Is activity in radioactive decay a differential?

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Discussion Overview

The discussion centers around the concept of activity in radioactive decay, specifically whether the expression for activity as a differential is appropriate. Participants explore the definitions and implications of the decay constant, the relationship between decay probability and the number of decays per unit time, and the interpretation of differentials in this context.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Mathematical reasoning

Main Points Raised

  • Some participants assert that the decay constant (λ) represents the probability of decay per unit time, leading to the expression for activity as λN, which they argue is a rate of change.
  • Others question the use of differentials, suggesting that dN/dt implies an instantaneous rate of change rather than an average over a time interval, which they believe should be represented as ΔN/Δt.
  • One participant emphasizes that the activity should be understood as an average number of decays per second, raising concerns about the phrasing of the definition of activity.
  • Another participant points out that the randomness of decay means that the number of decays cannot be precisely determined at any instant, suggesting that the activity should be treated as an average rate.
  • There is a discussion about whether the probability of decay is constant over time intervals or if it should be considered in the limit as the time interval approaches zero.

Areas of Agreement / Disagreement

Participants do not reach a consensus on whether the activity can be accurately described using differentials or if it should be expressed as an average over time intervals. Multiple competing views remain regarding the interpretation of decay probability and the nature of activity in radioactive decay.

Contextual Notes

Participants express uncertainty regarding the definitions and implications of the decay constant and activity, highlighting potential limitations in understanding the relationship between average and instantaneous rates of decay.

Acid92
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According to my textbook, the decay constant is the probability that a radionuclide will decay in any second (hence the unit s^-1) and so the total number of radionuclides decaying at any second, i.e. the activity, is λN but this is also the rate of change of N thus

dN/dt = -λN

Surely dN/dt can't be a differential since λN is the number of radionuclides decaying in a time interval of 1 second and not an instant?
 
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Naty1 said:
The decay constant — λ, "lambda" is the inverse of the mean lifetime...so it's unit is the reciprocal of time, but not one second...

See total activity A here:

http://en.wikipedia.org/wiki/Radioactive_decay#Radioactive_decay_rates

Yes I've looked at that but I am interested in its definition as the probability of decay (I looked around the internet and noticed this term being used but without the explanation I am looking for) as this is what my textbook introduces it as. My original post is how my textbook introduces it and the activity differential but it doesn't make sense to me...

Even that doesn't make sense to me, why isn't it (total activity) the number of decays at instantaneous time and not per unit time if we are talking about differentials?
 
The equation dn/dt = -λN stated in words means 'the number of decays per second is proportional to the number of nuclei present'.
This is the natural law of decay (or growth)
If you have 1000 nuclei in a sample and there are 10 decays per second it sounds obvious that in a sample of 2000 nuclei there would be 20 decays per second...this is 'natural.
In this case λ = 1/100 which means that the probability of a nucleus decaying is 1 in 100
λ is the probability of a decay.
 
truesearch said:
The equation dn/dt = -λN stated in words means 'the number of decays per second is proportional to the number of nuclei present'.
This is the natural law of decay (or growth)
If you have 1000 nuclei in a sample and there are 10 decays per second it sounds obvious that in a sample of 2000 nuclei there would be 20 decays per second...this is 'natural.
In this case λ = 1/100 which means that the probability of a nucleus decaying is 1 in 100
λ is the probability of a decay.

If λ is the probability of decay per unit time then the total decay per second will be λN, I understand that but then surely the activity can only be stated as

ΔN/Δt = λN NOT dN/dt which is a differential

If we want dN/dt, surely you would need the probability of decay in an instant (the limit as Δt tends to 0?) unless the probability of decay is independent of time intervel.

I don't understand how you say the total activity per unit time is dN/dt because this is a differential and gives the rate of change of N instantaneously and not per unit time as λN would be.
 
You seem to be misunderstanding the prase "total number of radionuclides decaying at any second"- and perhaps it is phrased badly. Precisely because nucleotides is random, that cannot actually be the number of radionuclides that decay. It has to be an "average" which is then saying that we are treating the rate at which radionuclides decay as if it were a constant. The slope of a straight line is the derivative.
 
HallsofIvy, I am not quite sure what youre saying.

Ok let me put this as simply as possible: Is the activity the average total number of decays per second, in other words the change in N per second, yes or no?

If yes, why is this (ΔN/Δt) the same as the instantaneous activity dN/dt?

If I were to draw a graph of N against t, wouldn't λN give me the gradient of the secant line over an intervel of 1 second and not the actual gradient at any point?
 

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