Instantaneous Decay rate per unit Volume

[Polarbear]

'A radioactive element has a 0.5 probability of decaying to a more stable element after a particular time-span T.

What is the instantaneous decay rate per unit volume? In other words determine a general expression for the number of decay events occurring per unit volume between t=t1 and t=t2 as the difference between these two times t2-t1 approaches zero.'

This is the last part of a group presentation we have to prepare and has us all stumped. Any thoughts?

 

[quantumdude]

'A radioactive element has a 0.5 probability of decaying to a more stable element after a particular time-span T.

You can interpret that statistically. If each nucleus has a 0.5 probability of decaying within time T, then you would expect half of a sample to have decayed by time T.

In other words, T is the half-life of the sample.

What is the instantaneous decay rate per unit volume? In other words determine a general expression for the number of decay events occurring per unit volume between t=t1 and t=t2 as the difference between these two times t2-t1 approaches zero.'

Use the standard model for exponential decay. Since they want the number of events per unit volume, I would divide both sides by the volume.

 

[quicknote]

The instantaneous decay rate per unit volume can be calculated by taking the limit as the difference between two times t2-t1 approaches zero. This can be represented by the expression:

dN/dV = lim(t2-t1->0) (N(t2)-N(t1))/(V(t2)-V(t1))

Where dN/dV represents the instantaneous decay rate per unit volume, N(t) represents the number of decays occurring at time t, and V(t) represents the volume at time t.

In the case of a radioactive element with a 0.5 probability of decaying after a time-span T, the number of decays occurring per unit volume can be represented by the expression:

dN/dV = 0.5 * N(t)

Where N(t) represents the number of radioactive atoms present at time t.

Therefore, the general expression for the instantaneous decay rate per unit volume between t=t1 and t=t2 is:

dN/dV = lim(t2-t1->0) (0.5 * N(t2)-0.5 * N(t1))/(V(t2)-V(t1))

This expression can be used to determine the instantaneous decay rate per unit volume at any given time interval.