# Physics question about half life / radioactive decay

• RRmy0440
In summary, the problem involves calculating the number of half-lives an unstable element has undergone based on the amount of stable daughter product present. The equation used is N=No (1/2)^(t/(t1/2)), where N represents the final activity, No is the original activity, t is the time, and t1/2 is the half-life. The key concept to understand is that as the unstable element decays, its mass decreases while the mass of the stable daughter product increases. Therefore, the mass of the daughter product can be used to determine the original mass of the unstable element.

## Homework Statement

You have 0.0625 grams of an unstable element and 0.9375 grams of the stable daughter product. How many half-lives has it undergone？

## Homework Equations

N=No (1/2)^(t/(t1/2))
In which
N represents the final activity for a period of time
No is the original activity
t represents the time
t1/2 represents the half life

(One could also be written as

(log N/No)÷(log1/2)＝t/(t1/2)

## The Attempt at a Solution

I was rather confused about this question from the very beginning, I was wondering why is there more mass of the stable daughter product than the mass of this unstable element. Shouldn 't it always be that the daughter product have lesser mass than the element originally have, since it was undergoing a radioactive decay？

The correct answer for this question is 4 half lives.

One of my friend told me that if he add this two variables together, and states the result as real original mass of this unstable element,which turns into:
No=0.9375+0.0625=1g
Then applied the equation on the above, he then got the answer. But it does not make sense to me.
Can anyone help me with this question and explain the principle behind that please？I would be super grateful if you do that.

The mass is not the mass of one nucleus of the element, it is the total mass in the sample of the element.

If all of the unstable element had decayed, then there would only be the daughter element left and it would have all the mass.

RRmy0440 said:
I was wondering why is there more mass of the stable daughter product than the mass of this unstable element. Shouldn 't it always be that the daughter product have lesser mass than the element originally have, since it was undergoing a radioactive decay？

No. Suppose you had 5 radioactive atoms. Now 3 of them decay, so you have 2 radioactive and 3 daughters.
Are you saying for some reason that you expect even after 3 of them decay, there are still more original than daughters? Why?
Now suppose all 5 of the original atoms decay. 0 original atoms left. 5 daughters. Are you saying that you STILL think you have more original than daughters? Why?

As the radioactive atom decays, their mass decreases. The mass of daughters increases. Sooner or later the second one is bigger than the first one.

RRmy0440 said:
One of my friend told me that if he add this two variables together, and states the result as real original mass of this unstable element,

So it appears the assumption is that the 0.9375 g of daughter came from 0.9375 g of unstable atom. The mass lost was negligible.

If you saw 2 unstable atoms and 3 of the stable decay product and wanted to know how many unstable atoms you started with, you would say "these 3 daughter atoms came from 3 unstable atoms. So I started with 3 more than I have now. There were 5."

You see 0.9375 g of daughter. You assume that came from 0.9375 g of unstable atom that decayed. So there was originally 0.9375 g more than there is now.

Orodruin said:
The mass is not the mass of one nucleus of the element, it is the total mass in the sample of the element.

If all of the unstable element had decayed, then there would only be the daughter element left and it would have all the mass.
Thank you very much, I got it now.

RPinPA said:
No. Suppose you had 5 radioactive atoms. Now 3 of them decay, so you have 2 radioactive and 3 daughters.
Are you saying for some reason that you expect even after 3 of them decay, there are still more original than daughters? Why?
Now suppose all 5 of the original atoms decay. 0 original atoms left. 5 daughters. Are you saying that you STILL think you have more original than daughters? Why?

As the radioactive atom decays, their mass decreases. The mass of daughters increases. Sooner or later the second one is bigger than the first one.
So it appears the assumption is that the 0.9375 g of daughter came from 0.9375 g of unstable atom. The mass lost was negligible.

If you saw 2 unstable atoms and 3 of the stable decay product and wanted to know how many unstable atoms you started with, you would say "these 3 daughter atoms came from 3 unstable atoms. So I started with 3 more than I have now. There were 5."

You see 0.9375 g of daughter. You assume that came from 0.9375 g of unstable atom that decayed. So there was originally 0.9375 g more than there is now.
Your reply is pretty helpful to me, thank you so much.

## What is half life?

Half life is the amount of time it takes for half of a radioactive substance to decay into a stable form.

## What is radioactive decay?

Radioactive decay is the process by which an unstable atom releases energy and particles to become a more stable element.

## How is half life calculated?

Half life is calculated using the formula t(1/2) = ln(2)/λ, where t(1/2) is the half life, ln(2) is the natural logarithm of 2, and λ is the decay constant.

## What factors affect the half life of a substance?

The half life of a substance can be affected by various factors such as temperature, pressure, and the chemical environment. It is also specific to each type of radioactive substance.

## Why is half life important in nuclear medicine?

Half life is important in nuclear medicine because it allows for accurate dosing and timing of radiation treatments. It also helps determine the decay rate of radioactive isotopes used in medical imaging and diagnostic procedures.