### 3) Law of radioactive Decay

• Radioactivity is a nuclear phenomenon
• When a nucleus disintegrates by emitting a particle ( α and β) or by capturing an electron from the atomic shell( K-shell) ,the process is called radioactive decay. This decay process is spontaneous.
• Let us take a radioactive sample containing N0 at time t=0 i.e, at the beginning. We wish to calculate the number N of these nuclei left after time t.
• The number of nuclei of a given radioactive sample disintegrating per sec is called the activity of that sample is
dN/dt=rate of decrease of nuclei with time=Activity of sample at time t                              --(1)
• Experimentally it is found that the activity at any instant of time t is directly proportional to the number N of parent type nuclei present at that time

Where λ > 0 is proportionality constant and negative sign indicates that N decreases as t increases
• From equation (2) we get

i.e. ,λ is fractional change in N per sec
=> λ is not merely a proportionality constant ,but it gives us the probability of decay per unit interval of time
• Hence λ is called the probability constant or decay constant or disintegration constant
• dN is the no of parent nuclei that decay between t and t+dt and we have taken N as continuous variable
• From (2)

N0=No of radioactive nuclei at t=0
• From (4) we see that law of radioactive decay is exponential in character

• From figure it can be noted that only half the amount of radon present initially after 3.83 days and 1/4 after 7.66 days and so on
• Plot shows that in a fixed time interval a fixed fraction of the amount of radioactive substance at the beginning of interval decays
• This faction is independent of the amount of radioactive substance and depends only on the interval of the time
• The decay constant λ is a characteristics of radioactive substance and it depends in no way on the amount of the substance present
##### a) Half Life
• Time interval during which half of a given sample of radioactive substance decays is called its half life. It is denoted by T

##### b) Mean Life τ
• Individual radio atomic atoms may have life spans between zero and infinity
• Average or mean life τ is defined as
τ=Total life time of all nuclei in a given sample/Total no of nuclei in that sample                        --(6)

• From curve one can see that each of dN number of radioactive nuclei has lived a life of t sec i.e. the total life span of a dN nuclei is (dN.t) sec . Therefore equation (6) can be written as