Rate Constant Calculator
Enter the initial and final concentrations plus the reaction time to find the rate constant k for zero, first, or second order reactions using the integrated rate law. You can also solve directly from an experimental rate and concentrations. The calculator shows the half-life, a step-by-step derivation, and a live concentration-vs-time chart so you can see the decay curve for your reaction.
Formula
Worked example
A first-order reaction starts with [A]₀ = 1.0 M and after 100 s the concentration falls to 0.25 M. k = ln(1.0/0.25) / 100 = ln(4) / 100 = 1.386 / 100 = 0.01386 s⁻¹. The half-life is ln(2)/0.01386 = 50.0 s.
What is the rate constant?
The rate constant k is a proportionality factor that links the concentrations of reactants to the speed of a chemical reaction. For a generic reaction A → products with rate law rate = k[A]^n, the larger k is the faster the reaction proceeds at a given concentration. The rate constant is specific to a reaction at a given temperature: it changes when you change the temperature (described by the Arrhenius equation, k = A·exp(-Ea/RT)) but remains the same regardless of the concentrations you use. Chemists determine k experimentally either by measuring the rate directly at known concentrations, or by tracking how concentration changes with time and fitting the integrated rate law for the appropriate reaction order.
How to determine reaction order
The reaction order is the exponent of the concentration term in the rate law. A zero-order reaction (rate = k) proceeds at a constant rate regardless of how much reactant is present - the concentration drops linearly with time and the graph of [A] vs. time is a straight line. A first-order reaction (rate = k[A]) is proportional to concentration - the graph of ln[A] vs. time is a straight line, and the half-life is constant (it does not depend on how much A you started with, a key diagnostic). A second-order reaction (rate = k[A]²) slows dramatically as the reactant is consumed - the graph of 1/[A] vs. time is a straight line and the half-life doubles each time the concentration halves. To determine the order experimentally, plot all three linearized forms and see which gives the best straight line.
The Arrhenius equation and temperature dependence
The rate constant calculated here applies at one temperature. To see how k changes with temperature, the Arrhenius equation is used: k = A·exp(-Ea/RT), where A is the pre-exponential (frequency) factor, Ea is the activation energy in J/mol, R is the gas constant (8.314 J/mol·K) and T is the absolute temperature in kelvin. If you measure k at two temperatures T1 and T2 you can isolate Ea: ln(k2/k1) = -(Ea/R)(1/T2 - 1/T1). A higher activation energy means k is more sensitive to temperature changes. Most reactions roughly double in rate for every 10 K rise in temperature near room temperature.
Units of k and how to check them
The units of k depend on the overall reaction order because the rate must always have units of M/s (concentration per time). For a zero-order reaction k has units of M/s; for first order, s⁻¹; for second order, M⁻¹s⁻¹ (or L/mol/s); for third order, M⁻²s⁻¹. A quick check: multiply k by the concentration term raised to the appropriate power and confirm the result has units of M/s. If the units do not work out, the assumed order is wrong.
Integrated rate law summary by reaction order
| Order | Integrated rate law | k from data | Half-life (t₁/₂) | Units of k |
|---|---|---|---|---|
| 0th | [A]ₜ = [A]₀ - kt | k = ([A]₀ - [A]ₜ) / t | [A]₀ / (2k) | M/s (mol L⁻¹ s⁻¹) |
| 1st | ln[A]ₜ = ln[A]₀ - kt | k = ln([A]₀/[A]ₜ) / t | ln(2) / k = 0.693 / k | s⁻¹ |
| 2nd | 1/[A]ₜ = 1/[A]₀ + kt | k = (1/[A]ₜ - 1/[A]₀) / t | 1 / (k[A]₀) | L mol⁻¹ s⁻¹ |
Formulas to find k from concentration-time data and the half-life expression for each order.
Frequently asked questions
What is the difference between the rate and the rate constant?
The rate of a reaction is how fast the concentration of a reactant or product changes over time, measured in mol/(L·s). It varies as the reaction proceeds because concentrations change. The rate constant k is fixed at a given temperature; it tells you how fast the reaction is on a per-concentration basis. For a first-order reaction, rate = k[A], so even though rate decreases as [A] decreases, k itself does not change.
Why does the half-life of a first-order reaction not depend on concentration?
From the first-order integrated rate law, t₁/₂ = ln(2)/k. Because k and ln(2) are both constants, the half-life is fixed regardless of starting concentration. This is in contrast to zero-order reactions (where t₁/₂ = [A]₀/(2k), so it decreases as the reaction proceeds) and second-order reactions (where t₁/₂ = 1/(k[A]₀), so it increases as the reactant is consumed). Constant half-life is a reliable diagnostic for first-order kinetics.
How do I know which reaction order to select?
The simplest approach is to linearize your concentration-time data three ways: plot [A] vs. time (linear for zero order), ln[A] vs. time (linear for first order), and 1/[A] vs. time (linear for second order). Whichever graph gives the straightest line is the correct order. Alternatively, if you have rate-vs-concentration data from initial-rate experiments, plot log(rate) vs. log([A]) - the slope gives the order with respect to A.
What are the units of the rate constant for each reaction order?
For zero order: mol/(L·s) or M/s. For first order: s⁻¹ (per second). For second order: L/(mol·s) or M⁻¹s⁻¹. In general for an nth-order reaction, the units are M^(1-n)/s. You can always verify the units are correct by checking that k multiplied by the concentration term(s) gives a result in M/s.
Can I use this calculator for the Arrhenius equation?
This calculator finds k from concentration or rate data at a single temperature. To apply the Arrhenius equation you need k values at two or more temperatures, then you can calculate the activation energy Ea using ln(k2/k1) = -(Ea/R)(1/T2 - 1/T1), where R = 8.314 J/(mol·K) and temperatures are in kelvin. Many university chemistry courses have a dedicated Arrhenius calculator for that step.