeroth, first, and second order reactions, each with different integrated rate laws and half-life formulas. Use the radioactive decay calculator specifically for nuclear half-life problems.