MEASUREMENT OF RADIOACTIVITY

Units of Radioactivity
The International systems of units (SI) unit of radioactive activity is the becquerel (Bq), named in the honour of the scientist Henri Becquerel. One Bq is defined as one transformation (or decay or disintigration) per second. An older unit of radioactivity is the Curie, Ci, which was originally defined as "the quantity or mass of radium (element) emanation in equillibrium with one gram of radium. Today, curie is defined as 3.7 x 10^10 disintegrations per second, so that 1 curie = 3.7 x 10^10 Bq. The effect of ionizing radiation is often measured in units of grey for mechanical or sievert for damage to tissue.

Mathematics of Radioactive Decay:
1. Universal law of Radioactive Decay

Radioactive Decay 

The law describes the statistical behaviour of a large number of nuclides, rather than individual atoms. In following formalism, the number of nuclides or the nuclide population N, is of course a discrete variable (a natural number) - but for any physical sample N is so large that it can be treated as a continous variable. Differential calculus is used to model the behaviour of nuclear decay.
2. One-Decay Process:

One Decay of Cobalt - 60

Consider the case of a nuclide A that decays into another B by some process A→B (emission of other particles, likeelectron neutrinos  and electrons e- as in beta decay, are irrelevant in what follows). They decay of an unstable nucleus is entirely random and it is impossible to predict when a particular atom will decay. However, it is equally likely to decay at any instant of time. Given sample of a particular radioisotope, the number of decay events -dN expected to occur in a a small interval of time dt is proportional to the number of atoms present N, that is
                            -dN/dt α N
Particular radionuclides decay at different rates, so each has its own decay constant λ. The expected decay -dN/N is proportional to an increment of time, dt: -dN/N = λdt.
The negative sign indicates that N decreases as time increases, as the decay events follow one after another . The solution to this first order differential equation is the function
N(t) = N0e^-λt = N0e^-t/τ
N0 is the value of N at time t = 0.
We have for all time t:
Na + Nb = N (total) = Nao
If the number of non-decayed A nuclei is: Na = Naoe^-λt
then the number of nuclei of B, i.e., the number of decayed A nuclei, is Nb = Nao - Na = Nao - Naoe
3. Chain-Decay Processes

Chain Decay of Uranium 

(i) The law for a decay for two nuclides: dNb/dt = - λbNb + λaNa. The rate of change of Nb, that is dNb/dt, is related to the changes in the amounts of A and B, Nb can increase as B is produced from A and decrease as B produces C. dNb/dt = -λbNb + λaNaoe^-λat
(ii) Chain of any number of decays: dNj/dt = -λjNj = λj-iN(j-1)oe^-λt
(iii) Alternative decay mode:
We have for all time t;
N = Na = Nb = Nc
4. Half-Life
A more commonly used parameter is the half-life. Given a sample of a particular radionuclide, the half-life is the time taken for half the radionuclide's atom to decay. For the case of one-decay nuclear reactions: 
N = Noe^-λt = Noe^-t/τ
the half-life is related to the decay constant as follows: set N = No/2 and t = T12 to obtain.  t1/2 = ln2/λ = τln2.
This relation between the half-life and the decay constant shows that highly radioactive substances are quickly spent, while those radiate weakly endure longer. Half-life of radionuclides vary widely, from more than 10^19 years, such as for the very nearly stable nuclide 209Bi, to 10^-23 seconds for highly unstable ones. The factor of ln(2) in the above expression results from the fact that the concept of 'half-life' is merely a way of selecting a different base other than the natural base e for the lifetime expression. The time constant τ is the e^-1 -life, the time until only 1/e remains, about 36.8%, rather than the 50% in the half-life of a radionuclide. Thus, τ is longer than t1/2. The following equation can be shown to be valid: N(t) = Noe^-t/τ = No2^-t/t1/2
Since, radioactive decay is exponential with a constant probability, each process could as easily be described with a different constant time period that (for example) gave its  "(1/3) -life" or "(1/10) -life" and so on. Thus, the choice of τ and t1/2 for marker-times, are only for conenience, and from convention. Mathematically, the nth life for the above situation would be found in the same way as above -by setting N = No/n, t =T1/n and substituting into the decay solution to obtain: t1/n = ln n/τ = τln n.
Example: A sample of 14 C has a half-life of 5,730 years and a decay rate vod 14 disintegration per minute (dpm) per gram of natural carbon. If an artifact is found to have radioactivity od 4 dpm per gram of its present C, we can find the approximate age of the object using the above equation: N = Noe^-t/τ
where, N/No = 4/14 ~ 0.286, τ = (T1/2)/ln2 ~ 8267 years; t = -τlnN/No ~ 10356 years.

Half-Life of different types of Reactions
1. Zero Order Reaction;

Graphical representation of Zero Order Reaction 

The rate of reaction doesnot depend on the substrate concentration i.e., saturating the amount of substrate doesnot speed up the rate of the reaction. Half-Life:
t1/2 = [Ao]/2k
The t1/2 formula for a zero order reaction suggests the half-life depends on the amount of initial concentration and rate constant.
2. First Order Reaction:

Graphical representation of First Order Reaction

Half-Life:
t1/2 = ln2/k
3. Second Order Reaction

Graphical representation of Second Order Reaction 

Second order reactions decrease at a much faster rate. The length of half-life increase while the concentration of sunstrate constantly decrease, umlike zero and first order reaction.

t1/2 = 1/k[A]o