Nuclear Decay Equation, Derivation & Rearrangements

Core idea

Overview

The nuclear decay equation models the statistical decrease of radioactive nuclei within a sample over time. It demonstrates that the rate of decay is proportional to the number of nuclei present, resulting in a predictable exponential decay curve.

When to use: Apply this equation when calculating the remaining mass or activity of a radioactive isotope after a specific duration. It assumes a sufficiently large sample size where the constant probability of decay (L) remains uniform across all atoms.

Why it matters: This formula is essential for carbon-14 dating to determine the age of organic artifacts and for nuclear medicine to calculate precise patient dosages. It also informs safety protocols for the storage and management of hazardous nuclear waste products.

Symbols

Variables

N = Remaining Nuclei, N_0 = Initial Nuclei, \lambda = Decay Constant, t = Time

N

Remaining Nuclei

V a r iab l e

N_{0}

Initial Nuclei

V a r iab l e

λ

Decay Constant

s^{-} 1

t

Time

s

Walkthrough

Derivation

Derivation of the Nuclear Decay Equation

Derives exponential decay from the assumption that decay rate is proportional to the number of undecayed nuclei.

Decay of a single nucleus is random.
Large N so the model is statistical.

1

State the Rate Equation:

The decay rate is proportional to N; $λ$ is the decay constant.

\frac{d N}{d t} = - λ N

2

Separate Variables and Integrate:

Integrate both sides to solve the differential equation.

\int \frac{1}{N} d N = - \int λ d t

3

Solve for N(t):

Using N= $N_{0}$ at t=0 gives the exponential decay law.

ln N = - λ t + C ⟹ N = N_{0} e^{- λ t}

Result

ln N = - λ t + C ⟹ N = N_{0} e^{- λ t}

Source: AQA A-Level Physics — Nuclear Physics

Free formulas

Rearrangements

Solve for $N$

Make N the subject

N = N_{0} e^{- λ t}

N is already the subject of the formula.

Difficulty: 1/5

Solve for $N_{0}$

Make N0 the subject

N_{0} = N e^{λ t}

Start from the Nuclear Decay equation. To make N0 the subject, divide by the exponential factor and simplify the negative exponent.

Difficulty: 2/5

Solve for $λ$

Make lambda the subject

λ = - \frac{1}{t} ln (\frac{N}{N _{0}})

To make $λ$ the subject, start with the nuclear decay equation, divide by $N_{0}$ , take the natural logarithm of both sides, and then divide by -t.

Difficulty: 2/5

Solve for $t$

Nuclear Decay: Make t the subject

t = - \frac{1}{λ} ln (\frac{N}{N _{0}})

To make time (t) the subject of the nuclear decay equation, first isolate the exponential term, then take the natural logarithm of both sides, and finally rearrange to solve for t.

Difficulty: 3/5

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

Visual intuition

Graph

The graph is an exponential decay curve that starts at N0 when time is zero, dropping rapidly as time increases toward an asymptote at the x-axis. For a physics student, this shape shows that the rate of decay is highest when time is small and slows down significantly as time becomes large. The most important feature is that the curve never reaches zero, meaning that while the number of remaining nuclei constantly decreases, the substance is never fully exhausted according to this model.

Graph type: exponential

Why it behaves this way

Intuition

Imagine a smooth curve starting at an initial value on the y-axis and continuously falling towards zero on the x-axis, where the rate of descent is always proportional to the current height of the curve.

N

Number of undecayed radioactive nuclei remaining in a sample at time t.

The quantity of radioactive material still present after a certain period.

N_{0}

The initial number of radioactive nuclei in the sample at time t = 0.

The starting amount of radioactive material before any decay occurs.

λ

The decay constant, representing the probability per unit time for a single nucleus to decay.

A measure of how quickly a radioactive isotope decays; a larger value means faster decay.

t

The elapsed time since the initial measurement.

The duration over which the decay process is observed.

Signs and relationships

-λ t: The negative sign in the exponent signifies an exponential decrease in the number of radioactive nuclei over time. As 't' increases, the exponent becomes more negative, causing ' $e^{- λ t}$ ' to approach zero

Free study cues

Insight

Canonical usage

Ensures dimensional consistency, particularly that the product of the decay constant and time is dimensionless, and that the initial and final quantities of nuclei are expressed in consistent units.

Common confusion

Using inconsistent units for the decay constant (λ) and time (t), which leads to an incorrect dimensionless exponent and therefore an incorrect result.

Dimension note

The ratio N/ $N_{0}$ is dimensionless. The exponent -λt must also be dimensionless, requiring λ and t to have inverse units.

Unit systems

$N$ count | mass | activity · Represents the number of nuclei, mass, or activity. Must be consistent with N_0 (e.g., both in counts, grams, or becquerels).

$N_{0}$ count | mass | activity · Represents the initial number of nuclei, mass, or activity. Must be consistent with N (e.g., both in counts, grams, or becquerels).

$λ$ time^-1 · The decay constant. Its units must be the inverse of the units used for time (t) to ensure the exponent is dimensionless.

$t$ time · Time elapsed. Its units must be consistent with the inverse units of the decay constant (λ) to ensure the exponent is dimensionless.

One free problem

Practice Problem

A laboratory sample initially contains 1000 radioactive nuclei. If the isotope has a decay constant of 0.05 s⁻¹, how many nuclei will remain after 10 seconds?

Initial Nuclei1000

Decay Constant0.05 s^-1

Time10 s

Solve for: $N$

Hint: Calculate the exponent by multiplying the decay constant and time, then use the eˣ function.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Estimating remaining nuclei in a radioactive sample.

Study smarter

Tips

Ensure the units for the decay constant (L) and time (t) are reciprocals, such as s⁻¹ and s.
The value N can represent mass, number of atoms, or activity (becquerels).
Use the natural logarithm (ln) to move the variable 't' or 'L' out of the exponent when solving for them.

Avoid these traps

Common Mistakes

Using base⁻10 logs instead of natural logs.
Mixing half-life and decay constant directly.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

Derives exponential decay from the assumption that decay rate is proportional to the number of undecayed nuclei.

Apply this equation when calculating the remaining mass or activity of a radioactive isotope after a specific duration. It assumes a sufficiently large sample size where the constant probability of decay (L) remains uniform across all atoms.

This formula is essential for carbon-14 dating to determine the age of organic artifacts and for nuclear medicine to calculate precise patient dosages. It also informs safety protocols for the storage and management of hazardous nuclear waste products.

Using base⁻10 logs instead of natural logs. Mixing half-life and decay constant directly.

Estimating remaining nuclei in a radioactive sample.

Ensure the units for the decay constant (L) and time (t) are reciprocals, such as s⁻¹ and s. The value N can represent mass, number of atoms, or activity (becquerels). Use the natural logarithm (ln) to move the variable 't' or 'L' out of the exponent when solving for them.

References

Sources

Halliday, Resnick, Walker - Fundamentals of Physics
Wikipedia: Radioactive decay
IUPAC Gold Book: Decay constant
Halliday, Resnick, Walker, Fundamentals of Physics
AQA A-Level Physics — Nuclear Physics

Nuclear Decay

Overview

Variables

Derivation

State the Rate Equation:

Separate Variables and Integrate:

Solve for N(t):

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Half-life

Frequently Asked Questions

Sources