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

To solve this problem, we will evaluate each statement related to radioactivity and determine their accuracy.

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, meaning it occurs without external influence. However, it is independent of physical and chemical conditions. The rate of decay is not affected by factors like temperature or pressure.
   • Conclusion: Statement (A) is incorrect.
2. Statement (B): "The number of un-decayed nuclei in the radioactive sample decays exponentially with time."
   • This statement is correct as it follows the law of radioactive decay, where the number of undecayed nuclei \( N(t) \) at time \( t \) can be expressed as: N(t) = N_0 e^{-\lambda t}, where \( N_0 \) is the initial number of nuclei and \( \lambda \) is the decay constant.
   • Conclusion: Statement (B) is correct.
3. Statement (C): "Slope of the graph of log_e (number of undecayed nuclei) Vs. time represents the reciprocal of mean life time (τ)."
   • The slope of this graph represents the decay constant \( \lambda \) as \log_e(N(t)) = \log_e(N_0) - \lambda t.
   • The mean lifetime \( \tau \) is given by \tau = \frac{1}{\lambda}, so the slope (which is \( -\lambda \)) corresponds to the reciprocal of mean lifetime with a negative sign.
   • Conclusion: Statement (C) is correct.
4. Statement (D): "Product of decay constant (λ) and half-life time (T_1/2) is not constant."
   • The relation between decay constant and half-life is given by \lambda \cdot T_{1/2} = \ln(2), which is a constant.
   • Conclusion: Statement (D) is incorrect.

Based on the analysis above, the correct statements are (B) and (C). Therefore, the correct answer is: (B) and (C) only.