Question

The half-life period of a radioactive element A is 62 years. It decays into another stable element B. An archaeologist found a sample in which A and B are in 1 : 15 ratio. The age of the sample is:

• A. 248 years
• B. 186 years
• C. 124 years
• D. 310 years

Hint:

- The decay of radioactive substances follows an exponential decay model based on half-life.
- The ratio of remaining substance to decayed substance can be used to find the age of the sample.

Answer 1

The Correct Option is A

We are given that the half-life of A is 62 years, and the ratio of A to B in the sample is 1:15. The relationship between the amount of radioactive substance left and its half-life is governed by the equation: \[ N = N_0 \left( \frac{1}{2} \right)^{t/T} \] Where: - \( N \) is the amount of the radioactive substance left after time \( t \),
- \( N_0 \) is the initial amount of the substance,
- \( T \) is the half-life of the substance, and
- \( t \) is the time elapsed.
Here, we know the ratio of the remaining substance A to the product B is 1 : 15, which implies: \[ \frac{N_A}{N_B} = 1 : 15 \] Using the decay formula and the fact that B is formed as A decays, we can calculate the age of the sample: \[ \frac{N_A}{N_0} = \left( \frac{1}{2} \right)^{t/62} \] Given the ratio of 1:15, we substitute into the equation: \[ \left( \frac{1}{2} \right)^{t/62} = \frac{1}{16} \] This gives \( t = 248 \) years. Thus, the age of the sample is 248 years.

Answer 2

To find the age of the sample, we need to use the concept of half-life in radioactive decay. Given that the half-life of element A is 62 years and the ratio of A to stable element B is 1:15, we can solve for the age.

The decay formula for radioactive elements is:

N(t) = N₀e^-λt

Where:

• N(t) is the remaining quantity of the substance after time t.
• N₀ is the initial quantity of the substance.
• λ is the decay constant.

The decay constant λ relates to the half-life (T_1/2) as follows:

λ = ln(2) / T_1/2

For T_1/2 = 62 years:

λ = ln(2) / 62

Now, we know the ratio of A to B is 1:15, meaning 1 part of A remains and 15 parts of B are formed. Initially, let's assume there was 1 + 15 = 16 parts of A.

Thus, the amount of A remaining after time t is 1 unit:

N(t) = N₀ e^-λt = 16 e^-λt = 1

Rearrange and solve for t:

e^-λt = 1/16

-λt = ln(1/16)

t = -ln(1/16) / λ

Substitute λ = ln(2) / 62:

t = -ln(1/16) / (ln(2) / 62)

Simplifying further:

t = 62 × ln(16) / ln(2)

Since ln(16) = 4 ln(2), we have:

t = 62 × 4

t = 248 years

Therefore, the age of the sample is 248 years.