Rate of Reaction Calculator — Reaction Rate, Rate Laws, and Half-Life

Chemical kinetics answers the question that thermodynamics ignores: not just whether a reaction can happen, but how fast it does. A reaction that takes a millisecond and one that takes a million years are both "spontaneous" by thermodynamic standards — kinetics is what distinguishes them. The CalcHub Rate of Reaction Calculator handles concentration-vs-time calculations for all three common reaction orders.

Basic Rate Definition

For a reaction A → products:

Rate = −Δ[A] / Δt

The negative sign is there because reactant concentration decreases over time. Rate is always expressed as a positive number, in units of mol/(L·s) or similar.

Rate Laws by Order

Order	Rate Law	Integrated Form	Units of k
Zero	Rate = k	[A]t = [A]₀ − kt	mol·L⁻¹·s⁻¹
First	Rate = k[A]	ln[A]t = ln[A]₀ − kt	s⁻¹
Second	Rate = k[A]²	1/[A]t = 1/[A]₀ + kt	L·mol⁻¹·s⁻¹

How to Use the Calculator

1. Open CalcHub and select the Rate of Reaction Calculator.
2. Choose the reaction order (or let the tool determine it from data points).
3. Enter initial concentration, time, and rate constant k.
4. The calculator returns concentration at time t, half-life, and plots the concentration profile.

Worked Example: First-Order Reaction

A drug degrades in the body with rate constant k = 0.0462 h⁻¹. Starting concentration is 100 mg/L. What's the concentration after 6 hours?

Using first-order integrated law:
[A]t = [A]₀ × e^(−kt) = 100 × e^(−0.0462 × 6) = 100 × 0.757 = 75.7 mg/L

Half-life: t₁/₂ = ln(2)/k = 0.693/0.0462 = 15 hours

Concentration vs. Time Summary

Time	Zero Order [A]	First Order [A]	Second Order [A]
0	[A]₀	[A]₀	[A]₀
1 half-life	[A]₀/2	[A]₀/2	[A]₀/3
2 half-lives	0	[A]₀/4	[A]₀/5

Note that for zero-order reactions the half-life changes over time (it depends on current concentration). For first-order, it's constant — which is why first-order kinetics is so important in pharmacology.

Factors Affecting Reaction Rate

• Temperature: Higher T → more energy → more successful collisions → faster rate. The Arrhenius equation quantifies this.
• Concentration: More molecules → more collisions (hence the rate law)
• Catalysts: Lower activation energy, increase rate without being consumed
• Surface area: For heterogeneous reactions, more surface = more reaction sites

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How do I determine the order of a reaction experimentally?

Compare initial rates at different starting concentrations. If doubling [A] doubles the rate, it's first order. If it quadruples the rate, it's second order. If rate doesn't change, it's zero order in A. Alternatively, plot ln[A] vs. time (straight line = first order) or 1/[A] vs. time (straight line = second order).

What's the difference between rate constant and reaction rate?

Rate is a measured quantity that changes as reactant concentrations change. Rate constant k is a fixed value (at a given temperature) that characterizes the speed of a particular reaction regardless of concentrations. They're related by the rate law.

What is the Arrhenius equation, and can this calculator use it?

The Arrhenius equation (k = A·e^(−Ea/RT)) relates rate constant to temperature and activation energy. If you input k at two temperatures, the calculator can back-calculate the activation energy — useful for understanding temperature sensitivity.

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Related calculators: Half-Life Calculator · Enthalpy Calculator · Stoichiometry Calculator