Time dependence of radioactive decay: a radioactive isotope’s activity decreases with time according to which mathematical law?

Introduction / Context:
Radioactive decay is a stochastic process characterized by a constant probability of decay per nucleus per unit time. Despite the randomness at the microscopic level, the macroscopic behavior of large populations of nuclei follows a predictable mathematical form. Recognizing the exponential law underpins concepts such as half-life and decay constant.

Given Data / Assumptions:

• Nuclei decay independently.
• Decay constant (lambda) is time-invariant.
• No production or removal of the parent isotope during the interval considered.

Concept / Approach:
The rate equation for radioactive decay is dN/dt = −lambda * N. Solving this first-order differential equation yields N(t) = N0 * exp(−lambda * t). Activity A(t) is proportional to N(t), so it also decays exponentially. This elegant law explains why the fraction remaining after equal time intervals is constant (leading to the concept of half-life), unlike linear or inverse-square processes.

Step-by-Step Solution:
Write decay rate: dN/dt = −lambda * N.Separate variables: dN/N = −lambda dt.Integrate to obtain: N(t) = N0 * exp(−lambda * t).Relate activity: A(t) ∝ N(t) → exponential as well.

Verification / Alternative check:
Plotting ln(N) versus time produces a straight line of slope −lambda. This diagnostic linearization is a standard verification in radiochemistry and nuclear physics labs.

Why Other Options Are Wrong:
Logarithmic: implies very slow variation, not physically consistent here.Linear: would yield constant-rate decrease; not observed for decay.Inverse-square: pertains to intensity versus distance for point sources.Sinusoidal: oscillatory behavior, not applicable to decay.

Common Pitfalls:
Confusing exponential decay with half-life that depends on the starting amount; half-life is constant for first-order decay.Mixing activity decay (counts per second) with detector efficiency changes.

Final Answer:
Exponential law