Self Assessment Exercise 21.11 error

[Zellmer, Robert]

Someone asked about SA Exercise 21.11.  It asks the following:

"It takes 45 hours for a 6.00 mg sample of sodium-24 to decay to
0.750 mg.  What is the half-life of sodium-24?"

There's a couple of ways to do this:

1)  You can use the 1st-order integrated rate eqn to find k (the decay constant,
i.e. the rate constant) and then use the half-life eqn to get t_1/2.

ln([N]t/[No]) = -k*t

k = - ln([N]t/[No])/t = - ln(0.750/6.00)/45 hr = 4.621 x 10^-2 hr^-1

t_1/2 = 0.693/(4.621 x 10^-2 hr^-1) = 14.996 hr = 15 hr

2) This way is maybe a little easier but harder to see right off.

Take the ratio of the final mass to initial mass:

0.750/6.00 = 0.125

That's 12.5%.  That's one of the "magic" numbers.  To get to 12.5% it
takes 3 half-lives,

100% --> 50% --> 25% --> 12.5% --> 6.25% --> 3.125% etc.

Thus, since the ratio of the masses (concentrations) is 0.125 (12.5%)
that signifies 3 half-lives.

The half-life for 1st-order processes is constant.  That means if the
total time is 45 hours and that represents 3 half-lives the half-life
is 15 hours.

Either way that's not one of the possible answers.  I've reported
it to Pearson.

Dr. Zellmer
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