Question

How many half-lives would it take for one mole of a substance to be reduced to 6.25% of the original amount?

We know that one mole is equal to 6.02 * 10^23 and that 6.25% of that (the remaining amount) is .376*10^23

Answer 1

In radioactive decay, number of molecules remaining after time 't' is given by:

N = N0*exp(-*t)

= decay constant = (ln 2)/T_half

Now we need time when amount remaining is 6.25% of the original amount, So

N = 6.25% of N0 = 0.0625*N0

So,

0.0625*N0 = N0*exp(-(ln 2)*t/T_half)

0.0625 = exp(-(ln 2)*t/T_half)

take logarithm on both sides

ln 0.0625 = -ln (2)*t/T_half

t = T_half*(-ln (0.0625)/ln 2)

t = T_half*4

t = 4 half-lives

OR

from 100 to 50% --> 1 half-life

from 50 to 25% --> 1 half-life

from 25 to 12.5% --> 1 half-life

from 12.5 to 6.25% --> 1 half-life

So total half-lives required = 4

Let me know if you've any query.