Nuclear Half Life: Calculations
Summary
TLDRThis educational script explains the concept of half-life in radioactive decay using examples of radium to radon, tritium to helium-3, and thallium to lead. It covers calculations for remaining mass, percentage, and fraction after specific half-lives, emphasizing a step-by-step approach. The script also introduces the process of determining half-life duration when given initial and final masses over a known time.
Takeaways
- 📚 The half-life of radium is 11 days, and after 44 days (4 half-lives), 7.5 grams of a 120-gram sample would remain.
- 🔢 After 44 days, 6.25% of the original radium sample would be left.
- 🔄 The fraction remaining after 44 days of radium's half-life is 1/16, which is equivalent to 6.25%.
- 🕒 Tritium has a half-life of 12.3 years, and it takes 61.5 years for an 80-gram sample to decay to 2.5 grams.
- 📉 If 3.125% of tritium remains, it also takes 61.5 years for the decay, indicating 5 half-lives have passed.
- 🔢 To find the half-life of thallium-207, which decays to lead, start with 200 grams and observe it takes 20 minutes to reduce to 12.5 grams, indicating a half-life of 5 minutes.
- 📊 The process of decay can be visualized through charts showing the amount of substance left after each half-life.
- 🧮 Half-life problems can be solved using simple division for time or multiplication for fractions, but more complex problems may require exponents and logarithms.
- ⏳ The concept of half-life is crucial for understanding radioactive decay and can be applied to various isotopes like radium, tritium, and thallium.
- 🔬 Radioactive decay problems can be approached by considering the physical amount, percentage, or fraction remaining after a given number of half-lives.
Q & A
What is the half-life of radium-222 decaying into radon?
-The half-life of radium-222 decaying into radon is 11 days.
If you start with 120 grams of radium, how much will be left after 44 days?
-After 44 days, which is equivalent to 4 half-lives, 7.5 grams of radium will be left.
How can you calculate the percentage of a substance left after a certain number of half-lives?
-You start at 100% and reduce it by half for each half-life that passes. After 4 half-lives of radium-222, 6.25% of the original amount will be left.
What fraction of the original amount of radium-222 will be left after 44 days?
-After 4 half-lives, 1/16 of the original amount of radium-222 will be left.
How long does it take for 80 grams of tritium to decay to 2.5 grams?
-It takes 61.5 years for 80 grams of tritium to decay to 2.5 grams, considering each half-life is 12.3 years.
What is the half-life of tritium decaying into helium-3?
-The half-life of tritium decaying into helium-3 is 12.3 years.
If you start with 200 grams of thallium-207, how long does it take for it to decay to 12.5 grams?
-It takes 20 minutes for 200 grams of thallium-207 to decay to 12.5 grams.
What is the half-life of thallium-207 decaying into lead?
-The half-life of thallium-207 decaying into lead is 5 minutes.
How can you determine the half-life of a substance if you know the initial and final amounts and the time taken for the decay?
-You can determine the half-life by dividing the total time taken by the number of half-lives that occurred, as shown with thallium-207 where 20 minutes divided by 4 half-lives equals a 5-minute half-life.
What is the relationship between the number of half-lives and the remaining percentage of a substance?
-The remaining percentage of a substance after a certain number of half-lives is the original percentage divided by 2 for each half-life that has passed.
How can you calculate the time it takes for a substance to decay to a certain fraction of its original amount?
-You multiply the half-life of the substance by the number of half-lives needed to reach the desired fraction, as shown with tritium where 12.3 years times 5 half-lives equals 61.5 years.
Outlines
🔬 Radioactive Decay Calculations
This paragraph explains the concept of radioactive decay using the example of radium decaying into radon. The half-life of radium is given as 11 days. Starting with a 120-gram sample, the speaker calculates the remaining amount after 44 days, which is 4 half-lives. They use a step-by-step chart to show the decay process, ending with 7.5 grams of radium left. The explanation also covers how to determine the percentage and fraction of the original amount remaining after a certain number of half-lives. The speaker then introduces a problem involving tritium decaying into helium-3 with a half-life of 12.3 years, starting with an 80-gram sample and ending with 2.5 grams after an unknown number of half-lives. They calculate the time taken by considering the half-lives and the half-life duration, resulting in 61.5 years.
🕒 Determining Half-Life Through Decay
The second paragraph discusses how to determine the half-life of a radioactive decay process when it's not given. Using thallium-207 decaying into lead as an example, the speaker starts with a 200-gram sample and observes that after 20 minutes, only 12.5 grams remain. By creating a chart showing the decay over four half-lives, they calculate the half-life to be 5 minutes. The explanation covers the process of calculating half-life using both the actual amount of substance decayed and the percentage or fraction of the original amount. The speaker also mentions that more complex calculations involving exponents and logarithms will be discussed in the next lesson.
Mindmap
Keywords
💡Alpha Decay
💡Half-life
💡Radium
💡Radon
💡Tritium
💡Beta Decay
💡Helium-3
💡Percent
💡Fraction
💡Thallium
💡Lead
Highlights
Radium undergoes alpha decay to form radon with a half-life of 11 days.
Starting with 120 grams of radium, after 44 days (4 half-lives), 7.5 grams remain.
The percentage of radium left after 44 days is 6.25%.
The fractional amount of radium left after 44 days is 1/16.
Hydrogen-3 (Tritium) undergoes beta decay to form Helium-3 with a half-life of 12.3 years.
An 80-gram sample of tritium decays to 2.5 grams over 5 half-lives, taking 61.5 years.
If 3.125% of tritium remains, it also takes 5 half-lives or 61.5 years.
The process of decaying to 1/32 of the original amount takes 5 half-lives.
Thallium-207 undergoes beta decay to form Lead, but its half-life is unknown.
Starting with 200 grams of Thallium-207, after 20 minutes, only 12.5 grams remain.
The half-life of Thallium-207 decay is calculated to be 5 minutes.
Half-life calculations can be done using simple division or more complex methods involving exponents and logarithms.
The process of decay can be visualized and calculated using charts and tables.
The concept of half-life is crucial for understanding radioactive decay.
The percentage and fractional amounts left after decay can be calculated using simple mathematical operations.
The time it takes for a substance to decay to a certain amount can be determined by understanding half-life.
The examples provided illustrate the application of half-life in radioactive decay calculations.
Transcripts
so here's the equation for radium doing
alpha decay to make radon and the
half-life for this process is 11 days
our question is if you start with a 120
gram sample of radium how much will be
left after 44 days the first thing let's
do is figure out how many half-lives 44
days is going to be okay so one
half-life is 11 days so 44 days is going
to be 4 half lives okay now that we know
it's going to be 4 half lives let's just
make kind of a little chart and figure
out how much we're going to have at each
step so we're starting with 120 grams
after one half-life we'll have half of
that
so we'll have 60 grams okay that's the
first half-life now a second half-life
will go from 60 down to 32 half-lives
three will go to 15 grams and now our
fourth half-life will give us seven
point five grams so seven point five
grams is how much radium will have left
after 44 days or 4 half lives
now a lot of times when people ask
questions about half-lives they want to
know about percent and fractions too so
let's see how we'd answer this if we
were asked um what percentage will be
left after 44 days okay instead of what
amount this is this is really simple
it's actually easier than this okay so
for the percent left we'll do the same
thing but we'll just assume that we
start at a hundred percent okay so a
hundred percent is where we start after
one half-life how much will be left well
half of it which is 50 percent so
there's our first half-life now our
second half-life will go from 50
percents down to 25% our third half-life
will go to 12.5
okay and then for our fourth half-life
will go to 6.25% and this is what
percentage of the starting amount I'd
have left after 44 days or for
half-lives
now finally if you were asked to find
the fraction that was left after 44 days
here's how you do it keep in mind again
this is going to be for half-lives
so after one half-life we have one half
left okay now we multiply that after two
half-lives we lose another half so now
we have 1/2 times 1/2 1/4 left after two
half-lives we're going to lose another
half so now it's 1/2 times 1/2 times 1/2
we have 1/8 left and finally 1/4
half-life we're going to have 1/16 of
the original amount left and if you do
1/16 and turn that into a percent it's
six point two five so that's how you can
solve a problem like this for the actual
amount the percent and for the
fractional amount here's our next
question
hydrogen 3 which is also known as
tritium undergoes beta decay to make
helium 3 and this process has a
half-life of twelve point three years ok
so an 80 gram sample of tritium decays
leaving 2.5 grams of tritium how long
would this take okay let's figure it out
by just making a chart like we did
before so we're starting with 80 grams
of tritium one half-life is going to
give us how much 40 grams okay one
half-life now we're going to do another
half-life now it's down to 20 grams two
half-lives down one more 10 grams
another one 5 grams and finally down to
2.5 grams so that is 1 2 3 4 5
half-lives
okay so five half-lives and how much
does each half-life take each half-life
takes twelve point five years so we're
going to do 12 five sorry twelve point
three years so we're going to do twelve
point three years times five is going to
give us sixty one point five years
that's how long this whole process would
take okay now what if the question
involved percentages asking how much how
long it would take if we were left with
three point one percent I'm just going
to do this really fast so that you can
do it on your own all right
but we'd start with a hundred percent
take that down fifty percent and then
another half-life would give us 25
percent 12.5 percent
six point two five percent and finally
three point one two five percent that's
pretty close to the three point one
they're talking about and it's the same
answer it's one two three four five
half-lives sixty one point five years
now finally what would happen if you
were given this amount as a fraction
asking how long would take to get down
to 1/32 of the original amount we just
multiply one halves together and see how
many we'll need so do 1/2 times 1/2
that's 1/4 1/8 1/16 1/32 each one of
these one half's represent one half-life
so that's how we could get five
half-lives with fractions 5 t1 halves
okay let's do one more here's the
equation for thallium undergoing beta
decay to make lead but we don't know the
the half-life here we're going to have
to figure out what it is
so the question asks us we start with
200 grams of thousand 207 here after 20
minutes
there is only twelve point five grams of
thallium left what is a half-life of the
decay process as usual let's make a
chart that shows how much this is
decaying so
we start with 200 grams one half-life is
going to knock us down to 100 grams now
another half-life will take us down to
50 grams then we'll get down to 25 grams
and finally a fourth half-life we'll
take it down to twelve point five grams
so we have one two three four half-lives
for half lives now it's said that this
whole process to go from 200 grams down
to twelve point five grams takes 20
minutes in in that 20 minutes
there have been four half-lives so to
figure out the length of one half-life
we're just going to do 20 the total time
divided by four half lines which is
going to give us five minutes for the
length of one half-life and as I've
shown you in the previous examples you
could also do this with percentages by
starting at 100% and working your way
down or if the number had to do with
fractions just multiply one half
together for each half life that you
have now the calculations that we've
done for all these problems so far you
could probably do them in your head
pretty well you just take numbers and
cut them in half a bunch of times and do
some relatively simple math but there
are a bunch of half-life problems that
require trickier math that use exponents
and logarithms so we'll now talk about
those in the next lesson
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