[exodus]

Hi,

My teacher had everyone do a "Pennium" lab to help us understand the half-life concept. However, he gave us a set of 3 questions that I can't figure out.

1. Based on Reference Table N, what mass of an original 10.00 gram sample of C-14 will reamin unchanged after a period of 22,920 years?
I understand that C-14 has a halflife of 5730 years and the decay mode is...a positron?  Do I do some sort of equation here?

2. Refer to Reference Table N to answer this problem. A sample of P-32 is found to contain 6.25 grams of the isotope. The sample was originally massed at 100.0 grams. How long has the sample been sitting? The half life is 14.3 days and it's a beta particle. How do you find out how long something has been sitting?
 
3. A patient recieves an injection of I-311 for a thyroid disorder. What period of time will be required for the radioactive emissions of this radioisotope to be reduced to 1/32 of its original potency? Okay the halflife for this is 8.07 days...32 divided by 8 = 4. Is that the right answer?

Can someone help me out with these?
I have attempted to try these problems. Unfortunetly, I'm not so great at this subject.

Thanks so much

[Donaldson Tan]

the time that has been given to you in the question don't really mean a thing, other than to test your understanding of half-life. radioactive substance decay with a constant half-life. eg. if substance A decays radioactively to form substance B, and its half-life is 8days, then if you have 8g of substance A at day 1, then it means by 8th day, your sample would now contain 4g (ie. 1/2 of 8g) of substance A, but the sample you have now is a mixture of A and B. radioactive is a continuous decay phenomena.

are you familiar with calculus? radioactive decays is a first order rate equation, which you can integrate to obtain the decay profile.

radioactivity is directly proportional number of radioactive atoms
let N be number of radioactive atoms
dN/dt = -kN
(1/N)dN/dt = -k
integrate LHS and RHS with respect to time
ln N = -kt + C (constant of integration)
when t = 0, ln N_o = C where N_o is the initial number of radioactive atoms.

therefore, ln N - ln N_o = -kt => ln (N/N_o) = -kt

half-life of a substance (t_1/2) is given by:
N/N_o = 1/2 => - ln2 = -kt_1/2
t_1/2 = (ln 2)/k
k = ln 2 / t_1/2

the decay profile is thus:
ln (N/N_o) = -kt
ln (N/N_o) = -(ln 2 / t_1/2)t

thus by knowing the half life, you can calculate the decay constant (k) and thus obtain the fill decay profile of the substance of interest.

[Garneck]

Hi,

My teacher had everyone do a "Pennium" lab to help us understand the half-life concept. However, he gave us a set of 3 questions that I can't figure out.

1. Based on Reference Table N, what mass of an original 10.00 gram sample of C-14 will reamin unchanged after a period of 22,920 years?
I understand that C-14 has a halflife of 5730 years and the decay mode is...a positron?  Do I do some sort of equation here?

2. Refer to Reference Table N to answer this problem. A sample of P-32 is found to contain 6.25 grams of the isotope. The sample was originally massed at 100.0 grams. How long has the sample been sitting? This question I just don't understand!   My chem teacher never really explained how to find this.

3. A patient recieves an injection of I-311 for a thyroid disorder. What period of time will be required for the radioactive emissions of this radioisotope to be reduced to 1/32 of its original potency? Okay the halflife for this is 8.07 days...I don't understand where the 1/32 comes in!

Can someone help me out with these?

Thanks so much

1.
22920/5730 = 4. So that means C-14 decayed 4 times.
First time: 10/2 = 5
Second time: 5/2 = 2,5
Third time: 2,5/2 = 1,25
Fourth time: 1,25/2 = 0,625

So that means that after 22920 years you will have 0,625 out of the 10 original grams.


2.
6,25*2 = 12,5
12,5*2 = 25
25*2 = 50
50*2 = 100

So that means it has decayed 4 four times. But you don't have the half time for this element. If you find that half-time, multiply it by 4 and you have your answer.


3.
Your isotope is wrong. It's I-131 but you probably pressed the wrong key.

(1/2)^5 = 1/32

So that means it decayed 5 times. 5*8,07 days = 40,35 days

[exodus]

the time that has been given to you in the question don't really mean a thing, other than to test your understanding of half-life. radioactive substance decay with a constant half-life. eg. if substance A decays radioactively to form substance B, and its half-life is 8days, then if you have 8g of substance A at day 1, then it means by 8th day, your sample would now contain 4g (ie. 1/2 of 8g) of substance A, but the sample you have now is a mixture of A and B. radioactive is a continuous decay phenomena.

are you familiar with calculus? radioactive decays is a first order rate equation, which you can integrate to obtain the decay profile.

radioactivity is directly proportional number of radioactive atoms
let N be number of radioactive atoms
dN/dt = -kN
(1/N)dN/dt = -k
integrate LHS and RHS with respect to time
ln N = -kt + C (constant of integration)
when t = 0, ln N_o = C where N_o is the initial number of radioactive atoms.

therefore, ln N - ln N_o = -kt => ln (N/N_o) = -kt

half-life of a substance (t_1/2) is given by:
N/N_o = 1/2 => - ln2 = -kt_1/2
t_1/2 = (ln 2)/k
k = ln 2 / t_1/2

the decay profile is thus:
ln (N/N_o) = -kt
ln (N/N_o) = -(ln 2 / t_1/2)t

thus by knowing the half life, you can calculate the decay constant (k) and thus obtain the fill decay profile of the substance of interest.

Thank you so much for replying!  

I don't really know calculus...I'm in the very very basic chemistry. I really don't understand anything you said either...   Do you think you could try explaining one of the questions and how it's done

[Vette Freak]

Half life problems involve simple math and a bit of thinking.  I'll give you a few questions to get you started...

1)  How many C-14 half-lives have passed in 22,920 years?  After one half-life (5730 years) you have 5.00 grams of C-14 (half of the original amount) that remains unchanged.  How much is left after the second half-life? Continue this process.
2)  This is the same concept as the first problem but in a different order.  So after one half-life, 50.0 grams of P-32 is left. After two half-lives, 25.0 grams is left.  Continue this process until you reach the amount remaining.  Using the number of half-lives and the half-life of P-32 (which I assume you were given in Table N), calculate the total time elapsed.
3)  After one half-life, you have 1/2 the original amount of I-131 left. After the second half-life you have (1/2)/2 = 1/4 left.  Continue, add up the total number of half-lives elapsed, and use the I-131 half-life to calculate the total time elapsed.

Yes, there are very simple equations you can apply to these situations but in this case (with nice numbers) it's just as effective (and more helpful) to think about the half-life process and solve things "by hand".  Try to derive the equations on your own. I hope this helps.

[exodus]

Thank you everyone! You've all been VERY helpful.

[Donaldson Tan]

a simpler way of looking at half-life mathematically is:

N/N_o = (1/2)ⁿ where n is the number of half-lives that is required for the radioactive substance to decay to reach its current amount of mass

time taken for the decay process is therefore n times t_1/2