Question

Following statements related to radioactivity are given below:

(A) Radioactivity is a random and spontaneous process and is dependent on physical and chemical conditions.
(B) The number of un-decayed nuclei in the radioactive sample decays exponentially with time.
(C) Slope of the graph of log_e (no. of undecayed nuclei) Vs. time represents the reciprocal of mean life time (τ).
(D) Product of decay constant (λ) and half-life time (T_1/2) is not constant.
Choose the most appropriate answer from the options given below:

• (A) and (B) only
• (B) and (D) only
• (B) and (C) only
• (C) and (D) only

Answer 1

The Correct Option is C

Let's examine each statement related to radioactivity to determine which are correct and which are incorrect:

1. Statement (A): "Radioactivity is a random and spontaneous process and is dependent on physical and chemical conditions."
   • Radioactivity is indeed a random and spontaneous process; however, it is independent of physical and chemical conditions. External factors such as temperature or pressure do not affect the process of radioactivity.
   • Therefore, Statement (A) is incorrect.
2. Statement (B): "The number of un-decayed nuclei in the radioactive sample decays exponentially with time."
   • This is a correct statement. The decay of radioactive nuclei follows an exponential law specified by the equation: \(N(t) = N_0 \cdot e^{-\lambda t}\) where \(N(t)\) is the number of undecayed nuclei at time \(t\), \(N_0\) is the initial number of nuclei, and \(\lambda\) is the decay constant.
   • Thus, Statement (B) is correct.
3. Statement (C): "Slope of the graph of \(\log_e\) (no. of undecayed nuclei) vs. time represents the reciprocal of mean life time (\(\tau\))."
   • The slope of the graph \(\log_e(N)\) versus time is indeed related to the decay constant: \(\frac{d}{dt}(\log_e(N)) = -\lambda\). Since \(\tau = \frac{1}{\lambda}\), the statement is true as the negative slope provides \(\lambda\), reciprocal of mean life time.
   • Therefore, Statement (C) is correct.
4. Statement (D): "Product of decay constant (\(\lambda\)) and half-life time (\(T_{1/2}\)) is not constant."
   • This statement is incorrect. In radioactive decay, the product of the decay constant and the half-life is a constant: \(\lambda \cdot T_{1/2} = \log_e(2)\).
   • Thus, Statement (D) is incorrect.

After evaluating each statement, the correct answer is (B) and (C) only, since both are true in the context of radioactivity.