Radioactive Decay Rate Explained

The rate of radioactive decay is determined by the number of radioactive nuclei present and the decay constant. The fundamental law of radioactive decay states that the rate of disintegration is directly proportional to the number of nuclei ($N$) at that instant ($t$).

Radioactive Decay Law

Mathematically, this relationship is expressed as:

$-\frac{dN}{dt} \propto N$

Introducing the decay constant ($\lambda$) as the proportionality constant, we get the equation for the rate of decay:

$-\frac{dN}{dt} = \lambda N$

Where:

• $-\frac{dN}{dt}$ represents the rate of decay (number of nuclei decaying per unit time).
• $N$ is the number of radioactive nuclei present at time $t$.
• $\lambda$ is the decay constant, a characteristic property of the radioactive isotope.

Analyzing the Options

Comparing the derived formula with the given options:

• Option 1: $-\frac{dN}{dt} = \lambda N^2$ is incorrect as the rate is proportional to $N$, not $N^2$.
• Option 2: $-\frac{dN}{dt} = \lambda N$ correctly represents the law of radioactive decay.
• Option 3: $-\frac{dN}{dt} = \frac{\lambda}{N^2}$ is incorrect.
• Option 4: $-\frac{dN}{dt} = \frac{\lambda}{N}$ is incorrect.

Therefore, the correct representation of the rate of decay is $-\frac{dN}{dt} = \lambda N$.