Models use assumptions to Simplify Problems.
Without simplification, real life situations would be too complicated to solve. Models take out the negligible factors so an equation can be made.
As well as simplifying things to a practical level, equations are transferable. Radioactive decay and capacitors, for example, have very similar equations.
Any question could ask you about what assumptions you have made, so just pay attention to what you're ignoring. (uniform, spherical, etc.)
Radioactivity can be Modelled by Exponential Decay
Atoms are unstable when they have too many or too few neutrons or too much energy.
An individual atoms' decay cannot be predicted. But on a large scale with millions (or billions) (...or trillions) of atoms, the decay shows an exponentially declining pattern.
Remember, this is based on lots of atoms acting the same way like a crowd mentality. So we're predicting how the crowd acts, not each atom.
The Rate of Decay is Measured by the Decay Constan
Decay constant (λ)(per second) is a property of an isotope indicating how quickly it will decay.
Activity, measured in becquerels (Bq)(Decays/second), is how many decays per second happen at a single point. As time goes on, activity decreases, hence the exponential curved graph.
Equation Time! :D
Activity = Decay constant x number of atoms. A=λN
The "Rate of change" of Activity ~ dN/dt=-λN
p.s. "per second" isn't a typo... "/s" makes the activity equation work...
Half-Life and The most important radioactive equat
You can find the half life from the graph. Get the initial activity (t=0), half it, draw a horizontal line from there to the graph, draw a vertical line from the graph to the time axis, read off the half life and double check your units.
Half life = Ln2/Decay constant T(1/2)=Ln2/λ
To find out how many atoms left: N=N0e^(-λt)
N0 = Start no. of atoms, λ = decay constant and t = time (in seconds)
N=N0e^(-λt) is the most important equation in radioactivity... don't forget it... say it over and over... N=N0e^(-λt)... N=N0e^(-λt)...
...N=N0e^(-λt) ...N=N0e^(-λt) ...N=N0e^(-λt)...
Graph of Half-Life