From rate to concentration
A rate law describes how the rate of consumption changes with concentration. This tool considers one reactant A and three possible orders: zero, first and second. Reaction order is determined experimentally; it cannot generally be inferred from the coefficients of a chemical equation.
−dtd[A]=kA[A]n,n∈{0,1,2} The calculator’s k is k_A, the disappearance rate constant for A. For a reaction a A → products, IUPAC defines the reaction rate as v = −(1/a)d[A]/dt. If v = kᵥ[A]ⁿ, then k_A = a·kᵥ. For example, in 2 C₄H₆ → C₈H₁₂, a constant defined using reaction rate must be multiplied by two before entering it here. The butadiene example’s k already describes consumption of C₄H₆ in the OpenStax integrated law, so it is used without doubling.
Integration and units
Separating variables gives ∫d[A]/[A]ⁿ = −k∫dt. The condition [A](0) = [A]₀ sets the integration constant. With concentration in mol/L and time in seconds:
| Order | Integrated law | Unit of k |
|---|
| 0 | [A] = [A]₀ − kt | mol·L⁻¹·s⁻¹ |
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| 1 | [A] = [A]₀e⁻ᵏᵗ | s⁻¹ |
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| 2 | 1/[A] = 1/[A]₀ + kt | L·mol⁻¹·s⁻¹ |
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For zero order, the linear expression applies only until t_depletion = [A]₀/k. Concentration and consumption rate are displayed as zero afterward. A measurement of zero at a given time cannot establish when depletion occurred: it only gives k ≥ [A]₀/t.
Conversion is x = 1 − [A]/[A]₀. For first order, t = −ln(1 − x)/k; for second order, t = x/((1 − x)k[A]₀). Reaching 100% conversion requires infinite time for first- and second-order laws.
The half-life staircase
The initial half-life is the time needed to go from [A]₀ to [A]₀/2. Each later stage consumes half of what remains, not half of the initial amount.
t1/2(0)=2k[A]0,t1/2(1)=kln2,t1/2(2)=k[A]01 For zero order, each stage lasts half as long as the previous one. For first order, all stages take the same time. For second order, each stage takes twice as long. The chart guides and the successive half-lives table show both stage duration and cumulative time.
Example: k = 9.2×10⁻³ s⁻¹ for cyclobutane at 500 °C gives t₈₀ = ln(5)/k ≈ 174.9 s. Initial concentration does not affect this first-order conversion time.
Determining order from data
The graphical method looks for a straight line in [A] versus t (zero order), ln([A]/c_ref) versus t (first order) or 1/[A] versus t (second order). Their slopes are −k, −k and +k, respectively. c_ref is a concentration unit: a logarithm requires a dimensionless argument.
The Linear Regression module can perform these transformations and fit straight lines. Transforming measurements also transforms their errors: a small absolute error is amplified when taking ln[A] or 1/[A] near zero. This tool therefore fits all three models directly in concentration space, estimating [A]₀ and k; its linear views simply display the same fit.
All models use the same rows. Times are treated as known, and concentration errors as independent with constant variance. Negative or zero measurements can represent measurement noise around zero and are retained in fitting; predicted concentrations are always nonnegative. A value recorded as zero because of a detection limit is censored data and needs a different treatment.
Comparison and uncertainty
The sum of squares SSE = Σ([A]ᵢ − [A]̂ᵢ)² measures error in the same space for all three orders. RMSE = √(SSE/N) retains concentration units. With the same number of parameters and Gaussian errors, AIC differences are:
ΔAICj=Nln(SSEminSSEj) At least five points and three admissible fits are required. A most compatible order is declared only when every alternative has ΔAIC ≥ 15. This is a conservative tool-specific criterion, not a confidence level. With few points, limited conversion or noise, “inconclusive” is a useful answer. The AICc correction cancels between these models when defined; counting estimated variance adds the same parameter to all three.
Relative preference does not establish that a model adequately describes the chemistry. Examine the points against the curves and consider repeated measurements over a wider conversion range. R² is calculated in concentration space and can be negative: the model then predicts worse than a constant concentration equal to the mean.
The standard error of k is estimated locally from the derivative matrix and SSE/(N−2). It does not represent uncertainty across models, time errors or known instrumental uncertainties. It is not reported when the model passes exactly through every measurement, when the data do not determine its parameters well or when a measurement coincides with the zero-order depletion kink. Displayed digits describe calculation precision, not significant figures guaranteed by the experiment.
Conditions and connections
The model assumes constant temperature and volume, one reactant and an irreversible reaction during the observed interval. Models are compared over the measured interval, so shifting every time does not change the comparison. If the first measurement comes after t = 0, concentration and half-life are reported at that time; [A]₀ at t = 0 is an extrapolation that a second-order model may not admit. Changing units re-expresses quantities; changing order defines a different rate law and changes the dimension of k.
Apparent zero order may arise from surface or enzyme saturation; saturation limits are explored in Enzyme Kinetics. A reactant in large excess may produce pseudo-first-order behavior with an apparent rate constant.
Arrhenius describes the temperature dependence of k. Check definitions and units before transferring a constant. ICE Table describes equilibrium composition; these laws do not include a reverse reaction.
For absorbance series, use Beer–Lambert to obtain concentration and correct the blank. If absorbance = εℓ[A], the first-order k is unchanged, the zero-order slope scales by εℓ, and the second-order constant requires knowing εℓ. Inputs here accept molar concentrations, not instrumental signals.