23.2.2Quantum and nuclear physics (A Level)

Activity and The Decay Constant

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Lesson summary

23.2.2 Activity and The Decay Constant — AS PHYSICS 9702 notes: Define activity and the decay constant and use A = λN and the exponential decay equations.Decay constant (λ) = the probability that an individual nucleus will decay per unit of time.Its unit is s−1, or h−1 or year−1 …

Interpretation
Activity and count rate
Size of activity
Number of nuclei
What the symbols mean
Inverse functions

Section 1 of 9Decay constant and activity

Decay constant

(λ) = the probability that an individual nucleus will decay per unit of time.

Its unit is s⁻¹, or h⁻¹ or year⁻¹ as convenient.

Activity

Activity the number of nuclei that decay per unit time.

Its SI unit is the becquerel (Bq), equal to one decay per second.

Equation

A = ΔN/Δt = −λN, where N is the number of undecayed nuclei.
The minus sign shows N decreases, and it can be omitted in calculations.

Interpretation

A larger decay constant gives a larger activity and a shorter-lived nuclide.

Activity and count rate

A detector sees only a fraction of the radiation, so count rate is not the same as activity.

Size of activity

A small sample of a short-lived isotope can be more active than a large sample of a long-lived one.

Section 2 of 9Exponential decay

Exponential decay

Rule

The rate of decay is proportional to the number of nuclei remaining, dN/dt = −λN.

Number of nuclei

N = N₀e^−λt.
What the symbols mean: N = number of nuclei left after time t · N₀ = number at the start · λ = decay constant (s⁻¹) · t = time (s)
N₀ is the initial number, N the number at time t, and e = 2.718.

Activity

A = A₀e^−λt, since A is proportional to N.

Count rate

C = C₀e^−λt for corrected count rate.

Inverse functions

Inverse functions ln and e^x are inverse functions, so if e^x = y then x = ln y.

Section 3 of 9Graph of decay

Shape

N, A or C falls steeply at first and flattens, approaching zero but never reaching it.

Start

The curve begins on the vertical axis at N₀.

Gradient

The gradient at any point is −λN, so a steeper slope means a larger λ.

Logarithmic form

Logarithmic form ln N = ln N₀ − λt.
A graph of ln N (or ln A) against t is a straight line.
The gradient is −λ and the intercept is ln N₀.

Finding λ

Plot ln of the corrected count rate against time, draw the best-fit line and take the magnitude of the gradient.

Section 4 of 9Units

Matching units

The units of λ must match those of t in the exponent, because λt is dimensionless.

Conversion

If λ is in h⁻¹ and the activity is needed in becquerels, convert λ to s⁻¹ by dividing by 3600.

Section 5 of 9Number of nuclei from a mass

Equation

N = (m/M) × N_A, where m is the mass, M the molar mass and N_A = 6.02 × 10²³ mol⁻¹.

Route to activity

Convert mass to moles, then to the number of nuclei N, then use A = λN with λ in s⁻¹.

Intermediate numbers

They are large, so state each step clearly to avoid power-of-ten errors.

Section 6 of 9Link to half-life

Relation

λ = ln 2 / t_½ = 0.693 / t_½.
What the symbols mean: t_½ = half-life (s) · λ = decay constant (s⁻¹) · ln 2 ≈ 0.693

Two roles

λ is a probability for one nucleus and a rate constant for the whole sample.

Section 7 of 9More from the specification

1Radioactive Decay as a Random and Spontaneous Process

Radioactive decay is a spontaneous nuclear process unaffected by external physical conditions,
and its random nature means it is impossible to predict which specific nucleus decays at any given moment.

2Activity, Decay Constant, and Rate of Decays

The activity A of a radioactive source measures the number of decays or counts per unit time (in Bq), governed by the equation A = λ·N,
where λ is the decay constant (s⁻¹) and N represents the quantity of undecayed nuclei in the sample.

3Half-Life Equations and Time Conversions

The half-life t₁/₂ is defined as the time taken for half the radioactive nuclei in a sample to decay; for long-lived isotopes whose half-lives are given in years, quantities must be converted to seconds to determine decay constants.
- (t₁/₂ = ln 2 / λ)

4Radioactive

Describes a nucleus that decays spontaneously and randomly.

5Random

Without a pattern in speed or direction.

6Spontaneous

A process which cannot be influenced by environmental factors such as temperature or pressure.

Section 8 of 9Key terms

Term	Meaning
Activity	The number of nuclei that decay per unit time, A = λN, measured in becquerel.
Becquerel (Bq)	The SI unit of activity, equal to one decay per second.
Decay constant	The probability per unit time that an individual nucleus will decay.
Exponential decay	A decrease in which the rate of decay is proportional to the number of nuclei remaining.
Count rate	The number of counts detected per unit time, proportional to activity but smaller.

Section 9 of 9Exam technique

Using λ in per hour with t in seconds in the exponent — use the same time unit for both.

Saying activity is the number of particles detected — it is the number of decays per second in the sample.

Believing the decay graph reaches zero at a finite time — an exponential decay never reaches exactly zero.

Using log₁₀ in place of ln in ln N = ln N₀ − λt — use natural logarithms.

Forgetting the minus sign in the exponent — N = N₀e^−λt, not e^+λt.