Exponential decay is usually represented by an exponential function of time with base #e# and a negative exponent increasing in absolute value as the time passes:
#F(t) = A*e^(-K*t)#
where #K# is a positive number characterizing the speed of decay. Obviously, this function is descending from some initial value at #t=0# down to zero as time increases towards infinity.
For example, this function can represent a radioactive decay of certain quantity of plutonium-239 and describes the amount of plutonium-239 left after a time period #t#.
Half life is the value of #t# when there will be left only half of what was in the beginning at #t=0#.
At #t=0# the value of our function equals to
#F(0)=A*e^(-K*0)=A*e^0=A*1=A#
If at time #t=T# there is only half of the initial amount that is left, we have an equation:
#A/2=F(T) = A*e^(-K*T)#
The above represents an equation with #T# being an unknown.
Solution is:
#A/2=A*e^(-K*T)# (reduce by #A#)
#1/2 = e^(-K*T)# (take natural logarithm)
#-K*T=ln(1/2)=-ln(2)# (now we can resolve for #T#)
#T=ln(2)/K#
So, all we need to know to find half life is the speed of a decay #K#. It can be determined experimentally for most practical situations since it depends on inner physical and chemical characteristics of a decaying substance.
For instance, half life of plutonium-239 is #24110# years, half life of caesium-135 is #2.3# million years, half life of radium-224 is only a few days.
