Activation Energy Calculator
Calculate activation energy using the Arrhenius equation with two-temperature rate data. Supports Ea from rate constants, frequency factor estimation, and temperature dependence analysis.
Rate Constant Calculator
Calculate reaction rate constants from experimental data for zero, first, and second order reactions. Determine reaction order, half-life, and rate law parameters.
Presets
M
M
s
Rate Constant (k)⧉
0.011552
Units: s⁻¹
Half-Life (t₁/₂)⧉
60.0000 s
Constant for first-order reactions
Time to 90% Completion⧉
199.316 s
Time for 90% of reactant to be consumed
Time to 99% Completion⧉
398.631 s
Time for 99% of reactant to be consumed
Integrated Rate Law⧉
ln[A]t = ln[A]₀ - kt
Order 1 — plot ln[A] vs t for straight line
Rate at t=0⧉
0.001155 M/s
Initial rate = k[A]₀ⁿ
Order Comparison (same data)
| Order | k value | Units | Integrated Form | Selected |
|---|---|---|---|---|
| 0 | 8.3333e-4 | M·s⁻¹ | [A]t = [A]₀ - kt | |
| 1 | 0.011552 | s⁻¹ | ln[A]t = ln[A]₀ - kt | ✓ |
| 2 | 0.166667 | M⁻¹·s⁻¹ | 1/[A]t = 1/[A]₀ + kt |
Concentration vs Time Profile
| Time (s) | [A] (M) | % Remaining | Decay Progress |
|---|---|---|---|
| 0.00 | 0.1000 | 100.0% | |
| 30.00 | 0.0707 | 70.7% | |
| 60.00 | 0.0500 | 50.0% | |
| 90.00 | 0.0354 | 35.4% | |
| 120.00 | 0.0250 | 25.0% | |
| 150.00 | 0.0177 | 17.7% | |
| 180.00 | 0.0125 | 12.5% | |
| 210.00 | 0.0088 | 8.8% | |
| 240.00 | 0.0063 | 6.3% | |
| 270.00 | 0.0044 | 4.4% | |
| 300.00 | 0.0031 | 3.1% |
Planning notes, formulas, and examples
About the Rate Constant Calculator
The rate constant (k) is the proportionality constant in a rate law that relates the reaction rate to the concentrations of reactants. For a reaction A → products, the rate law takes the form rate = k[A]^n, where n is the reaction order. The rate constant encapsulates all the factors affecting rate that aren't concentration: temperature, catalyst, medium, and molecular properties.
Different reaction orders have different integrated rate laws, different half-life expressions, and different units for k. For zero-order reactions, k has units of M/s and the half-life depends on initial concentration. For first-order reactions, k has units of s⁻¹ and the half-life is constant (t₁/₂ = ln2/k). For second-order reactions, k has units of M⁻¹s⁻¹ and the half-life depends inversely on initial concentration.
This calculator determines k from concentration-time data for zero, first, and second order reactions. It computes the half-life, plots the appropriate linear relationship, and helps determine the reaction order by comparing the linearity of different plots.
When This Page Helps
Determining rate constants from experimental data requires applying the correct integrated rate law for the reaction order. This calculator automates the process and helps identify the reaction order from concentration-time data.
How to Use the Inputs
- Select the reaction order (0, 1, or 2) or let the calculator determine it.
- Enter the initial concentration [A]₀.
- Enter the concentration [A]t at time t.
- Enter the elapsed time and its units.
- Or enter multiple data points for order determination.
- Use presets for common reaction type examples.
- Review k, half-life, and the integrated rate law plots.
Formula used
Zero order: [A]t = [A]₀ - kt, t₁/₂ = [A]₀/(2k)\nFirst order: ln[A]t = ln[A]₀ - kt, t₁/₂ = ln(2)/k\nSecond order: 1/[A]t = 1/[A]₀ + kt, t₁/₂ = 1/(k[A]₀)\n\nUnits of k:\n Zero order: M·s⁻¹\n First order: s⁻¹\n Second order: M⁻¹·s⁻¹ This keeps planning practical and lowers the chance of preventable errors.Example Calculation
Result: k = 0.01155 s⁻¹, t₁/₂ = 60.0 s
For a first-order reaction with [A]₀ = 0.100 M and [A]₆₀ = 0.050 M: k = ln(0.100/0.050)/60 = ln(2)/60 = 0.01155 s⁻¹. The half-life = ln(2)/k = 60.0 s, consistent with the concentration halving in 60 s.
Tips & Best Practices
- First-order reactions are the most common in chemistry — radioactive decay, many organic reactions, and enzyme kinetics all follow first-order kinetics.
- If the half-life is constant regardless of initial concentration, the reaction is first-order.
- For first-order: t₁/₂ = 0.693/k. For 90% completion: t₉₀ = 2.303/k. For 99%: t₉₉ = 4.605/k.
- The units of k are a dead giveaway for reaction order: M/s (0th), s⁻¹ (1st), M⁻¹s⁻¹ (2nd).
- In biochemistry, enzyme kinetics uses the Michaelis-Menten model, which transitions from zero-order (saturated) to first-order (dilute substrate).
- Rate constants at different temperatures can be used to find the activation energy via the Arrhenius equation.
Integrated Rate Laws
Each reaction order has a characteristic integrated rate law that relates concentration to time. These equations are derived by separating variables and integrating the differential rate law. The integrated forms are the basis for determining k from experimental data and for predicting concentrations at any future time.
Half-Life Relationships
The half-life (t₁/₂) is the time for the reactant concentration to decrease by half. For zero-order: t₁/₂ = [A]₀/(2k) — it depends on the initial concentration and gets shorter as the reaction proceeds. For first-order: t₁/₂ = ln(2)/k — constant and independent of concentration. For second-order: t₁/₂ = 1/(k[A]₀) — inversely proportional to initial concentration.
Practical Applications
Rate constants are essential for reactor design in chemical engineering, shelf-life prediction for pharmaceuticals and food, environmental fate modeling of pollutants, and pharmacokinetic modeling in drug development. The temperature dependence of k (via the Arrhenius equation) allows extrapolation from accelerated testing conditions to real-world temperatures.
Sources & Methodology
Last updated:
Frequently Asked Questions
Plot [A]t vs t (zero-order), ln[A]t vs t (first-order), and 1/[A]t vs t (second-order). The plot that gives a straight line reveals the order.
The rate constant k depends on temperature (Arrhenius equation), the presence of catalysts, the solvent/medium, and the reaction mechanism. It does NOT depend on concentration.
Because the integrated rate law is exponential: [A]t = [A]₀e^(-kt). The time to halve is always ln(2)/k regardless of starting concentration. This is unique to first-order kinetics.
When one reactant is in large excess, its concentration barely changes, and the rate law simplifies to first-order in the limiting reagent. k_obs = k[B]₀ where [B]₀ is the excess reactant concentration.
The Arrhenius equation describes this: k = A·exp(-Ea/RT). As temperature increases, k increases exponentially. A 10°C increase typically doubles to triples the rate for reactions with Ea ≈ 50-100 kJ/mol.
No, k is always positive. A negative value would indicate a calculation error or incorrect reaction order assumption.
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