Integrated Rate Law (1st Order) Equation, Derivation & Rearrangements

Core idea

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.

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.

Symbols

Variables

$ln$ [A] = ln(Concentration), k = Rate Constant, t = Time, $ln$ [A]_0 = Initial ln[A]0

ln [A]

ln(Concentration)

Variable

k

Rate Constant

s^{-} 1

t

Time

s

ln [A]_{0}

Initial ln[A]0

Variable

Walkthrough

Derivation

Derivation of First-Order Integrated Rate Law

Gives concentration as a function of time for a first-order reaction.

Reaction is first order in A.

1

Start with the Differential Rate Law:

Rate of disappearance of A is proportional to [A].

- \frac{d [ A ]}{d t} = k [A]

2

Separate Variables and Integrate:

Integrate from t=0 to t and [A]_0 to [A]_t.

\int_{[A]_{0}}^{[A]_{t}} \frac{1}{[ A ]} d [A] = - k \int_{0}^{t} d t

3

State the Integrated Form:

Rearranges to $ln ([A]_{t} / [A]_{0}) = - k t$ . A plot of $ln$ [A] vs t is a straight line of gradient -k.

ln [A]_{t} = - k t + ln [A]_{0}

Result

ln [A]_{t} = - k t + ln [A]_{0}

Source: OCR A-Level Chemistry A — Reaction Rates

Free formulas

Rearrangements

Solve for $ln [A]$

Make lnA the subject

ln [A] = - k t + ln [A]_{0}

lnA is already the subject of the formula.

Difficulty: 1/5

Solve for $k$

Rearranging the Integrated Rate Law (1st Order) to solve for k

k = \frac{ln [ A ] _{0} - ln [ A ]}{t}

Rearrange the first-order integrated rate law to solve for the rate constant, k. This involves isolating the term with k and then dividing.

Difficulty: 2/5

Solve for $t$

Make t the subject

t = \frac{ln [ A ] _{0} - ln [ A ]}{k}

Rearrange the Integrated Rate Law (1st Order) to make time (t) the subject. This involves isolating the term containing t, then dividing to solve for t.

Difficulty: 2/5

Solve for $ln [A]_{0}$

Make ln[A]0 the subject

ln [A]_{0} = ln [A] + k t

Rearrange the Integrated Rate Law for a first-order reaction to make the initial natural logarithm of concentration, $ln$ [A]_0, the subject.

Difficulty: 2/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

This graph displays a straight line with a negative slope equal to the negative rate constant, where the y-intercept represents the natural log of the initial concentration. For a chemistry student, a small time value indicates the reaction has just begun with high concentration remaining, while a large time value shows the reaction has progressed significantly toward completion. The most important feature of this linear relationship is that the natural log of concentration decreases at a constant rate relative to time, allowing the rate constant to be determined directly from the slope.

Graph type: linear

Why it behaves this way

Intuition

Picture a straight line with a negative slope when the natural logarithm of the reactant concentration is plotted against time, illustrating the exponential decay of the reactant.

[A]

Concentration of reactant A at time t

This value decreases over time as reactant A is consumed in the reaction.

k

First-order rate constant

A larger 'k' means the reaction proceeds faster, causing the concentration of A to decrease more rapidly.

t

Elapsed time since the start of the reaction

As time progresses, more reactant A is consumed, and its concentration decreases.

[A]_{0}

Initial concentration of reactant A at time t=0

This value sets the starting amount of reactant A from which the decay begins.

Signs and relationships

-kt: The negative sign indicates that the natural logarithm of the reactant concentration decreases linearly with time, reflecting the continuous consumption of reactant A.

Free study cues

Insight

Canonical usage

The arguments of the natural logarithms must be dimensionless ratios, and the product of the rate constant and time must be dimensionless.

Common confusion

Students often mistakenly apply units like mol dm^-3 s^-1 to the first-order rate constant k, which are actually the units for a zero-order rate constant.

Dimension note

The ratio [A]/[A]0 is dimensionless, which is a requirement for the argument of a logarithmic function. Consequently, the term kt must also be dimensionless.

Unit systems

[A]mol dm^-3 · Can be any concentration unit or partial pressure, provided [A]0 uses the same unit.

$k$ s^-1 · The unit of the first-order rate constant is always the reciprocal of the time unit used.

$t$ s · Must match the reciprocal time unit of the rate constant k.

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.

Initial ln[A]01.5

Rate Constant0.025 s^-1

Time20 s

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.

Where it shows up

Real-World Context

In how much drug remains in the body after a certain time, Integrated Rate Law (1st Order) is used to calculate ln(Concentration) from Rate Constant, Time, and Initial ln[A]0. The result matters because it helps connect measured amounts to reaction yield, concentration, energy change, rate, or equilibrium.

Study smarter

Tips

Ensure the units for time (t) and the rate constant (k) are consistent (e.g., seconds and s⁻¹).
The value of 'c' represents the natural log of the initial concentration, ln[A]₀.
The negative slope of the line in a ln[A] vs. time plot represents the rate constant k.
A constant half-life, independent of initial concentration, is a signature of first-order kinetics.

Avoid these traps

Common Mistakes

Forgetting to use natural log (ln), not log10.
Using [A] instead of ln[A] for the graph.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

Gives concentration as a function of time for a first-order reaction.

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.

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.

Forgetting to use natural log (ln), not log10. Using [A] instead of ln[A] for the graph.

In how much drug remains in the body after a certain time, Integrated Rate Law (1st Order) is used to calculate ln(Concentration) from Rate Constant, Time, and Initial ln[A]0. The result matters because it helps connect measured amounts to reaction yield, concentration, energy change, rate, or equilibrium.

Ensure the units for time (t) and the rate constant (k) are consistent (e.g., seconds and s⁻¹). The value of 'c' represents the natural log of the initial concentration, ln[A]₀. The negative slope of the line in a ln[A] vs. time plot represents the rate constant k. A constant half-life, independent of initial concentration, is a signature of first-order kinetics.

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

Integrated Rate Law (1st Order)

Overview

Variables

Derivation

Start with the Differential Rate Law:

Separate Variables and Integrate:

State the Integrated Form:

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Half-Life (1st Order)

Integrated Rate Law (2nd Order)

Frequently Asked Questions

Sources