The half-life of a radioactive isotope is the time it takes for a quantity of the isotope to be reduced to half its initial mass. Starting with grams of a radioactive isotope, how much will be left after half-lives

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The half-life of a radioactive isotope is the time it takes for a quantity for the isotope to be reduced to half its initial mass. Starting with 150 grams of a radioactive isotope, how much will be left after 6 half-lives

Explanation:

Let analyse the question generally first,

The the mass of the radioactive element be M.

We want to know it mass after n half life

Then,

After first half life, it mass is

M1=M*½

After second half life, it mass is

M2 = M * (½) ²

After third half life, it mass is

M3 = M * (½) ³

But now we can see a pattern developing, because for each new half-life we are dividing the quantity by 2 to a power that increases as the number of half-lives.

Then we can take the original quantity and quickly compute for

nth half-lives:

So after nth half life will be

Mn = M * (½) ⁿ

Generally,

Now, let apply it to our questions

Give that the mass of the radioactive isotope is 150grams

It mass after 6th half life

Then, n=6

So applying the formula

Mn = M * (½) ⁿ

M6 = 150 * (½) ^6

M6 = 150*1/64

M6=2.34grams

The mass of the radioactive isotope after 6th half life is 2.34grams