Question about isotope halflife and decay

[furface]

First off I just wanted to say something quickly. I am a Visual Arts major taking a Introductory Physics class for fun, (probably my favorite class that I have ever taken, also the most rewarding. I have learned so much that I wish I had taken it sooner so I could have pursued it more)

My physics professor gave out a study guide for our final with questions and their respective answers. I have been able to complete every question flawlessly except for one. I have tried everything that I have been capable of to solve this problem but I can not seem to figure it out (even researched online, but all results went over my head). So here we go.

Homework Statement

The isotope 90Sr has a half-life of 28 years. How many years will it take for 7/8 of a sample of this material to decay (so that 1/8 is left)?

The solution to the problem listed in the answer key is 84 years.

Homework Equations

This is where I have become confused, I really do not have notes for this type of problem, only for problems with a similar subject matter. The equation I have is $$\alpha$$= (2.7x10^-11)n [I am not sure how I would use these though, I feel like they do not work for this problem, I have used it on others]

The Attempt at a Solution

I have tried quite a few things. I assumed that 28 years being the half life, is half of its life. So I thought maybe the full length of decay would be 28*2 which would be 56. Then I took 7/8 of that number to find out how long it will take to have 1/8 of it left. My answer for that was 49 years.

Is my train of thought correct in any way when it comes to this problem or am I missing something completely? I just need to be guided in the right direction so I can solve this myself when a similar problem comes along on the final.

 

[ehild]

Half-life of an isotope means the time period after the amount of the isotope is half of the original amount. If you had one million 90Sr atoms, you will have only half million after 28 year, so half million of the isotopes has decayed. After 56 years, you have a quarter million. This is halved again during the next 28 years, and so on.
If =7/8 of the isotopes decayed what fraction of the original amount of 90Sr is still present?

ehild

 

[furface]

There would be 1/8 of the decayed isotopes present. If there were 1 million there would be 125,000 left.

So is the teachers solution to the problem incorrect? I see what you are saying but when I keep getting 84 years as 6/8 of its life having past, while for 7/8 I get 112 years.

 

[ehild]

After 28 years, you have half the original amount. After the next 28 year, 1/4 remains. It is halved again during the next 28 year. So after 3 times 28 years you have 1/8 of the original number of atoms. That 84 years is correct. The half life-time does not mean that a single atom lives for 56 years. It means that the probability that it still exist after a period of 28 years is 50 %.

ehild

 

[furface]

Oh wow, thank you so much. I understand what I was doing wrong. Maybe it was because I was working so late I just was confusing myself.

Is this the correct train of thought?
1/2 life is 28 years
half of that half life is 1/4, which is 56 years
and half of 1/4 is 1/8 (which is how much I am looking for) which is now 84 years

 

[ehild]

Well, it is better to say "half alive" instead of 1/2 life... And do not forget, it is probability. Valid for high number of atoms. If you have only two atoms, you cannot say that one of them will decay exactly after 28 years.

In case of a more general problem, for example: how long have to wait till an isotope decays to 1/100 of the original amount, you can use the equation
$$N(t)/N(0)=2^{-t/\tau}$$
where N(0) is the initial amount of the atoms and N(t) is their number at time t, and tau is the half-life.



ehild