Question

How long does it take for the radioactive isotope to decay to 5.4×10⁻³μg ? (Assume no excretion of the nuclide from the body.)

Answer 1

The more information needed about the half life of radioactive isotope with initial quantity.

So I assume the information needed is below.

0.050 μg of a radioactive isotope with a half-life of about 6.0 hours.

Explanation:

As you know, a radioactive isotope's nuclear half-life tells you exactly how much time must pass in order for an initial sample of this isotope to be halved.

The equation that establishes a relationship between the initial mass of a radioactive isotope, the mass that remains undecayed after a given period of time, and the isotope's half-life looks like this

A=A₀⋅(1/2)ⁿ , where

A - the initial mass of the sample
A0 - the mass remaining after a given time
n - the number of half-lives that pass in the given period of time

the initial amount of radioactive isotope is said to be equal to 0.050 μg. find the amount of time needed to reduce this initial sample to 5.4*10^-3 μg.

Use the above equation to find the value of n

5.4*10^-3μg=0.050μg * (1/2)ⁿ

This is equivalent to

0.1080=1/2ⁿ

Take the natural log of both sides of the equation to get

ln(0.1080)=n⋅ln(1/2)

n=ln(0.1080)/ln(1/2)

n = 3.21

Since n represens the number of half-lives that pass in a given period of time, you can say that

n=t/(t1/2) ⇒t=n*t1/2

t1/2=6.0 h, which means that

t=3.21*6.0 h=19.26 h

Whatever the initial amount of radioactive decay you have been given, the process to solve the problem will remain same.