Question

For a radioactive material, if N is the number of nuclei present at time t, and \(\lambda\) is decay constant, which of the following is/are CORRECT representation(s) of the rate of decay?

• A. \( -\dfrac{dN}{dt} = \lambda N^2 \)
• B. \( -\dfrac{dN}{dt} = \lambda N \)
• C. \( -\dfrac{dN}{dt} = \dfrac{\lambda}{N^2} \)
• D. \( -\dfrac{dN}{dt} = \dfrac{\lambda}{N} \)

Hint:

Radioactive decay is a first order process, the decay rate is directly proportional to N, not to N squared or to 1 over N.

Answer 1

The Correct Option is B

Step 1: Recall the basic law of radioactive decay:
Radioactive decay is a first order process. The physical statement of the decay law is that the rate at which nuclei disintegrate at any instant is directly proportional to the number of undecayed nuclei present at that instant. This is written as \( -\frac{dN}{dt} \propto N \), which becomes \( -\frac{dN}{dt} = \lambda N \) once the constant of proportionality, the decay constant \(\lambda\), is introduced. The negative sign shows that N decreases with time.
Step 2: Analyze option (A):
Option (A) proposes \( -\frac{dN}{dt} = \lambda N^2 \), which would describe a second order decay process where the rate depends on the square of the number of nuclei present, similar to a second order chemical reaction. Radioactive decay is a spontaneous single nucleus event and does not depend on interaction between pairs of nuclei, so this form is incorrect.
Step 3: Analyze option (B):
Option (B) proposes \( -\frac{dN}{dt} = \lambda N \), which is exactly the standard first order radioactive decay law derived in Step 1. Integrating this equation gives the familiar exponential decay result \( N = N_0 e^{-\lambda t} \), which correctly matches observed radioactive decay behavior. So option (B) is correct.
Step 4: Analyze options (C) and (D):
Options (C) and (D) propose that the decay rate is inversely related to N or to N squared. Physically this would mean the decay rate increases as fewer nuclei remain, which is the opposite of what is observed, since fewer nuclei available to decay means a smaller absolute decay rate, not a larger one. Both of these forms are inconsistent with the physics of radioactive decay, so options (C) and (D) are incorrect.
Step 5: Conclude on the correct representation:
Only option (B), the first order linear relationship \( -\frac{dN}{dt} = \lambda N \), correctly represents the rate of radioactive decay. Even though the question is phrased as is/are, only one option satisfies the actual physical law, so this behaves as a single correct answer question.
Final Answer:
\[ \boxed{-\dfrac{dN}{dt} = \lambda N} \]