Nuclear Half Life: Calculations

Summary

This 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

00:00

🔬 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.

05:00

🕒 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.

Keywords

💡Alpha Decay

Alpha decay is a type of radioactive decay in which an atomic nucleus emits an alpha particle (consisting of two protons and two neutrons) and transforms into a different element. In the video, alpha decay is used to describe how radium transforms into radon, with the process having a half-life of 11 days.

💡Half-life

The half-life of a radioactive substance is the time it takes for half of the substance to decay. It's a fundamental concept in nuclear physics and is used throughout the video to calculate the remaining amount of a substance after a certain period. For example, the half-life of radium is given as 11 days.

💡Radium

Radium is a radioactive chemical element that undergoes alpha decay to form radon. In the video, radium is used as an example to demonstrate how to calculate the remaining amount after a certain number of half-lives.

💡Radon

Radon is a radioactive gas that is produced from the decay of radium. It is mentioned in the video as the product of radium's alpha decay.

💡Tritium

Tritium, also known as hydrogen-3, is a radioactive isotope of hydrogen that undergoes beta decay to form helium-3. The video uses tritium to explain how to calculate the time taken for a certain amount of substance to decay to a specific amount.

💡Beta Decay

Beta decay is a type of radioactive decay in which a beta particle (an electron or a positron) is emitted from an atomic nucleus, changing the atom into a different element. The video discusses beta decay in the context of tritium transforming into helium-3.

💡Helium-3

Helium-3 is an isotope of helium produced by the beta decay of tritium. It is mentioned in the video as the result of tritium's radioactive decay process.

💡Percent

Percent is used in the video to express the remaining amount of a substance in terms of percentage after a certain number of half-lives. It helps to understand the decay process in a relative sense, such as calculating 50%, 25%, etc., of the original amount.

💡Fraction

A fraction is used in the video to represent the proportion of the original amount of a substance that remains after each half-life. It is used to calculate the exact amount of substance left after several half-lives, such as 1/2, 1/4, 1/8, etc.

💡Thallium

Thallium is a chemical element that undergoes beta decay to form lead. In the video, thallium is used to illustrate how to calculate the half-life of a decay process when given the initial and final amounts and the total time.

💡Lead

Lead is a chemical element that can be produced from the beta decay of thallium. It is mentioned in the video as the product of thallium's decay.

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