A half-life is the time required for a quantity to reduce to half of its initial value. The term is commonly used in nuclear physics to describe how quickly unstable atoms decay, but it can be applied to any quantity that follows a set exponential-decay model.
The original definition of “half-life” was based on the observation that the logarithm (base 10) of the ratio of two identical quantities decays at a constant rate over time. This relationship between the decay rate and time is known as an exponential decay law, and it can be written as: =ln(2)/ 1/2
This equation says that k, the Decay Constant, is equal to the natural logarithm of 2 divided by the half-life t1/2. The natural logarithm has a base of e ≈ 2.718281828… So we can also write this equation as: =e/(2× 1/2)No matter what physical quantity you are measuring—whether it’s radioactive decay or something else—you will always find that its decrease over time obeys this same mathematical relationship.
In most cases, we are interested in knowing how long it will take for a given quantity to decay completely. We can use the above equation to solve for t1/2 if we know k and vice versa. For example, let’s say we want to know how long it will take for a sample of 100 nuclei to decay down to 50 nuclei. We would plug these values into our equation like so: =ln(2)/t1/2⇒t1/2=ln(2)/kNow all we need is a value for k, which we can look up in a table of radioactive isotopes or calculate from first principles if we know enough about the decaying process. Once we have k,plugging it into our equation gives us t1/2 = 0.693 seconds (this number comes from ln(2)/k = 0.693 where k = 0.693 s-1). So our sample will take just under one second to go from 100 nuclei down to 50 nuclei—half its original size! After another 0.693 seconds have passed, it will be down 25 nuclei (one quarter its size), and so on until there are no nuclei left at all—a process known as “complete” or “total” radioactivity decay 1 .
Interestingly, you don’t needto know anything about radioactivity or even exponential equationsto understand why half-lives exist; all you needis some basic arithmetic skills 2 . Suppose someone gave you a bag containing 100 marbles and told you that every minute , 10%of the remaining marbles would vanish forever 3 . You could work out how long it would take for allthe marbles toget eaten upby solvingthis simple proportions problem:100marbles______10minutes→xmarbles______60minutes Cross multiplyinggives us x = 600 minutes ,or 10 hours . In other words ,it takes 10 hoursto loseall your marblesat this rate ! Now suppose insteadthat they told youthat every minute20%of th remainin gmarbl eswouldvanish forever 4 .You couldagainworkout h owlongittakesforthebagtomarrybe emptied by solvinga simil arproportionsproblem :100mar bles → x ma rb les 20 minu tes 60 m i n u t e S Crossmultiplying now givesusx= 300 minutes ,or 5hours .It only takes5hoursto loseall your ma rb les atthisfasterrate ! Of courseinthereal world thingsrarelydecayaccordingto suchsimple rules ;however ,the importantthingisthatthey do obeyan overall trendof halvingin size after each fixedamountoftime haselapsed —justlikethese bags o fma rb les did 5 ! This general ideaalso appliestomanyothersituations outsideofthe realm offundamentalparticlephysics 6 .For instance financiersusehalf – liveswhen working outhowlongittakestoan investmentto fall tonew worth 7 ;medicine uses themwithpatientswhohavebeen treated withradioactiveisotopes 8 ;and geologists oftenestimate thenumberoftimes amaterialhas been throughaparticularprocessbymeasuringits currentratioagainstwhatits originalratio was 9 !