Half-life Equation, Derivation & Rearrangements

Core idea

Overview

The half-life represents the time required for a radioactive substance to diminish to one-half of its original quantity. It is a fundamental property of unstable atomic nuclei, where the decay rate is proportional to the number of atoms present, governed by the decay constant.

When to use: Use this formula when modeling first-order decay processes where the probability of decay is constant per unit time. It is applicable for calculating the longevity of radioactive isotopes or determining the decay constant from observed temporal data.

Why it matters: Understanding half-life is essential for carbon dating ancient artifacts, managing nuclear waste safety, and calculating medical tracer dosages. It allows scientists to predict how long hazardous materials will remain active in a given environment.

Symbols

Variables

T_{1/2} = Half-life, \lambda = Decay Constant

T_{1/2}

Half-life

s

λ

Decay Constant

s^{-} 1

Walkthrough

Derivation

Derivation of the Half-Life Equation

Relates half-life to decay constant using the exponential decay model.

Exponential decay model holds.

1

Start with the Decay Law:

At t= $T_{1/2}$ , by definition N= $N_{0}$ /2.

N = N_{0} e^{- λ t}

2

Substitute Half-Life Condition:

Cancel $N_{0}$ from both sides.

\frac{1}{2} = e^{- λ T_{1/2}}

3

Take Natural Logs and Solve:

Use $ln (1/2) = - ln 2$ to obtain the final result.

ln (\frac{1}{2}) = - λ T_{1/2} ⟹ T_{1/2} = \frac{ln 2}{λ}

Result

ln (\frac{1}{2}) = - λ T_{1/2} ⟹ T_{1/2} = \frac{ln 2}{λ}

Source: Edexcel A-Level Physics — Nuclear and Particle Physics

Free formulas

Rearrangements

Solve for $T_{1/2}$

Make T the subject

T_{1/2} = \frac{ln 2}{λ}

T is already the subject of the formula.

Difficulty: 1/5

Solve for $λ$

Make lambda the subject

λ = \frac{ln 2}{T _{1/2}}

To make the decay constant ` $λ$ ` the subject, start with the half-life formula, multiply both sides by ` $λ$ `, then divide by ` $T_{1/2}$ `.

Difficulty: 2/5

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

Visual intuition

Graph

The graph displays an inverse curve where the half-life decreases rapidly as the decay constant increases, creating a vertical asymptote at zero. For a physics student, this means that a large decay constant represents a substance that decays very quickly with a short half-life, while a small decay constant indicates a very stable substance with a long half-life. The most important feature is that the curve never reaches zero, meaning that no matter how large the decay constant becomes, the half-life will always re

Graph type: inverse

Why it behaves this way

Intuition

Visualize a declining curve where the initial quantity of a substance repeatedly halves itself over equal successive time intervals, with each interval being the half-life.

T_{1/2}

The time interval during which half of a given quantity of a radioactive isotope decays.

A longer half-life means the radioactive material decays slowly, remaining active for a longer duration.

λ

The decay constant, representing the probability per unit time for an individual nucleus to decay.

A larger decay constant signifies a higher probability of decay, leading to a faster reduction in the number of radioactive nuclei and thus a shorter half-life.

Signs and relationships

\ln 2: The constant $ln$ 2 arises from the mathematical definition of half-life for an exponential decay process. It ensures that when the number of decaying entities has reduced to exactly half, the time elapsed is $T_{1/2}$ .

Free study cues

Insight

Canonical usage

Ensures dimensional consistency between half-life (a time unit) and the reciprocal of the decay constant (an inverse time unit).

Common confusion

Students often forget that the decay constant ( $λ$ ) has units of inverse time (e.g., s-1) and must be consistent with the time unit used for $T_{1/2}$ .

Unit systems

$T_{1/2}$ s, min, h, d, yr (any unit of time) · The choice of time unit for half-life must be consistent with the unit used for the decay constant.

$λ$ s-1, min-1, h-1, d-1, yr-1 (any unit of inverse time) · The decay constant's unit must be the reciprocal of the half-life's unit for dimensional consistency.

$ln 2$ dimensionless · A mathematical constant, approximately 0.693.

One free problem

Practice Problem

A radioactive isotope of Cobalt-60 used in medical radiotherapy has a decay constant (L) of 0.131 per year. Calculate its half-life (T) in years.

Decay Constant0.131 s^-1

Solve for: $T$

Hint: Divide the natural log of 2 (approximately 0.693) by the given decay constant.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Finding decay constant from a half-life.

Study smarter

Tips

The value of ln(2) is approximately 0.693.
Ensure the units for half-life and the decay constant are reciprocals, such as seconds and s⁻¹.
Half-life is constant for a specific isotope and does not depend on the initial amount of material.
If the decay constant increases, the half-life decreases proportionally.

Avoid these traps

Common Mistakes

Using log10 instead of ln.
Mixing seconds and years.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

Relates half-life to decay constant using the exponential decay model.

Use this formula when modeling first-order decay processes where the probability of decay is constant per unit time. It is applicable for calculating the longevity of radioactive isotopes or determining the decay constant from observed temporal data.

Understanding half-life is essential for carbon dating ancient artifacts, managing nuclear waste safety, and calculating medical tracer dosages. It allows scientists to predict how long hazardous materials will remain active in a given environment.

Using log10 instead of ln. Mixing seconds and years.

Finding decay constant from a half-life.

The value of ln(2) is approximately 0.693. Ensure the units for half-life and the decay constant are reciprocals, such as seconds and s⁻¹. Half-life is constant for a specific isotope and does not depend on the initial amount of material. If the decay constant increases, the half-life decreases proportionally.

References

Sources

Halliday, Resnick, Walker - Fundamentals of Physics
Wikipedia: Half-life
Atkins' Physical Chemistry
Halliday, Resnick, and Walker, Fundamentals of Physics
Wikipedia: Carbon-14
IUPAC Gold Book: Half-life
Halliday, Resnick, Walker Fundamentals of Physics
Edexcel A-Level Physics — Nuclear and Particle Physics

Half-life

Overview

Variables

Derivation

Start with the Decay Law:

Substitute Half-Life Condition:

Take Natural Logs and Solve:

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Nuclear Decay

Frequently Asked Questions

Sources