Chem143 Half Lives of Radioactive Nuclei

Half-lives of radioactive nuclei. What a half-life is and various calculations involving half-lives.

>> In this video, we're
talking about half-lives

of radioactive nuclei.

Well, the first thing
is, what is a half-life?

Well, it's just how much time
it takes for 1/2 of a sample

of any radioactive nucleus to
decay to where there's only 1/2

of the amount that
we started with left.

And one nice thing is
that the half-lifes

of any given isotope
is constant.

It doesn't matter how much of
that stuff we have at any time,

it takes -- whatever
that half-life is,

that amount of time
for it to decay

to half the original amount.

So let's say we have
iodine-131, 20 mg.

After 1 half-life, we'll
have half of that 20,

which is 10 left, 10 mg.

After another half-life,
2 half-lives,

we'll have half the
10, which is 5 mg left.

And after the third half-life,

we'll have of 5,
which is 2.5 mg.

And whatever this time is,
whatever the half-life is,

it's the same from here to here,
here to here, here to here.

It stays constant
for that isotope.

Of course, different
isotopes have vastly different

half-lives, from billions of
years to fractions of a second.

So let's look at some examples.

And as we look at
these examples,

it's going to be
important to figure

out what we're being given
and what we're being asked

because that's going

to determine how you
set the problem up.

In this problem, we're
told that we have carbon-15

and we're told what its
half-life is, 2.4 seconds.

What we want to find is
how much of a given amount

of the sample is left after
a given amount of time.

So we're told what the
half-life is, how much we start

with (240 g), and how long
-- how much time has passed,

12 seconds have gone by.

So this is going to
be a 2-step problem.

First step's always going to
be, in a problem like this

where we want to know how
much of this stuff is left,

is we want to find out how
many half-lives have gone by.

All you do to do that is take
the amount of time that's passed

and divide by the half-life.

So 12 seconds have gone
by, divided by 2.4 seconds

for each half-life, and that
tells us 5 half-lives have

gone by.

Now that we know that, we can
approach it one of two ways.

Either, we can take how
much we started with

and cut in half, 5 times.

So 240 divided by 2 is
120, that's 1 half-life --

divided by 2 again, that's 2
half-lives is 60 -- 30, 15.

And after 5 half-lives, we
see that we have 7.5 g left

and that's the answer.

Another way to do that
-- and this always works.

Again, when the problem is
we want to find out how much

of the sample is left and you're
told what the half-life is

and how much time has gone

by so you can figure how
many half-lives there are

and how much you start with,
that was 240 g, times 1/2 raised

to the power of however many
half-lives have gone by.

In this case, 5 --
the power of 5 here is

because 5 half-lives
have gone by.

And it's always 1/2 here though.

So if you plus this
into your calculator,

you'll get the answer --

7.5 g. You can do this
type of problem either way,

whatever is easier for you.

The other type of
problem we can --

another type of problem we
can do this is we can figure

out how long it takes
for a given sample

to decay to a certain amount.

So we have cesium-137,
this isotope,

and we're told what its
half-life is, 30 years.

We want to know how long it
takes for a 60 g sample to decay

to 7.5 g. So we're told
what the half-life is.

We're told how much we start
with, how much we end up with,

and we want to find
out how long it takes.

So, once more, the first
thing, we're going to want

to do is find out how many
half-lives have gone by.

So if we take how much we
start with, cut it in half,

keep cutting it in
half until we get

to the amount we're told we have
-- so 60 divided by 30 is --

I mean, excuse me -- 60 divided
by 2 is 30, that's 1 half-life,

30 divided by 2 is 15,
that's 2 half-lives.

And 15 divided by 2 is 7.5.

So 3 half-lives have gone by.

So now that we know how many
half-lives have gone by,

we want to find out how
much time has gone by.

Well, we know how much time goes

by for each half-life
-- 30 years, right?

So we just take the number

of half-lives times
the half-life itself --

in other words, the
time per 1 half-life.

And that tells us
that, in this case,

90 years will have gone by.

So another type of
problem we can do

with half-lives is like this.

So we have 1.2 mg
of phosphorous-32.

And we know that it
decays to .30 mg --

1.2 mg to .30 mg in 28.6 days.

What's the half-life?

So here, we're told how much
we start with, how much we end

up with, how long has
gone by, and we're trying

to find the half-life.

So, once more, we figure out how
many half-lives have gone by.

So we start with 1.2.

Cut it in half.

That gives us .60,
that's 1 half-life.

Cut that in half, .30.

That's 2 half-lives.

That's how much we have.

So we know that 2
half-lives have gone by.

We also know that the
total time is 28.6 days.

So if we just take the
total time, 28.6 days,

divided by the number of
half-lives, 2 in this case,

that gives us the
time per half-life,

14.3 days per half-life, or
we say that the half-life

for t 1/2 is equal to 14.3 days.