Integrated Rate Law Calculator

Solve integrated rate laws for zero, first, and second order reactions. Find concentration at any time, or calculate time to reach a target concentration from initial conditions.

⏱️ Kinetics📐 [A]t = [A]0·e^(-kt) (first order)🧪 Chemistry

Initial concentration [A]0 (mol/L)

Rate constant k

Time t (same units as 1/k)

Reaction order

Please enter valid values.

Formula & Reference

Variable	Symbol	Formula	Units
Integrated Rate Law Calculator	See formula	[A]t = [A]0·e^(-kt) (first order)	mol/L

Step-by-Step Examples

Example 1

First Order Decay

A to B, first order: [A]0=1.00 M, k=0.0500 s^-1, t=20.0 s.

[A] = [A]0 x e^(-kt) = 1.00 x e^(-0.0500x20.0)
= 1.00 x e^-1.00 = 1.00 x 0.368
[A] = 0.368 M

✓ [A] = 0.368 M (36.8% remains after 1/k seconds)

Example 2

Second Order

A to B, 2nd order: [A]0=0.500 M, k=0.100 L/mols, t=5.00 s.

1/[A]t = 1/[A]0 + kt = 1/0.500 + 0.100x5.00
= 2.00 + 0.500 = 2.50 M^-1
[A]t = 1/2.50 = 0.400 M

✓ [A] = 0.400 M

Example 3

Zero Order

[A]0=2.00 M, k=0.100 mol/Ls, t=10.0 s.

[A]t = [A]0 - kt = 2.00 - 0.100x10.0
[A]t = 2.00 - 1.00 = 1.00 M

✓ [A] = 1.00 M (zero order: linear decay)

Real-World Applications

💊

Pharmacokinetics

Drug concentration vs time curves follow integrated rate laws. Most drug elimination: first order.

🧪

Chemical Kinetics Lab

Integrated rate laws allow identification of reaction order from concentration-time data.

🏭

Reactor Design

Batch reactor design equations are integrated rate laws integrated over reactor volume.

🌍

Environmental Fate

Pollutant degradation kinetics modeled with integrated rate laws for risk assessment.

Common Mistakes to Avoid

⚠️

Choose the correct order first

Method: plot [A] vs t (zero order), ln[A] vs t (first order), 1/[A] vs t (second order). Linear plot identifies the order.

⚠️

Second order: 1/[A] not [A]

Second order integrated law is in terms of 1/[A]. Rearrange to find [A] at end.

⚠️

Zero order: concentration can hit zero

For zero order: [A] reaches 0 at t = [A]0/k. After this time, [A] stays at 0.

Frequently Asked Questions

How do I determine reaction order graphically? ▾

Plot data three ways: [A] vs t: if linear, zero order. ln[A] vs t: if linear, first order (slope=-k). 1/[A] vs t: if linear, second order (slope=+k).

What are half-lives for different orders? ▾

Zero order: t1/2=[A]0/(2k) (depends on [A]0). First order: t1/2=0.693/k (constant). Second order: t1/2=1/(k[A]0) (depends on [A]0).

What is a pseudo-first-order reaction? ▾

When [B] >> [A] in A + B to products, [B] ~ constant, and rate = k_eff[A]. Integrated law: first order. k_eff = k[B].

Can I use integrated rate law to find time? ▾

Yes. First order: t = -ln([A]/[A]0)/k. Zero order: t = ([A]0-[A])/k. Second order: t = (1/[A] - 1/[A]0)/k.

What is the relationship between integrated rate law and rate law? ▾

Integrated rate law is obtained by integrating the differential rate law: d[A]/dt = -k[A]^n over time.

Related Chemistry Calculators

Formula Explorer connections

Interpretation: This formula connects concentration, time, temperature or transport to the speed of a chemical process. Assumption: The reaction order and mechanism must match the model. Temperature, catalyst, mixing and mass-transfer limitations can alter the observed rate.

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