Question

What is the half‑life of an isotope that decays to 12.5% of its original activity in 64.9 h?

Answer 1

The half life can be obtained as 21.6 hr.

In the context of radioactive substances, the half-life is the time it takes for half of the radioactive atoms in a sample to decay into a more stable form.

The half-life of a substance is the time required for half of a quantity of that substance to undergo a specific process, such as decay or transformation.

We have that;

N/No =
`(1/2)^{t/t1/2`

N = amount at time t

No = Initial amount

t = time taken

t1/2 - half life

N= 0.125No

0.125No/No =
`(1/2)^{64.9/t1/2`

ln0.125 = 64.9/t1/2ln 0.5

64.9/t1/2 = ln0.125/ln 0.5

64.9/t1/2 = 3

t1/2 = 64.9/3

= 21.6 hr

Answer 2

The half-life of an isotope is the time taken for half of the original activity to decay.

We know that the isotope decays to 12.5% of its original activity. So, the fraction of the original activity remaining is:

0.125 = 1/2^3

This means that the isotope has undergone three half-lives.

The time taken for three half-lives is:

3 x half-life = 3 x 64.9 h = 194.7 h

Therefore, the half-life of the isotope is:

time/half-life = number of half-lives

64.9 h/half-life = 1

194.7 h/half-life = 3

Solving for half-life:

64.9 h/half-life = 1

half-life = 64.9 h

Therefore, the half-life of the isotope is 64.9 hours.