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

The question involves understanding the properties of radioactivity and evaluating which statements among the given options are correct. Let's examine each statement individually: 

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. This is a key characteristic of radioactivity, as changes in temperature, pressure, or chemical state do not affect the radioactive decay of a specific isotope. Thus, this statement is incorrect.
2. Statement (B): "The number of un-decayed nuclei in the radioactive sample decays exponentially with time."
   • This statement is correct. The decay of radioactive nuclei follows an exponential decay law, expressed by the equation \( N(t) = N_0 e^{-\lambda t} \), 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.
3. Statement (C): "The slope of the graph of \( \log_e \) (no. of undecayed nuclei) vs. time represents the reciprocal of mean lifetime (\( \tau \))."
   • Considering \( N(t) = N_0 e^{-\lambda t} \), taking the natural logarithm results in \( \log_e N(t) = \log_e N_0 - \lambda t \). This is a linear relationship where the slope is indeed \(-\lambda\). Since the mean lifetime \( \tau \) is given by \( \tau = \frac{1}{\lambda} \), the slope of this plot being \(-\lambda\) means the slope is the negative reciprocal of mean lifetime. The statement correctly identifies the relationship when viewed in terms of the negative slope, so this statement is correct.
4. Statement (D): "Product of decay constant (\( \lambda \)) and half-life time (\( T_{1/2} \)) is not constant."
   • This statement is incorrect. The relationship between the decay constant and half-life is given by \(\lambda \cdot T_{1/2} = \ln(2)\), which is a constant value for any radioactive isotope.

After evaluating all statements, we conclude that the statements (B) and (C) are correct. Thus, the most appropriate answer is:

(B) and (C) only

Answer 2

The correct answer is (C) : (B) and (C) only
Radioactive decay is a random and spontaneous process it depends on unbalancing of nucleus.
N = N₀e^–λt …(B)
lnN = –λt + lnN₀
So, slope = – λ …(C)
\(t_{1/2}=\frac{ln2}{λ}\)
So t_1/2 × λ = ln2 = Constant