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>> In this video,
we're going to get
a closer look at half-life.
We learned that the time
a radioactive isotope takes
to decay to half its original
amount is called its half-life.
If we counted the time
it took for 100 grams of
a radioactive isotope
to decay to 50 grams,
that would be the half-life
of that particular isotope.
So half-life is a time,
measured in seconds,
and it's different for each
decaying isotope.
The half-life of carbon 14
is 1.8 times 10
to the eleventh seconds.
About 5,730 years.
While carbon 22 has a half-life
of only 6.2 milliseconds.
Let's get a bit of practice
with this idea of half-life.
Say we have 1,000 grams
of Dubnium-262, which has
a half-life of 34 seconds.
Take a look at this table
that shows how much of
the Dubnium-262 remains,
and how much time has passed,
and the number of
half-lives experienced.
Take a minute, press pause,
and try filling out
the empty boxes in the table.
Okay, how did it go?
If your table looks like this,
great.
If not, let's take a look
at the solution together.
One thing we saw earlier
is there was an equation
that could help us fill out
a table like this.
Remember that if we know
how long a certain isotope's
half-life is and how much
total time has passed,
we can figure out how many
half-lives have happened.
The number of half-lives
elapsed equals the total
time passed, divided by
the isotope's half-life.
We can use this equation
to figure out how much
of an original isotope remains
after some amount of time
has passed.
If we start with some initial
amounts of radioactive isotope,
call it m initial,
the mass that's left over,
call it m remaining,
at any given time
equals the original mass
times one half
raised to the number
of half-lives completed.
Let's try using these
relationships to help us
answer questions about
atomic half-life.
You have 75 grams
of Beryllium-14.
A radioactive isotope with
a half-life of five seconds.
How many grams of Beryllium-14
remain after 4.5 half-lives
have elapsed?
Our equation involving
initial and final mass
is a great start point
for our solution.
We're given the initial mass
of the sample and the number
of half-lives it experiences.
We're solving for
the mass remaining.
Let's plug in the initial mass,
75 grams, and the number of
half-lives, 4.5.
When we plug our given numbers
into the equation,
we see the mass of 75 grams
Beryllium-14 leftover after
4.5 half-lives is 3.3 grams.
Let's try another question about
radioactive isotope half-life.
As part of a radioactive
chemical identification project,
you are given a 200 milligram
sample of an unknown element.
After 14 days, the sample has
decreased to 7.5 milligrams.
Along with the sample,
you are provided a chart showing
radioactive isotope decay rates,
and are told the sample is made
of one of the elements listed.
What isotope is
the sample made of?
To answer this question,
we'll need to figure out,
what is the half-life of
the unknown sample?
Let's start with
our relationship between
the starting and ending
sample mass.
The sample mass remaining
equals the mass at the start
times one half, raised to the
number of half-lives passed.
We want to solve for
the number of half-lives.
We can rearrange this equation
to isolate one half to
the number of half-lives
on one side
by dividing by m initial.
Since the number of half-lives
is the exponent, we'll need
to use a special math function
called the logarithm to find it.
Mathematically, logarithms
and exponents go together.
If I take the logarithm,
or log, of both sides,
by the property
of the logarithm,
the right side becomes equal
to the number of half-lives
times the log of one-half.
Now I'll divide both sides
of the equation by the log
of one-half, which leaves us
with this equation.
The number of half-lives equals
the log of the mass remaining,
divided by the initial mass,
all divided by the log
of one half.
We plug in the final mass
of 7.5 milligrams,
and the initial mass
of 200 milligrams,
all divided by the log
of one half.
Then we use a calculator
to compute the final answer,
and find that the number
of half-lives elapsed
is equal to 4.73.
The next thing we want to know
is how long each one of these
4.73 half-lives is.
To figure that out,
we can use our equation relating
time and half-lives passed.
The number of half-lives
equals the time passed
divided by the time
for each half-life.
We multiply both sides
of the equation by the time
of each half-life, t sub HL,
and divide both sides by
the number of half-lives passed,
number HL.
We find that each half-life
is equal to the time passed,
divided by the number
of half-lives.
When we plug in our numbers,
we find that the time passed,
which is 14 days, divided by
the number of half-lives,
which is 4.73, gives us
a total time for the half-life
of 2.96 days.
That's about 2.6 times 10
to the fifth seconds.
Now we know how long
our half-life of
our unknown sample is.
So, what isotope is it?
Looking at the given table
of isotopes, we see a match
between the half-life
we calculated and Fermium-253.
Based on its half-life,
the mystery sample is made
of the radioactive isotope,
Fermium-253.
These radiation events
that change an atom's identity
are called transmutations.
Basically, these nuclear
reactions transform one element
into another.
In order for the identity
of an atom to change,
the number of protons
must change.
Here are two
transmutation examples.
In the first example,
Radium-226 turns into Radon-222,
plus an alpha particle.
During this transmutation,
the atomic number decreases
by two, and the mass number
decreases by four.
We know Radon is formed
because it is element number 86
on the periodic table.
In the second example,
a transmutation occurs
involving beta minus radiation.
Here, a neutron changes into
a proton and emits an electron.
Even though the mass number
doesn't change,
the atomic number
increases by one.
In this example,
when a beta particle
is released,
Carbon-14 becomes Nitrogen-14.
That's a closer look
at half-life.
For more "Physics in Motion",
go to our homepage,
where the entire series
is available to you.
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