Differential equation radioactive decay. Does this problem make any sense?

[studentMCCS]

I'm going over our sample exam, and the following is one the problems:

Suppose that 10 grams of plutonium Pu-239 was released. The half-life of Pu-239 is 24,100 years. How long will it take for the 10 grams to decay to 1 gram?

The answer is that 10 grams of Pu-239 will never decay into 1 gram right? Isn't half life the amount of time it takes for half of the atoms in a sample to decay, not to an atom half the weight, but just to a different isotope?

 

[wjm11]

Suppose that 10 grams of plutonium Pu-239 was released. The half-life of Pu-239 is 24,100 years. How long will it take for the 10 grams to decay to 1 gram?

The answer is that 10 grams of Pu-239 will never decay into 1 gram right? Isn't half life the amount of time it takes for half of the atoms in a sample to decay, not to an atom half the weight, but just to a different isotope?

I do not follow your logic. Any radioactive substance will decay over time into more stable isotopes of other elements, eventually leaving none of the original sustance (although the mathematical model never reaches the zero point). Why would you think that Pu-239 would not decay to less than 10 grams no matter how much you start with?

After the first 24,100 years, half of the starting amount (10 gm) is gone, leaving 5 gm. After another half life of 24,100 years, half of the 5 gm is gone, leaving only 2.5 gm. And so it continues...

Most of the original 10 gm mass is retained in the form of other elements (some is lost in the form of radiation, ala E = mc^2), but the mass of Pu-239 decreases continuously.

 

[pappus]

I'm going over our sample exam, and the following is one the problems:

Suppose that 10 grams of plutonium Pu-239 was released. The half-life of Pu-239 is 24,100 years. How long will it take for the 10 grams of plutonium Pu-239 to decay to 1 gram of plutonium Pu-239?

The answer is that 10 grams of Pu-239 will never decay into 1 gram right? Isn't half life the amount of time it takes for half of the atoms in a sample to decay, not to an atom half the weight, but just to a different isotope?

I assume that you see more problems than there actually are:

1. To avoid any philosophical discussions I've added the necessary terms to clarify what you are asked to do.

2. This looks to me like a simple exponential problem:
Let A(t) denotes the amount at the time t, A(0) the amount at the start and k a spcific constant then you can express A(t) as:

\(\displaystyle \displaystyle{A(t)=A(0) \cdot e^{k \cdot t}}\)

3. You already know:

\(\displaystyle \displaystyle{5=10 \cdot e^{k \cdot 24100}}\) .......... Determine k

4. Now solve for t:

\(\displaystyle \displaystyle{1=10 \cdot e^{k \cdot t}}\) .......... Don't forget: You know the value of k.

 

[HallsofIvy]

I'm going over our sample exam, and the following is one the problems:

Suppose that 10 grams of plutonium Pu-239 was released. The half-life of Pu-239 is 24,100 years. How long will it take for the 10 grams to decay to 1 gram?

The answer is that 10 grams of Pu-239 will never decay into 1 gram right? Isn't half life the amount of time it takes for half of the atoms in a sample to decay, not to an atom half the weight, but just to a different isotope?

Yes, Plutionum decays into another atom. But not all the Plutonium decays at the same time. The question is asking how long it will take all of the Plutonium except for 1 gram to have decayed.

Saying "the half life of Pu-239 is 24,100 years" means that 1/2 of all the Plutonium will have decayed in 24,100 years- if the original amount of Plutonium is P, after 24,100 years, you will have P/2 left. In another 24,100 years, you will have \(\displaystyle (1/2)^2P= P/4\) left. In general, after N periods of 24,100 years, you will have \(\displaystyle (1/2)^NP\) left.

In this problem you start with P= 10 so first you need to find N such that \(\displaystyle 10(1/2)^N= 1\). N will not be an integer and you will need to use a logarithm to solve that. After you have found N, which counts the number of 24,100 year time periods. Multiply N by 24,100 to find the number of years.

 

[studentMCCS]

I do not follow your logic. Any radioactive substance will decay over time into more stable isotopes of other elements, eventually leaving none of the original sustance (although the mathematical model never reaches the zero point). Why would you think that Pu-239 would not decay to less than 10 grams no matter how much you start with?

After the first 24,100 years, half of the starting amount (10 gm) is gone, leaving 5 gm. After another half life of 24,100 years, half of the 5 gm is gone, leaving only 2.5 gm. And so it continues...

Most of the original 10 gm mass is retained in the form of other elements (some is lost in the form of radiation, ala E = mc^2), but the mass of Pu-239 decreases continuously.

Because even though it decays into another substance, it will never be 1 gram. I guess I'm thrown off by the language. It seams to me that literally the problem is incorrect. It should say, until there is only 1 gram of Pu-239 left.

If it decays into another isotope, and it also into decays nothing, as you have restated, there is a logical contradiction. The problem is just written incorrectly and had me assuming the teacher didn't know what a half life is.

 

[HallsofIvy]

Because even though it decays into another substance, it will never be 1 gram. I guess I'm thrown off by the language. It seams to me that literally the problem is incorrect. It should say, until there is only 1 gram of Pu-239 left.

?? That is what it says! It is not relevant what the Pu-239 decays into- it decays into something which is NOT Pu-239, so there is less and less Pu-239 as time goes by. Eventually, at some time there will be only 1 gram left.

If it decays into another isotope, and it also into decays nothing, as you have restated,

I certainly did not say "decays into nothing". I don't know what that could mean. It just decays into something that is not Pu-239.

there is a logical contradiction. The problem is just written incorrectly and had me assuming the teacher didn't know what a half life is.

No, the problem is written correctly.

 

[studentMCCS]

?? That is what it says! It is not relevant what the Pu-239 decays into- it decays into something which is NOT Pu-239, so there is less and less Pu-239 as time goes by. Eventually, at some time there will be only 1 gram left.


I certainly did not say "decays into nothing". I don't know what that could mean. It just decays into something that is not Pu-239.

No, the problem is written correctly.

But the problem says: How long will it take for 10 grams of plutonium 239 to decay into one gram. So how can something weighing 10 grams decay into another isotope, and into 1 gram. Logically, there is a contradiction with the language.

I understand it is just something where people know what your talking about anyways.

The problem was just an issue with the use of the words "decays into", using two different definitions, in one paragraph. I understand what is meant, that 90 percent of a sample decays into another isotope, leaving 1 gram of Pu 239 left.

If you think about it, it's just wrong on more than one level. ..."for ten grams of Pu 239 to decay into 1 gram." Firstly, this implies that all ten grams decayed. Secondly, it says decays into 1 gram. The one gram is what has not decayed. So Ten grams of stuff decays into 1 gram of non decayed stuff?

It's kind of like saying Pu -239 decays into Pu -239.

 

[studentMCCS]

Sorry if I have been annoying on this issue. Thanks for your replies. I don't mean to be rude, I just thought that the question didn't make sense the way it was worded. I know I shouldn't be bothered by it. I just think it is sort of misleading, especially for people who might not know anything about radio-active decay.