Chemistry

First-Order Integrated Rate Law

Track exponential decay of a reactant’s concentration.

C_{t} = C_{0} e^{- k t}

Study tools

Save, share, export, or report this formula.

Source & review

Difficulty

Intermediate

Read time

3 min

Prerequisites

Algebra, logarithms, and molar quantities

Review statusNeeds review

Source

OpenStax, Chemistry 2e, §12.4

Adapted from Teng-Jui Lin’s CHEM 155 equation sheet. The linked textbook reference was used to check the equation and its physical context. Pending editorial review.

Adapted from Teng-Jui Lin, Equation Sheets, under CC BY 4.0.

FormuLab adapted notation and chemical symbols, added variable meanings, units, applicability conditions, and an original worked example. Defined k using the reactant-disappearance rate, absorbing any stoichiometric normalization into k.

Definition

Understanding the core concept

In a first-order disappearance process, the rate is proportional to the current reactant concentration. The same fraction is consumed in equal time intervals, so concentration decreases exponentially.

Variables & Units

Understanding each component

Symbol	Meaning	Units
$C_{t}$	Reactant concentration at elapsed time t	$mol L^{- 1}$
$C_{0}$	Initial reactant concentration	$mol L^{- 1}$
$k$	First-order disappearance-rate constant	$s^{- 1}$
$t$	Elapsed time	$s$

Real-World Applications

Where this formula is used in practice

Chemical kinetics

Predict a reactant concentration from its first-order rate constant.

Concentration data

Compare measured concentration decay with an exponential model.

Worked Example

Step-by-step calculation with real numbers

Problem

A first-order reactant starts at 0.100 mol/L with k = 0.0200 s⁻¹. Find its concentration after 50.0 s.

Given

C_{0} = 0.100 mol L^{- 1}

k = 0.0200 s^{- 1}

t = 50.0 s

Solution

C_{t} = C_{0} e^{- k t}

C_{t} = (0.100 mol L^{- 1}) e^{- (0.0200 s^{- 1}) (50.0 s)}

C_{t} \approx 0.03679 mol L^{- 1}

Final Answer

C_{t} \approx 0.03679 mol L^{- 1}

Formula Guide

Info

DifficultyIntermediate

Read Time3 min

PrerequisitesAlgebra, logarithms, and molar quantities

Tools

Source & review

StatusNeeds review

Last reviewedPending

Source

OpenStax, Chemistry 2e, §12.4

Adapted from Teng-Jui Lin’s CHEM 155 equation sheet. The linked textbook reference was used to check the equation and its physical context. Pending editorial review.

Adapted from Teng-Jui Lin, Equation Sheets, under CC BY 4.0.

Assumptions

• Use a constant-volume system at fixed temperature with the stated disappearance-rate law and constant rate coefficient.
• Use consistent concentration and time units; apply the model only while the reactant remains present.
• Here −dC/dt = kC, with nonnegative k and elapsed time.

Languages

FormuLab is a study reference. Verify formulas, units, assumptions, and course-specific conventions before relying on them.

Browse by category

See all

Engineering Physics Chemistry Mathematics Fluid Mechanics Electrical

First-Order Integrated Rate Law

Chemical kinetics

Concentration data

Problem

Given

Solution

Final Answer

Contents

Info

Tools

Source & review

Languages

Browse by category

Product

Company

Legal