Radioactivity

Introduction:

Almost one year after the discovery of x-rays by Roentgen, Henry Becquerel in 1896 observed that uranium and some of its salts emit spontaneously some invisible radiations, which resembled x-rays in many respects (e.g. both could affect photographic plate, penetrate through thick opaque substances etc. ) But two types of radiations greatly differ with regard to their origin. X-rasy are produced when fast moving electrons strike a metal target of high atomic number in a discharge tube in which a large potential difference is applied between the electrodes. But the radiations emiited from uranium are spontaneous and continuous , and do not require any favourable condition. These radiations originate from the nucleus of uranium atom. This behaviour of uranium was found to be unaffected by temperature, pressure, electric and magnetic field etc. These radiations are called radioactive radiations or radioactive rays or Becquerel rays. The spontaneous emission of such powerful radiation from the nuclei of some substance is called radioactivity: the substance are called radioactive substance. Example of radioactive elements are uranium, polonium, radium, thorium etc.

Also these radioactivity from the respective substance are categorised into two types:

(1) Natural radioactivity and
(2) Artificial radioactivity

The phenomenon of spontaneous emission of highly penetrating radiations from heavy elements (i.e A > 206) occuring in the nature is called natural radioactivity. And the phenomenon in which radioactivity can be induced by artificial means through nuclear transmutation is called artificial radioactivity.

Laws of radioactive emission;

Mainly there are two laws of radioactive emissioon which are given below.

(1) It is not sure that which of the particle (i.e. \(\alpha\) , \(\beta\) , \(\gamma\) ) emits from the radioactive source (nucleus). That means theemission of these particle from the nucleus is independent to the physical conditions such as temperature , pressure, humidity etc. That is we can't control the emission of these particles from the nucleus.

(2) The rate of disintegraton (called activity ) is directly proportional to the number of nuclei at that time.

If N be the number of undecayed nuclei at time t , then rate of disintegratiion is, $$\frac{dN}{dt}\propto N$$

$$or,\;\;\;\frac{dN}{dt} =-\lambda N$$

where, \(\lambda\) is called decay constant and negative sign indicates the decay of nuclei ( decrease of number of nuclei. )

$$or\;\;\;\frac{dN}{N}=-\lambda dt$$

Integrating we get,

$$\int_{N _\circ}^N\frac{dN}{N}=\int_{0}^{t}-\lambda dt$$

$$or,\;\;\;\;\;\biggl[\log_eN\biggr]_{N_\circ}^{N}=-\lambda t$$

$$or,\;\;\;\log_eN-\log_eN_\circ=-\lambda t$$

$$or,\;\;\;\log_e\biggl(\frac{N}{N_\circ}\biggr)=-\lambda t$$

$$or,\;\;\;N=N_\circ e^{-\lambda t}$$

This is the required expression for the number of nuclei present at any time t.

Successive Radioactive Disintegration:

It is the chain radioactive disintegrtion. The first sampld of radioactive source disintegrate to from second sample. The second sample disintegrate to from the next one. The process is continued until the stable nuclei is achieved.

For eg; A \(\rightarrow\) B \(\rightarrow\) C \(\rightarrow\) \(\cdot\) \(\cdot\) \(\cdot\) \(\rightarrow\) stable nuclei.

Let us suppose, a sample A is disintegrate to form sample B and then sample C which is stable.

Let the number of atom in sample A at time t = 0 is \(N_\circ\). At time t = 0, the number of atom in sample B is zero. At time t, let \(N_1\) and \(N_2\) be the number of atoms in sample A and B.

Therefore, the rate of decay of A ( i.e. formation of B ) = \(\lambda_1N_1\)

and rate of decay of B = \(\lambda_2N_2\) where, \(\lambda_1\) and \(\lambda_2\) are decay constant for A and B

Therefore rate of disintegration of B is, $$\frac{dN_2}{dt}=\lambda_1 N_1-\lambda_2 N_2$$

$$or,\;\;\;\frac{dN_2}{dt}+\lambda_2 N_2=\lambda_1 N_1$$

$$or,\;\;\;\frac{dN_2}{dt}+\lambda_2 N_2=\lambda_1 N_0 e^{-\lambda_1t}$$

multiplying both sides by \(e^{-\lambda_2 t}\), we get

$$e^{\lambda_2 t}\frac{dN_2}{dt}+e^{\lambda_2t}\lambda_2 N_2=\lambda_1N_\circ e^{(\lambda_2-\lambda_1)t}$$

$$or,\;\;\;\frac{d}{dt}\biggl(N_2 e^{\lambda_2 t}\biggr)=\lambda_1 N_\circ e^{(\lambda_2-\lambda_1)t}$$

Integrating, we get

$$N_2 e^{\lambda_2 t}=\frac{\lambda_1 N_\circ e^{(\lambda_2-\lambda_1)t}}{\lambda_2-\lambda_1}+C\dotsm(1)$$

At t = 0, \(N_1\) = \(N_\circ\)

And \(N_2\) = 0

therefore; C = \(\frac{-\lambda_1 N_\circ}{\lambda_2-\lambda_1}\)

so from equation (1), we get

$$N_2 e^{\lambda_2 t}=\frac{\lambda_1 N_\circ}{\lambda_2-\lambda_1}\biggl[e^{(\lambda_2-\lambda_1)t} -1\biggr]$$

$$N_2=\frac{\lambda_1 N_\circ}{\lambda_2-\lambda_1}\biggl[e^{-\lambda_1 t}-e^{-\lambda_2 t}\biggr]\dotsm(2)$$

Here equation (2) is the required expression for the number of atoms present in the sample B.

Radioactive Equilibrium:

There are two type of equillibrium.Thtey are secular equillibrium and transition eqillibrium .

(1)Secular equillibrium(permanent equillibrium):

In this case, the parent nucleus lived infinitely long time that of daughter nucleus.That means the half life of parent nucleus is very very greater than that of daughter nucleus.$$i.e.T_1>T_2$$ $$\lambda_1=\frac{0.693}{T_1}\simeq0$$ $$\lambda_2=\frac{0.693}{T_2}\simeq\infty=very large$$We have known the epression $$

$$N_2=\frac{\lambda_1 N_\circ}{\lambda_2-\lambda_1}\biggl[e^{-\lambda_1t}-e^{-\lambda_2t}\biggr]$$ $$N_2=\frac{\lambda_1 N_\circ}{\lambda_2} \biggl[1-e^{-\lambda_2t}\biggr]$$After a long time t,\(N_\circ=N\),\(e^{-\lambda_2 t}=0\) $$N_2=\frac{\lambda_1 N_1}{\lambda_2}$$ $$\lambda_1N_1=\lambda_2N_2$$This is required expression for secular equillibrium.

(2)Transient euillibrium:

In thits equillibrium, the parent nucleus lived long period of time that of daughter nuclei ,That means the half life of parent nuclie is greater than that of parent nuclie.$$i.e.T_1>T_2$$ $$\lambda_1=\frac{0.693}{T_1}\simeq small$$ $$\lambda_2={0.693}{T_2}\simeq large$$ We have known the expression;$$N_2=\frac{\lambda_1 N_\circ}{\lambda_2-\lambda_1}\biggl[e^{-\lambda_1t}-e^{-\lambda_2t}\biggr]$$ $$N_2=\frac{\lambda_1 N_\circ}{\lambda_2-\lambda_1}\biggl[1-e^{-\lambda_2 t}\biggr]$$After a long time t,\(N_\circ=N\),\(e^{-\lambda_2 t}\) $$N_2=\frac{\lambda_1 N_1}{\lambda_2-\lambda_1}$$ $$\frac{N_2}{N_1}=\frac{\lambda_1}{\lambda_2-\lambda_1}$$This is the required expression for the trasient equillibrium.

Unit of radioactivity

The unit of radioativity is curie .It is defined as the quantity of radioactivity substance which gives \(3.70X10^{10}\) disintigration persec $$i.e.\frac{dN}{dt}\simeq1 curie=3.70X10^{10} dis persec$$ Another unit is rutherford$$i.e.\frac{dN}{dt}\simeq 1 Rutherford=10^{6} dispersec$$ Bacquerel(Bq)$$\frac{dN}{dt}\simeq 1Bq=1 dispersec$$