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Use calc to turn basic inputs into a working kinetic model. Enter Arrhenius parameters, reaction orders, initial concentrations and temperature. The tool computes k(T), integrates concentration profiles for common reactor types, plots rate and concentration vs time, and returns key numbers like initial rate r₀, half-life t½ and time to target conversion. This guide explains the math, shows practical examples with US-friendly temperatures and times, and gives tips to avoid the usual unit and stability traps.
📝 The workflow is straightforward. First compute k from Arrhenius. Plug k into the chosen rate law. Integrate the resulting differential equations over the time window you care about. Extract r₀, t½ when applicable, and times to 25, 50, 75 and 100 percent conversion. The simulator supports zero, first and second order closed batch modes and a simple perfectly mixed continuous reactor model.
Table of Contents
Core formulas
Arrhenius law
k(T) = A · exp(−Ea / (R · T))
Where A is the pre-exponential factor with units that match the reaction order, Ea is activation energy in J·mol⁻¹ or kJ·mol⁻¹, R = 8.314462618 J·mol⁻¹·K⁻¹, and T is absolute temperature in kelvin. Convert Celsius to kelvin by adding 273.15. For US users you can also check the Fahrenheit equivalent; for example 25 °C equals 77 °F which is 298.15 K.
General rate law
r = k(T) · [A]^{nA} · [B]^{nB}
r is reaction rate in mol·L⁻¹·s⁻¹. nA and nB are empirical reaction orders. Units of k depend on the sum of orders s = nA + nB. For a single reactant A with order s, k units are L^{s−1}·mol^{1−s}·s⁻¹.
Batch reactor mass balances
d[A]/dt = −νA · r
d[B]/dt = −νB · r
Stoichiometric coefficients νA and νB relate rate r to concentration consumption. Integrate these ODEs numerically using RK4 or a similar solver to get concentration vs time.
Closed-form formulas you can use by hand
Zero order r = k
[A](t) = [A]₀ − k·t
First order r = k·[A]
[A](t) = [A]₀ · exp(−k·t) ; t½ = ln(2)/k
Second order (A + A → products)
1/[A](t) = 1/[A]₀ + k·t
Units cheat sheet and dimensional checks
Always check units before you run a simulation. Typical lab units:
- Concentration: mol·L⁻¹ (M)
- Time: seconds or minutes; convert consistently
- Temperature: use K in Arrhenius; show °C and °F for readability
- Energy: J·mol⁻¹ (multiply kJ by 1000)
Example dimension rules: for first order k has units s⁻¹. For second order in a single-species A + A case k has units L·mol⁻¹·s⁻¹. If you mix minutes and seconds you will get nonsense numbers, so normalize to a single time unit before interpreting k or computing t½.
Worked example 1
Inputs:
- A = 1.0 × 10⁶ s⁻¹
- Ea = 50 kJ·mol⁻¹ (that is 50,000 J·mol⁻¹)
- T = 77 °F which is 25 °C or 298.15 K
- [A]₀ = 1.00 M and reaction is first order r = k·[A]
Steps:
- Compute R·T = 8.314462618 × 298.15 ≈ 2478.9 J·mol⁻¹
- Compute exponent −Ea/(R·T) = −50,000 / 2478.9 ≈ −20.1636
- exp(−20.1636) ≈ 1.7393 × 10⁻⁹
- k = A × exp(…) = 1.0 × 10⁶ × 1.7393 × 10⁻⁹ ≈ 0.001739 s⁻¹
- Initial rate r₀ = k × [A]₀ = 0.001739 × 1.00 = 0.001739 mol·L⁻¹·s⁻¹
- Half-life t½ = ln(2)/k ≈ 0.693147 / 0.001739 ≈ 398.5 s which is about 6.64 minutes
Worked example 2
Inputs:
- k = 0.10 L·mol⁻¹·s⁻¹
- [A]₀ = 1.00 M
- Reaction A + A → products
Use: 1/[A](t) = 1/[A]₀ + k·t
After t = 10 s: 1/[A] = 1/1.0 + 0.10 × 10 = 1 + 1 = 2 so [A] = 0.5 M.
Numerical integration and solver behavior
The simulator uses RK4 with adaptive step control tuned by characteristic time τ ≈ [A]₀ / r₀. The integrator attempts to place roughly 500 points across τ for a smooth curve. If concentrations drop below a practical floor such as 1×10⁻¹² M the solver stops to save time. If your t_end is too small the solver will simply return the last computed concentrations and you will see that 100% conversion was not reached within t_end.
Practical tips and common mistakes
Units mismatch is the top cause of bogus results. Always convert Ea from kJ to J if you use R in J·mol⁻¹·K⁻¹. If A is reported in per minute convert to per second or vice versa before combining with time. For multicomponent rate laws ensure the concentration units are consistent across components. If you see extremely short times to completion check whether A or Ea were entered in inconsistent units.
Parameter estimation tip: use two or more temperatures to build an Arrhenius plot of ln k vs 1/T. The slope yields −Ea/R and the intercept gives ln A. Fit using weighted linear regression when k estimates have different uncertainties.
Sensitivity and how to use the tool for planning
Run sensitivity sweeps over Ea, A and initial concentration to see which parameter most affects conversion time. For scale-up planning identify τ and the heat release rate. Fast reactions with large k may be heat-limited when scaled; always check energy balance and consider using a calorimeter measurement for the fastest systems.
Output table — what the calculator gives you
Typical outputs:
- k(T) with units
- Initial rate r₀
- Half-life t½ for first order
- Times to 25, 50, 75 and 100% conversion where meaningful
- Time series arrays for concentrations, rates and optionally conversion vs time
Quick reference — units for A and k
| Order | Rate form | Rate r units | k units |
|---|---|---|---|
| 0 | r = k | mol·L⁻¹·s⁻¹ | mol·L⁻¹·s⁻¹ |
| 1 | r = k·[A] | mol·L⁻¹·s⁻¹ | s⁻¹ |
| 2 | r = k·[A]² | mol·L⁻¹·s⁻¹ | L·mol⁻¹·s⁻¹ |
| general | r = k·[A]^{nA}·[B]^{nB} | mol·L⁻¹·s⁻¹ | L^{s−1}·mol^{1−s}·s⁻¹ where s = nA+nB |
Safety, validation and experimental notes
Treat the model as a planning tool. For reactive systems with heat release or side reactions run small scale experiments and monitor temperature and conversion. Validate fitted Ea and A with independent runs and check mass balances. When simulating hazardous chemistries consult safety data sheets and run a process hazards analysis before scale-up.
Control examples you can paste into the tool
- Example set 1: A = 1.0e6 s⁻¹; Ea = 50 kJ·mol⁻¹; T = 77 °F (25 °C); [A]₀ = 1.0 M; order = 1. Expect k ≈ 0.00174 s⁻¹ and t½ ≈ 6.6 min.
- Example set 2: k = 0.10 L·mol⁻¹·s⁻¹; [A]₀ = 1.0 M; second order A + A. At t = 10 s expect [A] ≈ 0.5 M.
It is a fast, no-nonsense tool for exploring reaction behavior, estimating timescales and planning experiments. Use it to compare scenarios, check unit consistency and prioritize what to measure in the lab. Always validate model outputs with experimental data before acting on them in process design.
Recommended books
- Chemical Kinetics and Reaction Dynamics by Paul L. Houston
- Elements of Chemical Reaction Engineering by H. Scott Fogler
- Reaction Kinetics: Homogeneous Gas Reactions by Manuel A. Loyola
- Physical Chemistry by Peter Atkins and Julio de Paula
David Parry — Senior Engineering Analyst
Specializing in electronics and physics-based simulations with 20+ years of engineering experience. David ensures the mathematical and physical accuracy of the tools at ProCalcLab.
