okay let's check out a problem regarding radioactive half-life cobalt-60 decays
with a half-life of 5.2 7 years to produce nickel 60 so the first question
is what is the decay constant for this disintegration and in order to answer
that we will need this expression and the second question is what fraction of
a sample of cobalt 60 nuclei will remain after 15.0 years and to answer that we
will need this expression so that's all the information we need go ahead and
give this a try.
so let's put our value for half-life up at the top that is 5.2
seven years and we have this expression where the half-life is equal to natural
log of 2 over the decay constant so if we want to get that decay constant let's
solve for it by bringing the decay constant up to the left side and then
bringing the half-life down to the denominator of the right side and then
we can plug in what we know so the natural log of 2 is 0.693 and the
half-life is 5.2 seven years and that will give us a decay constant of 0.1 3 2
years to the minus 1 now we want to find out some information about how many
nuclei will remain after a certain period of time and for that we will need
this expression so what are we going to do with this expression we have NT
equals N0 e to the negative lambda T and so we have the number of nuclei that
remain at some time T equally the number of nuclei that were initially there
right and N naught or N0 implies some initial condition and then we have e
raised to the negative decay constant times
time so what we can do is we can divide both sides by n zero and we'll get NT
over n0 on the left and that is important because that represents the
fraction left after time T right we're not asked for a specific number or a
specific amount we're asked for a relative amount we just want to know a
fraction or a ratio and so Nt over N0 is the amount that is left after time T
divided by the initial amount and that will be the fraction left after time T
so let's just plug in what we know we can plug in our decay constant and then
for T let's plug in 15.0 years because that's what the question asked we wanted
to know the fraction left after 15 years had elapsed so we just do the arithmetic
that is going to be equal to e to the negative one point nine eight and that
will be equal to 0.138 which expressed as a percentage will be thirteen point
eight percent so after fifteen years which is just a little bit under three
half-lives we will have thirteen point eight percent of the radioactive nuclei remaining.