Radioactive decay (may include math)

[ShawnD]

A question was originally put in the homework help forum
"1 gramm radiates 3,7*10^10 alpha-particles in a second. Find out the half-life"

At a glance it looks simple. You start with X number of AMU, it's radiating at a rate of Y, find out how long it takes to get to 0.5X; right? Then I thought about it a second time. Once 1 atom of this substance emits an alpha particle, that atom is no longer the same, so it doesn't have the same rate it had before. This would mean the overall decay rate is constantly changing. If you think back to radioactive decay graphs from school, things do not decay in straight lines. Decay is always, or usually, logarithmic.

This problem was in the easy physics homework section, which would imply no calculus is involved. I can't figure out how this problem can be done without it.

 

[cesiumfrog]

Is it presumed you'll just plug some numbers into some exponential-decay forumula? Otherwise, I think you need basic calculus to derive the result.

Note: once an atom has decayed, it is normally assumed that the product atom has zero significant decay rate.

 

[Marco_84]

Hey guys it is not too difficult the assumption is that the rate of change in particles number is Proportional to the present number: dN=-NdtY (where Y is the inverse of the half-life time). this ODE is easy too to integrate:

N=NoExp(-tY)

So the decay is alway exponentially decreasing... and it depeds upon two parameters No and Y...

bye Marco

 

[daveb]

If you need to derive the formula, you need calculus. But to use the formula is simple college algebra.
Activity = Number of atoms*Decay Constant (Decay Constant = ln2/Half-life)
Number of atoms in sample = Avogadro's Number*Mass of sample/Gram Atomic Weight
Activity(at time t) = Activity (at time = 0)exp(-decay constant*t)
Specific Activity = Avogadro's Number*Decay Constant/Gram Atomic Weight