How do you Find exponential decay half life?

Question

William Abdalla · Accepted Answer

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.

Luke Alvarez · Answer

undefined To find the half-life of exponential decay, you can use the formula:

$$ T_{1/2} = \frac{\ln(2)}{k} $$

Where:
- $ T_{1/2} $ is the half-life,
- $ \ln(2) $ is the natural logarithm of 2 (approximately 0.693),
- $ k $ is the decay constant.

The decay constant $ k $ is determined by the specific decay process. If you have the decay rate or decay constant provided, you can directly substitute it into the formula to find the half-life. If you're given the time it takes for a quantity to decrease to half its initial value, you can rearrange the formula to solve for the decay constant $ k $, and then use it to find the half-life.