Reaction Kinetics Calculator - Integrated Rate Laws

What reaction kinetics tells us

Chemistry doesn’t happen instantly. When two compounds react, the reaction proceeds over time, with reactant concentrations decreasing and product concentrations increasing.
How quickly that happens, and how the concentrations change as it does, is the field of chemical kinetics.

Kinetics has practical importance everywhere:

• Pharmaceuticals: how quickly drugs metabolize and clear
• Industrial chemistry: optimizing reaction conditions for efficiency
• Food science: shelf life prediction
• Environmental science: pollutant degradation rates
• Nuclear physics: radioactive decay timing
• Geology: radiometric dating of rocks
• Atmospheric chemistry: ozone destruction reactions
• Combustion: engine efficiency optimization
• Cooking: how quickly food browns or denatures

The rate law concept

A reaction’s rate depends on the concentration of reactants. The general rate law:

Rate = k × [A]^m × [B]^n

Where:

• Rate: how fast products form (or reactants disappear)
• k: rate constant (depends on temperature)
• [A], [B]: concentrations of reactants
• m, n: orders with respect to each reactant
• m + n: overall reaction order

Reaction order tells you how concentration affects rate. Most reactions are zero, first, or second order overall.

Zero-order reactions

Rate doesn’t depend on reactant concentration:

Rate = k

Integrated rate law: [A] = [A]₀ − k × t

The concentration drops linearly with time. Half-life: t₁/₂ = [A]₀ / (2k)

Where zero-order kinetics occurs:

• Enzyme-catalyzed reactions when enzyme is saturated (Michaelis-Menten kinetics)
• Surface-catalyzed reactions where catalyst surface is fully covered
• Photochemical reactions limited by light intensity
• Many industrial processes deliberately operated in zero-order regime

Example: alcohol elimination by liver alcohol dehydrogenase. The liver clears about 0.015% BAC (Blood Alcohol Concentration) per hour regardless of starting concentration, which is zero-order kinetics.
The enzyme system is already saturated at any BAC a person is likely to reach, so adding more alcohol does not make it work faster. It is the reason a heavy night takes so long to clear.

First-order reactions

Rate is proportional to one reactant:

Rate = k × [A]

Integrated rate law: [A] = [A]₀ × e^(−k × t) Or equivalently: ln[A] = ln[A]₀ − k × t

The concentration drops exponentially. Half-life: t₁/₂ = ln(2) / k = 0.693 / k

A unique property: half-life is independent of initial concentration. The reaction takes the same time to go from 1.0 M to 0.5 M as from 0.001 M to 0.0005 M.

Where first-order kinetics occurs:

• Radioactive decay (the textbook example)
• Many gas-phase reactions
• Many drug eliminations in the body
• Bond rotations and conformational changes
• Many unimolecular decompositions
• Most isotope decay

Famous first-order example: Carbon-14 decay. Half-life is 5,730 years. This means after 5,730 years, half the C-14 in a sample has decayed. After another 5,730 years (11,460 total), 1/4 remains. This consistent decay is the basis for carbon-14 dating.

Second-order reactions

Rate depends on two reactant concentrations:

Rate = k × [A]² (or k × [A][B])

Integrated rate law: 1/[A] = 1/[A]₀ + k × t

Half-life: t₁/₂ = 1 / (k × [A]₀), which depends on the initial concentration

Where second-order kinetics occurs:

• Bimolecular reactions (two molecules collide)
• Many enzyme-substrate reactions
• Many gas-phase reactions
• Polymerization steps
• Acid-base reactions
• Electron transfer reactions
• Decomposition reactions involving two molecules of the same compound

Example: 2 HI → H₂ + I₂. Two HI molecules must collide for reaction. Rate depends on [HI]².

Comparing reaction orders

How the three orders behave:

Determining reaction order experimentally

To find a reaction’s order:

Method 1: Initial rates. Measure the rate at different starting concentrations.

• If doubling [A] doubles rate → first order
• If doubling [A] quadruples rate → second order
• If rate unchanged → zero order

Method 2: Half-life. Measure t₁/₂ at different starting concentrations.

• t₁/₂ doubles when [A] halves → second order
• t₁/₂ constant → first order
• t₁/₂ halves when [A] halves → zero order

Method 3: Concentration vs time plots. Find which plot comes out as a straight line.

• [A] vs t linear → zero order
• ln[A] vs t linear → first order
• 1/[A] vs t linear → second order

Method 3 is the one to reach for with real lab data, because a straight line is easy to judge by eye and the slope hands you k directly. Methods 1 and 2 both need several separate experiments.

Temperature dependence: the Arrhenius equation

Rate constants increase with temperature, governed by:

k = A × e^(−Eₐ/RT)

Where:

• A: pre-exponential factor (collision frequency)
• Eₐ: activation energy (the “energy barrier”)
• R: gas constant (8.314 J/mol·K)
• T: absolute temperature in Kelvin

Rule of thumb: rate roughly doubles for every 10°C temperature increase.

This is why:

• Food spoils faster in summer
• Reactions in industrial plants are run at elevated temperatures
• Refrigeration extends shelf life
• Freezing essentially stops most reactions

The activation energy concept

Eₐ is the energy “hill” reactants must climb to react. Lower Eₐ = faster reaction.

Catalysts work by providing alternative pathways with lower Eₐ. They don’t change reaction extent (equilibrium), just rate.

Typical activation energies:

• 50 kJ/mol: relatively fast at room temperature
• 100 kJ/mol: slow at room temperature
• 200 kJ/mol: requires very high temperature
• 400 kJ/mol: requires combustion-level temperatures

Pseudo-first-order reactions

Many real reactions are second-order but appear first-order under specific conditions.

When [B] is in large excess relative to [A]:

• [B] remains essentially constant during reaction
• Rate appears proportional to [A] alone
• “Pseudo-first-order” behavior
• Useful for simplifying analysis

Example: hydrolysis of an ester in dilute aqueous solution. Technically: Rate = k[ester][H₂O]

Since H₂O is in vast excess, [H₂O] is essentially constant, simplifying to: Rate = k’[ester], where k’ = k × [H₂O]

This is how kinetic analysis of complex reactions becomes tractable.

Multi-step reactions

Real reactions often involve multiple steps:

Reaction: A + B → C + D (the overall equation)

Actual mechanism:

• Step 1: A → I (slow, rate-determining)
• Step 2: I + B → C + D (fast)

The slowest step (the rate-determining step) controls overall rate. Order may not match coefficients in the balanced equation.

This is why reaction order is determined experimentally, not from the balanced equation. The same overall reaction can have different orders depending on mechanism.

Practical applications

Pharmaceutical kinetics:

• Drug half-life predicts dosing schedule
• First-order elimination most common
• Dose × half-life relationship determines steady-state concentration

Food preservation:

• Microbial growth and food spoilage follow first-order kinetics
• Temperature dependence (Arrhenius) determines refrigeration vs freezing
• “Best by” dates predicted from kinetics

Environmental persistence:

• DDT half-life in soil: ~10-30 years (very slow first-order)
• Glyphosate: ~50 days
• Caffeine in water: ~25 days
• Plastics: hundreds to thousands of years

Nuclear waste:

• Uranium-238 half-life: 4.5 billion years
• Carbon-14 half-life: 5,730 years
• Iodine-131 half-life: 8 days (medical use)
• Half-life predicts storage requirements

Combustion and explosions:

• Branching chain reactions
• Different kinetic regimes
• Critical mass calculations

Atmospheric chemistry:

• Ozone depletion reactions
• Methane atmospheric lifetime: ~9 years
• CO₂ atmospheric lifetime: 300-1,000 years
• Predict climate impact persistence

Calculating remaining reactant

For any reaction order, after time t:

Zero order: [A] = [A]₀ − k × t (drops to zero when t = [A]₀/k)

First order: [A] = [A]₀ × e^(−k × t) = [A]₀ × (1/2)^(t/t₁/₂)

• After 1 half-life: 50% remains
• After 2 half-lives: 25%
• After 3 half-lives: 12.5%
• After 4 half-lives: 6.25%
• After 10 half-lives: 0.1%

Second order: [A] = [A]₀ / (1 + k × [A]₀ × t)

Common kinetics mistakes

1. Assuming order from balanced equation: must be measured
2. Ignoring temperature: Arrhenius dependence is critical
3. Treating all reactions as first-order: only common in some contexts
4. Confusing half-life behavior: zero/first/second have different patterns
5. Forgetting catalyst limits: catalysts speed up, don’t shift equilibrium
6. Mixing rate and equilibrium: very different concepts
7. Wrong units: k units depend on order
8. Linear extrapolation: many reactions aren’t truly first-order

Worked example: the same numbers through all three orders

Take [A]₀ = 1.0 mol/L, k = 0.05, t = 20, in whatever time unit k was measured in. Run it three ways:

Three very different answers from one set of numbers, which is the point. The zero-order reaction has already run to completion at t = 20, because it consumes reactant at a flat 0.05 mol/L per unit time and only had 1.0 to spend. The second-order reaction happens to be sitting exactly at its half-life. The first-order one is at e⁻¹ of its starting concentration, since k × t came out to exactly 1.

Notice how far apart the last column is: 18, 46 and 180. Getting the order wrong does not shift your answer by a few percent, it shifts it by an order of magnitude. This is why the order has to be measured rather than assumed.

Units of k, which trip up almost everyone

The rate constant does not have one unit. It has whatever unit makes Rate = k × [A]^order come out in mol/L per unit time:

• Zero order: mol/(L·time)
• First order: 1/time
• Second order: L/(mol·time)

A “k of 0.05” is therefore meaningless on its own. This calculator takes whatever time unit your k came in and reports every result in that same unit, so a k in reciprocal seconds gives a half-life in seconds and a k in reciprocal years gives one in years.