Integrated Rate Law (1st Order) Calculator
Concentration over time for 1st order.
Formula first
Overview
The first-order integrated rate law describes how the concentration of a reactant decreases over time in a reaction where the rate is proportional to the concentration of a single reactant. It provides a linear mathematical relationship between the natural logarithm of the concentration and the elapsed time.
Symbols
Variables
[A] = ln(Concentration), k = Rate Constant, t = Time, [A]_0 = Initial ln[A]0
Apply it well
When To Use
When to use: This equation applies to reactions where the rate is first-order, meaning the exponent in the rate law is one. Use it when analyzing radioactive decay, the elimination of certain drugs from the bloodstream, or when a plot of natural log concentration versus time yields a straight line.
Why it matters: Understanding first-order kinetics is crucial for determining the shelf-life of pharmaceuticals and predicting the decay of isotopes used in medical imaging or carbon dating. It allows chemists to calculate the time required for a pollutant to degrade to safe levels in environmental systems.
Avoid these traps
Common Mistakes
- Forgetting to use natural log (ln), not log10.
- Using [A] instead of ln[A] for the graph.
One free problem
Practice Problem
A first-order chemical reaction has an initial concentration natural log (c) of 1.50 and a rate constant (k) of 0.025 min⁻¹. Calculate the natural log of the concentration (lnA) remaining after 20 minutes.
Solve for: lnA
Hint: Multiply the rate constant by time, subtract that from the initial natural log value.
The full worked solution stays in the interactive walkthrough.
References
Sources
- Atkins' Physical Chemistry
- Wikipedia: Integrated rate law
- Bird, Stewart, Lightfoot: Transport Phenomena
- IUPAC Gold Book
- Wikipedia: Rate equation
- McQuarrie's Physical Chemistry
- IUPAC Gold Book (Kinetic order)
- OCR A-Level Chemistry A — Reaction Rates