Radioactive Activity-Decay Constant Relation

Equivalent forms

A = \lambda N_0 e^{-\lambda t}

A linear relation that turns an Avogadro-scale number of atoms into a measurable click rate on a Geiger counter.

Symbol	Name	SI unit	Dimension	Range
$A$	Activityoutput Decays per unit time	Bq	T⁻¹	0 – 1000000000000000
$λ$	Decay constant Per-atom decay probability per unit time	1/s	T⁻¹	1e-20 – 1
$N$	Number of nuclei Current number of undecayed nuclei	—	1	0 – 1e+25

Unit systems

Symbol	SI	CGS	Natural
$A$	Bq	Ci	MeV
$λ$	1/s	1/s	MeV
$N$	—	—	—

Where it holds

Valid when N is large enough for statistical averaging (typically N ≫ 1) and decays are independent events.

Dimensional analysis

A \to [T^{-1}]

\lambda N \to [T^{-1}] \times [1] = [T^{-1}]

Both sides are inverse time.

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Discovery

Ernest Rutherford & Frederick Soddy · 1902

Rutherford and Soddy formalized the statistical nature of radioactive decay, defining decay constant λ as the per-atom decay probability per unit time.

Try this

How 'hot' is one gram of radium?

Calculate the activity (in Bq) of 1.0 g of Ra-226 (λ = 1.37 × 10⁻¹¹ s⁻¹).

Research status: stable

Real-world applications

Nuclear medicine dosimetry (Tc-99m, I-131)
Radiocarbon dating (C-14, $\lambda \approx 3.83 \times 10^{-12} s^{-1})$
Nuclear reactor inventory and waste classification
Radon hazard assessment

Common misconceptions

Activity decreases over time as A $(t) = A_0\,\mathrm{e}^(-\lambda t)$ — it is not constant unless N is replenished.
Specific activity (activity per gram) depends on both $\lambda and$ the atomic mass.
The half-life $T_1/2 = \ln (2)/\lambda$ , not $1/\lambda$ .

Experimental verification

Liquid scintillation counters, Geiger-Müller tubes, and HPGe detectors directly measure A. The Curie

(3.7 \times 10^{10} Bq)

was historically defined as the activity of 1 g of Ra-226.

Derivation

Decay is a Poisson process: each nucleus has independent probability

\lambda dt

of decaying in time dt.

For N nuclei, expected decays per second

= -dN/dt = \lambda N

, which IS the activity A.

Limiting cases

N \to 0

⟶

A \to 0

Sample is exhausted; no further decays.

\lambda \to 0

⟶

A \to 0

Infinitely stable isotope has no activity.

\lambda

large, N large⟶ Sample 'hot'Short-lived isotopes in large numbers give intense radioactive sources.

What if…

What if half-life were doubled?

$\lambda$ halves → activity halves for same N.

What if N were Avogadro's number?

$A = \lambda \times N$ _A — for fast decayers like $Po-214 (\lambda \sim 10^{-3} s^{-1})$ , 1 mole would produce $\sim 10^{20} Bq$ , catastrophically radioactive.

1

Activity of 1 g of Ra-226

Given ·

λ:: 1.37e-11
N:: 2.66e+21

Find · A

Steps

Step 1: N for 1 g of $Ra-226 = (1/226) \times N$ _ $A = 2.66 \times 10^{21}$ atoms.
Step 2: $\lambda for Ra-226 = \ln (2)/T_1/2 = 0.693/(1600 \times 3.156 \times 10^{7}) = 1.37 \times 10^{-11} s^{-1}$ .
Step 3: $A = \lambda N = 1.37 \times 10^{-11} \times 2.66 \times 10^{21} = 3.64 \times 10^{10} Bq$ , matching the historical definition of 1 Curie.

Result ·

A = 1.37 \times 10^{-11} \times 2.66 \times 10^{21} = 3.64 \times 10^{10} Bq \approx 1\,\mathrm{Ci}

Radioactive Decay Law→half life relation