What is the derivation for N = R*tau for the number of radioactive atoms left in a quantity of radioactive material?

Answer 1

Radioactivity assumes an individual isotope will decay with a certain probability in a certain time interval. If you have N isotopes N · k will decay in a certain time interval. k would be probability an isotope will decay in the time interval and is also called the decay constant. Notice also that 1/k = tau. This makes the total change in isotopes

ΔN = N k Δt.

If you look at a small interval

dN = N k dt, and

dN/dt = N k.

dN/dt is the rate of decay. If you don't know calculus dN/dt is just the change of the isotopes over the change of time.

In your notation

R = dN/dt

k = 1/tau

so

R tau = N.