Question

A patient is given 0.050 mg of technetium-99 m (where m means metastable—an unstable but long-lived state), a radioactive isotope with a half-life of about 6.0 hours.
How long until the radioactive isotope decays to 1.3×10−2 mg ?

Answer 1

The correct answer is 9.6h.

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.

Using the formula , A = Ao. $1/2^n$

where , A- final mass after decay

Ao - initial mass

n - the number of half-lives that pass in the given period of time

Now, putting all the values, we get

1.3 × $10^-2$ mg = 0.050 mg × $1/ 2^n$

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

㏑$(1.3 . 10^{-2} / 0.050 )$ = ㏑$((1/2)^{n})$

㏑$(1.3 . 10^{-2} / 0.050 )$ = n. ln$(1/2)$

n = 1.6

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

$n = t / t _{1/2}$

t= 1.6 × 6 h

t = 9.6h
Hence, it will take 9.6 h  until the radioactive isotope decays.

Learn more about radioactive isotope and half life here:-

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