Half Life

Radioactive decay is a random and spontaneous process it depends on unbalancing of nucleus.
N = N₀e^–λt …(B)
ln N = –λt + lnN₀
So, slope = – λ …(C)
t_1/2=
So t_1/2 × λ = ln2 = Constant

To determine the time after which the radiation emitted by the radioactive source falls to a safe level, we begin by understanding radioactive decay characterized by half-life. Given:

• Initial intensity = 64 times the safe level.
• Half-life (T₁/₂) = 2.5 hours (2 hours 30 minutes).

The formula for radioactive decay is:

N(t) = N₀ × (1/2)^(t/T₁/₂)

where N(t) is the intensity at time t, N₀ is the initial intensity. We set N(t) equal to 1 (safe level) and solve for t.

1. Set N(t) = 1, N₀ = 64, thus:

1 = 64 × (1/2)^(t/2.5)

1. Rearrange the equation to isolate the exponential term:

(1/2)^(t/2.5) = 1/64

1. Recognize that 1/64 = (1/2)⁶, hence:

(1/2)^(t/2.5) = (1/2)⁶

1. Equate the exponents as the bases are identical:

t/2.5 = 6

1. Solve for t:

t = 6 × 2.5 = 15 hours

Thus, the minimum required time is 15 hours. This value lies within the expected range of 15 to 15 hours.

Understanding the Problem

We are given a radioactive substance with a half-life of 5 years. We need to find the fraction of the original sample that decays after 15 years.

Solution

1. Radioactive Decay Formula:

The number of remaining nuclei after time is given by:

where:

• is the initial number of nuclei
• is the remaining number of nuclei
• is the elapsed time
• is the half-life

2. Substitute Values:

Given and , we have:

3. Calculate Remaining Fraction:

This means that of the original sample remains.

4. Calculate Decayed Fraction:

The fraction that decayed is the difference between the initial fraction (1) and the remaining fraction ( ):

Final Answer

The fraction of the original sample that decays is .