Reaction Rate Law & Half-Life Calculator
Explore 0th, 1st, and 2nd order kinetics. Calculate concentrations over time, determine half-lives, and fit experimental data to find reaction order.
Reaction Kinetics Calculator
Enter reaction parameters or time-concentration data to explore integrated rate laws, half-lives, and concentration changes over time.
Integrated Rate Laws
0th: [A] = [A]₀ − kt
1st: ln[A] = ln[A]₀ − kt
2nd: 1/[A] = 1/[A]₀ + kt
Half-Life Formulas
0th: t₁/₂ = [A]₀/(2k)
1st: t₁/₂ = ln(2)/k
2nd: t₁/₂ = 1/(k[A]₀)
Data Fitting
Enter time-concentration data to determine reaction order using linear regression and R² analysis.
First-Order Kinetics
Only first-order reactions have a half-life independent of initial concentration.
Understanding Reaction Kinetics
What are Integrated Rate Laws?
While the rate law (rate = k[A]n) tells us how fast a reaction proceeds at any instant, the integrated rate lawtells us how concentration changes over time. By integrating the differential rate law, we obtain equations that relate [A] to t directly.
The reaction order (n) determines which integrated rate law applies and dramatically affects how the half-life behaves.
Comparison of Rate Law Orders
| Property | Zero-Order (n=0) | First-Order (n=1) | Second-Order (n=2) |
|---|---|---|---|
| Rate Law | rate = k | rate = k[A] | rate = k[A]² |
| Integrated Law | [A] = [A]₀ − kt | ln[A] = ln[A]₀ − kt | 1/[A] = 1/[A]₀ + kt |
| Linear Plot | [A] vs t | ln[A] vs t | 1/[A] vs t |
| Slope of Linear Plot | −k | −k | +k |
| Half-Life Formula | t₁/₂ = [A]₀/(2k) | t₁/₂ = ln(2)/k | t₁/₂ = 1/(k[A]₀) |
| Half-Life Depends On | [A]₀ (directly) | Nothing (constant!) | [A]₀ (inversely) |
| Units of k | M/s | s⁻¹ | M⁻¹s⁻¹ |
First-Order: The Special Case
First-order kinetics is unique because the half-life is constantregardless of initial concentration. This is why radioactive decay and many drug elimination processes follow first-order kinetics—the time to decrease by half is always the same.
Determining Order from Data
To find the reaction order experimentally, plot the data in three ways: [A] vs t, ln[A] vs t, and 1/[A] vs t. Whichever gives a straight line (highest R²) indicates the order. The slope of that line gives k.
Understanding Half-Life
The half-life (t₁/₂) is the time required for the concentration to decrease to half its initial value. Its behavior is a key indicator of reaction order:
- Zero-order: Half-life decreases as the reaction progresses (each successive half-life is shorter because there's less reactant)
- First-order: Half-life is constant throughout the reaction (the signature feature of first-order kinetics)
- Second-order: Half-life increases as the reaction progresses (each successive half-life is longer)
Real-World Examples
Zero-Order
- • Enzyme-catalyzed reactions at saturation
- • Photochemical reactions at constant light
- • Alcohol metabolism (at high BAC)
First-Order
- • Radioactive decay
- • Drug elimination from blood
- • N₂O₅ decomposition
Second-Order
- • Dimerization reactions
- • NO₂ decomposition
- • Many bimolecular reactions
Important Considerations
- • Temperature dependence: The rate constant k varies with temperature according to the Arrhenius equation: k = Ae−Ea/RT
- • Order ≠ Stoichiometry: Reaction order must be determined experimentally; it cannot be predicted from the balanced equation
- • Complex reactions: Many reactions have fractional or negative orders, or depend on multiple species—these require more advanced treatment
- • Initial rates: This tool uses integrated rate laws. For initial rate experiments, different analysis methods apply
Frequently Asked Questions
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