Question

A certain radioactive isotope has a half-life of approximately 1150 years. How

many years would be required for a given amount of this isotope to decay to
25% of that amount?

Answer 1

The amount of a radioactive isotope remaining after a certain amount of time can be modeled by the exponential decay equation:

N(t) = N0 * (1/2)^(t/T)

where:
N0 = the initial amount of the isotope
N(t) = the amount of the isotope remaining after time t
T = the half-life of the isotope

To find the time required for a given amount of the isotope to decay to 25% of that amount, we can set N(t) equal to 0.25N0 and solve for t:

0.25N0 = N0 * (1/2)^(t/T)

Taking the natural logarithm of both sides and solving for t, we get:

t = (ln 0.25) * T / (ln 2)

Substituting T = 1150 years, we get:

t = (ln 0.25) * 1150 / (ln 2) ≈ 287.5 years

Therefore, it would take approximately 287.5 years for a given amount of this isotope to decay to 25% of that amount.

Answer 2

If the isotope has a half-life of 1150 years, this means that every 1150 years the amount of the isotope is halved. After one half-life, the amount is reduced to 1/2, after two half-lives it is reduced to 1/4, after three half-lives it is reduced to 1/8, and so on.

To determine how many years are required for the isotope to decay to 25% of its original amount, we need to determine how many half-lives it takes to get from 100% to 25%.

25% is the same as 1/4, so we need to determine how many times we need to halve the original amount to get to 1/4.

1/4 = (1/2)^n, where n is the number of half-lives

Solving for n:

n = log(1/4) / log(1/2)

n = 2

This means that it takes two half-lives for the isotope to decay to 25% of its original amount.

Since the half-life is approximately 1150 years, the time required for two half-lives is approximately:

2 x 1150 years = 2300 years

Therefore, it would take approximately 2300 years for a given amount of this isotope to decay to 25% of that amount.