r = -(1/a) d[A]/dt = (1/p) d[P]/dt
One reaction rate, normalized by stoichiometry, describes every species consistently.
Stoichiometrically normalized reaction rate for a balanced reaction a A -> p P.
- a
- stoichiometric coefficient of reactant A
- p
- stoichiometric coefficient of product P
Use when — A balanced reaction a A -> p P is given.
Common trap — Comparing raw species rates directly without dividing by their stoichiometric coefficients.
r = k [A]^m [B]^n
Experiments, not the balanced equation, determine how concentration affects rate.
Empirical differential rate law with experimentally determined exponents.
- k
- rate constant
- m
- order with respect to A
- n
- order with respect to B
Use when — Fixed temperature and defined reaction conditions.
Common trap — Setting m and n equal to the overall reaction's stoichiometric coefficients without experimental justification.
[A]_t = [A]_0 - k t
Concentration falls linearly with time in a zero-order reaction.
Zero-order integrated rate law.
- [A]_0
- initial concentration of A
- [A]_t
- concentration of A at time t
Use when — Constant rate constant k and a confirmed zero-order regime.
Common trap — Extending the linear decrease beyond the point where [A] reaches zero.
t(1/2) = [A]_0 / (2k)
More starting material takes longer to halve in a zero-order reaction.
Half-life of a zero-order reaction.
- t(1/2)
- half-life (s)
Use when — Same zero-order conditions as the integrated law.
Common trap — Calling zero-order half-life independent of initial concentration, when it is not.
ln( [A]_0 / [A]_t ) = k t
The concentration ratio decays exponentially in a first-order reaction.
First-order integrated rate law.
- k
- first-order rate constant (s^-1)
Use when — Constant rate constant k and a confirmed first-order regime.
Common trap — Taking the logarithm of a dimensional concentration value instead of a concentration ratio.
t(1/2) = ln(2) / k
The half-life of a first-order reaction is independent of the starting concentration.
Half-life of a first-order reaction.
- t(1/2)
- half-life (s)
Use when — Confirmed first-order reaction.
Common trap — Applying this formula to a zero-order or other-order reaction.
k = A × e^(-Ea / (R T))
Higher temperature increases the fraction of molecules able to cross the activation barrier.
Arrhenius relation between the rate constant, temperature, and activation energy.
- A
- pre-exponential (frequency) factor
- Ea
- activation energy (J mol^-1)
- R
- gas constant
- T
- absolute temperature (K)
Use when — One dominant mechanism, with A and Ea approximately constant over the temperature range considered.
Common trap — Using Celsius temperature, or treating the pre-exponential factor as constant over an unjustified range.
ln(k2/k1) = -(Ea/R) × (1/T2 - 1/T1)
Comparing rate constants at two temperatures reveals the activation energy.
Two-temperature form of the Arrhenius relation, used to find activation energy from two rate constants.
- k1
- rate constant at temperature T1
- k2
- rate constant at temperature T2
Use when — Same mechanism and Arrhenius parameters apply at both temperatures.
Common trap — Reversing the two temperatures or rate constants without matching the corresponding sign.