Behaviour of activity with time: how does the disintegration rate (activity) of a radioactive sample change as time progresses?

Introduction / Context:
Activity, denoted A, is the number of nuclear disintegrations per unit time in a sample. It is directly proportional to the number of undecayed nuclei present (N). As nuclei decay, N falls, and so must A. Recognising this monotonic decrease is central to understanding radiometric dating, dosimetry, and inventory calculations.

Given Data / Assumptions:

• First-order decay: dN/dt = −λN.
• Activity definition: A = λN.
• Decay constant λ is time-independent for a given nuclide.

Concept / Approach:
Because A = λN and λ is constant, A follows the same exponential law as N. As time increases, N(t) = N0 * exp(−λt) declines, so A(t) = A0 * exp(−λt) also declines. There is no physical mechanism in spontaneous decay for A to increase with time in a single, isolated nuclide sample.

Step-by-Step Solution:
Start with A = λN.Use N(t) = N0 * exp(−λt) → A(t) = λN0 * exp(−λt) = A0 * exp(−λt).Exponential term decreases with time, so activity progressively decreases.

Verification / Alternative check:
Half-life reasoning: each t1/2 reduces both N and A by 1/2. After n half-lives, A = A0 * (1/2)^n, clearly showing a decrease.

Why Other Options Are Wrong:
“Progressively increases”: contradicts A = λN with λ constant and N decreasing.“Remains constant”: would require λ = 0 or replenishment of N.“May increase or decrease”: spontaneous decay in a closed system does not produce increases.

Common Pitfalls:
Confusing multi-nuclide chains (where daughter growth can temporarily increase a specific isotope’s activity) with the total sample activity of a single nuclide; mixing up count-rate instrument effects with true activity.

Final Answer:
Progressively decreases