Calculating Radioactive Decay with Simultaneous Particle Emission

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Homework Help Overview

The problem involves calculating the time after which one-fourth of a radioactive material remains, given that it decays through the simultaneous emission of two particles with different half-lives of 1620 and 810 years. The original poster expresses confusion regarding the nature of the half-lives and how to approach the problem involving multiple decay channels.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the interpretation of simultaneous decay and whether the half-lives pertain to the original material or the emitted particles. There is a focus on finding an expression for the remaining substance over time and the relationship between the decay of the parent material and the formation of daughter nuclei.

Discussion Status

The discussion includes various interpretations of the problem, with some participants suggesting that the decay occurs through two different channels. There is an attempt to derive a formula for the remaining quantity of the substance, and while one participant presents a calculation, the overall consensus on the approach remains unclear.

Contextual Notes

Participants are navigating the complexities of simultaneous decay and the implications of multiple half-lives, which may lead to differing interpretations of the problem setup. The original poster's uncertainty about the nature of the decay and the relationship between parent and daughter nuclei is a key point of discussion.

erisedk
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Homework Statement


A radioactive material decays by simultaneous emission of two particles with respective half lives 1620 and 810 years. The time, in years, after which one-fourth of the material remains is:

Homework Equations

The Attempt at a Solution


I'm confused whether the half lives mentioned are of the original radioactive material itself, or of the the particles which are formed. In any case, I'm really not sure of how to deal with two particle emission. I've only ever done single decays and basic half life calculations. Please help?
 
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I believe that you have a material that can decay through two different channels, one with a H.L. 1620 years and one with a H.L. of 810 years. The term simultaneous does not refer to the emission of two types at the same time.
 
Yeah that's what I was thinking too. But what do I do after that?
 
erisedk said:
Yeah that's what I was thinking too. But what do I do after that?
You need to find an expression which describes how much of tthe substance will remain after a given time.
 
How do you think the rate of decay of the parent related to the rate of formation of the daughters?
 
I'm extremely sorry for replying after so long!

gleem said:
How do you think the rate of decay of the parent related to the rate of formation of the daughters?
I suppose the rate of decay of the parent is equal to the rate of formation of the daughter. So, I'm guessing the rate of decay would be equal to the sum of rates of formation of the two nuclei. Here's what I did.
N = N0e-λt where N is the number of nuclei left after time t and the decay constant is λ.
Rate of formation of both the nuclei would be λiN.
So,
dN/dt = λN = λ1N + λ2N where λ1, λ2 are the decay constants of the two different daughter nuclei.
So, N cancels.
I get t1/2 = 1620*810/1620+810 = 540 years
So, 1/4th will remain after 540*2 = 1080 years.

Which is the right answer. Thank you!
 

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