Radioactivity

The correct answer is (D) : 35 Days




⇒ t = 35 days

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 , where is the number of undecayed nuclei at time , is the initial number of nuclei, and is the decay constant.
3. Statement (C): "The slope of the graph of (no. of undecayed nuclei) vs. time represents the reciprocal of mean lifetime ( )."
   • Considering , taking the natural logarithm results in . This is a linear relationship where the slope is indeed . Since the mean lifetime is given by , the slope of this plot being 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 ( ) and half-life time ( ) is not constant."
   • This statement is incorrect. The relationship between the decay constant and half-life is given by , 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

The correct answer is 20
N = N₀e^–λt


⇒ t = 4t_1/2
⇒ t = 20 years

The decay of radioactive material is described by the equation: where:
is the initial amount of the material,
is the amount remaining after time ,
is the decay constant, \item is the time.
Step 1: Determine the decay constant. After 3 days, the material is reduced to of its initial amount: Divide through by : Take the natural logarithm: Simplify:
Step 2: Relate the remaining material after 5 days. After 5 days, the remaining material is: Substitute and : Simplify the exponent: Substitute: Solve for :
Final Answer: The initial amount of the material is:

We are given that the half-life of A is the same as the average life of B. The average life and half-life are related to the decay constant by the equations: Since the half-life of A is the same as the average life of B, we have: Hence, the correct relation is .