1. Disintegration Constant
According to radioactive decay law,
dN/dt=-λN; λ=(dN/dt)/N numerically.
The disintegration constant of a radioactive substance is defined as the ratio of the number of atoms of the substance disintegrated in unit time to the number of atom of the substance present at the time of disintegration.

2. Theory
Let N₀ be the number of atoms of the radioactive substance initially present ie at the instant t=0 and N the number of atoms left after a time t ie at the instant t=t. According to the radioactive decay law,
-(dN/dt)α N; (dN/dt)= - λN;
therefore dN/N= - λt. On integration
log e N = - λt+C where C is the constant of integration. When t=0,N=N₀;therefore C= log e N₀.

3. Hence log e N =-λt+ log e N₀ ,or log e(N/N₀) =-λt; taking antilog(N/N₀)=e⁻λt or N=N₀ e⁻λt .this equation shows that the number of radioactive atoms of a given radio active substance decreases exponentially with time.This means that the rate of decay of the substance decreases exponentially with time. Theoretically an infinite time is required for the radioactive substance to

4. Theoretically an infinite time is required for the radio active substance to disintegrate

5. Activity of a radioactive substance
It is defined as the rate of decay of the radioactive substance.
A=(dN/dt)=λN (numerically)
If A₀ is the activity at the instant t=0,and A the activity at t=t,then A=A₀ e⁻λt

6. Units SI unit: Becquerel(Bq) defined as one decay per second.
Practical Units: Curie(Ci),defined as 3.7x10¹⁰ disintegrations per second.
1Ci= 3.7x10¹⁰ Bq.
Rutherford(Rd): Another unit of radioactivity is rutherford. It is defined as 10⁶ disintegrations per second.
One Rd=10⁶Bq.

7. Half life Period
The half period or Half-life period of a radioactive substance is the time in which the number or mass of the radioactive atoms is reduced to half of its initial value.
The concept of Half-life period is introduced to differentiate the activity of one radio active sample from another.

8. Expression for Half-life

9. Mean life(Average life)
In the case of the disintegration of a radioactive substance, it is not possible to predict which atom of the radioactive substance will disintegrate at a particular instant. The atom which disintegrate s first may have zero life time and the atom which disintegrates last may have infinite life time. Thus the life of every atom is different.
The mean life of an atom of a radioactive substance is defined as defined as the average time for which the atoms of the radioactive substance exist. It is the ratio of sum of the ages of all atoms to the number of atoms.

10. Expression for mean life

11. Radio active series
Almost all elements whose atomic number is greater than 90 exhibit the phenomenon natural radioactivity. The nuclei of these elements are unstable and they undergo disintegration by ejecting either an Alpha particle or a Beta particle to form a new atom.This chain of disintegration continues till an atom which is stable and not radioactive is formed. All natural chain of disintegration end with stable lead atoms. This chain of disintegration is called radio active serirs.

12. There are four important radio active series
There are four important radio active series. They are (1) Uranium series. (2) Thorium series (3)Actinium series & (4)Neptunium series. In uranium series, the parent element is ₉₂U²³⁸ and the end product is ₈₂Pb²⁰⁶.This is achieved by emission of 8 alpha particles and 6 beta particles. Similarly the Thorium series ends with another isotope of lead.In actinium series,the parent element is an isotope of Uranium (actino-uranium) ₉₂U²³⁵ and the end product is ₈₂Pb²⁰⁷.

13. In Neptunium series, the parent atom is Plutonium ₉₃Pu²³⁹
In Neptunium series, the parent atom is Plutonium ₉₃Pu²³⁹.As in the earlier cases,this series does not end with an isotope of stable lead but with an isotope of bismuth ₈₃Bi²⁰⁹.

14. Radioactive equilibriua 1.Secular or Permanent Equilibrium
In a radio actve series,if the parent has very very long half period,a state is reached after a very long time when the rate of decay of the parent becomes equal to the rate of decay of the daughter.In other words ,the rate of growth of the daughter becomes equal to the rate of decay of the daughter.In this stage ,the number of parent atom remains practically the same.At this stage,the parent and daughter are said to be in secular equilibrium.

15. (dN₂/dt)=λ₁N₁-λ₂N₂ & N₂=[λ₁/ λ₂- λ₁] N₀[e⁻ λ₁t- e⁻ λ₂t]

16. Transient equilibrium
In a radioactive series, if the ratio of the number of parent atoms to the number of daughter atoms remains a constant at any instant,then the parent and daughter are said to be in transient equilibrium.the transient equilibrium is reached,b/w parent and daughter only if the half life of parent is greater than that of the daughter.this means λ₁ <λ₂ ;but λ₁ cannot be neglected.

17. Transient equilibrium