## Nuclei

# Radioactivity

- α-decay can be written as,
_{Z}P^{A}→_{Z−2}D^{A− 4}+_{2}He^{4} - Ex:
_{92}U^{238}→_{90}Th^{234}+_{2}He^{4} - β – decay can be written in the form,
_{Z}P^{A}→_{Z+1}D^{A}+_{−1}e^{0} - Ex:
_{90}Th^{234}→_{91}Pa^{234}+_{−1}e^{0} - The emission of γ-rays from the nucleus does not alter either atomic number Z or mass number A.
- The wave lengths of γ-rays is less than 1Å
- RADIO ACTIVE DECAY LAW : \tt \frac{-dN}{dt} \propto N \Rightarrow \frac{dN}{dt} - \lambda N\ [N = N
_{0}e^{−λt}] - A = λN = λN
_{0}e^{−λt}= A_{0}e^{−λt} - A = λN ⇒ \tt A = \frac{0.693}{t_{1/2}}\ N

\tt \therefore A \propto \frac{N}{t_{1/2}} - Units of activity are curie and Rutherford
- 1 Curie = 3.7 × 10
^{10}disintegrations / sec - 1 Becquerel = 1 disintegration per second.
- The time taken by the atoms to decrease from N
_{0}to N is \tt t = \frac{1}{\lambda} \log_{e} \frac{N_{0}}{N} \Rightarrow t = \frac{2.303}{\lambda} \log_{10} \frac{N_{0}}{N} - The time taken by the radioactive element to disintegrate to half of the initial number of atoms is known as the half-life (t
_{1/2}) of a radioactive nuclei. - \tt t_{1/2} = \frac{2.303}{\lambda}\ \log_{10} (2) = \frac{0.693}{\lambda}
- The MEAN LIFE of a radioactive substance is equal to the average time for which the nuclei of atoms of the radioactive substance exist.
- The mean life of an atom of a radioactive nuclide is equal to the inverse of its decay constant.
- \tau = \frac{1}{\lambda} \Rightarrow \tau = 1.44 t_{1/2}
- Time required for disintegration of \tt \frac{3}{4} or 75% of the radioactive element is 2t
_{1/2} - t
_{87.5%}(or) t_{7/8}= 3t_{1/2}

t_{90%}= \tt \frac{10}{3} t_{1/2}

\tt t_{\frac{15}{16}} or t_{93.75%}= 4t_{1/2} - t
_{99%}= \tt \frac{20}{3} t_{1/2} - t
_{99.9%}= 10t_{1/2} - t
_{29.3%}= \tt \frac{1}{2} t_{1/2}

### View the Topic in this video From 00;20 To 09:42

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1. **Activity :** It is defined as the rate of disintegration (or count rate) of the substance (or the number of atoms of any material decaying per second) i.e., ratio active decay

A = -\frac{dN}{dt} = \lambda N = \lambda N_{0}e^{-\lambda t} = A_{0}e^{-\lambda t}

2. Half life (T_{1/2}): Time interval in which the mass of a radioactive substance or the number of it's atom reduced to half of it's initial value is called the half life of the substance.

i.e., if N = \frac{N_{0}}{2}

then *t* = *T*_{1/2}

Hence form N = N_{0}e^{−λt}

\frac{N_{0}}{2} = N_{0}e^{-\lambda(T_{1/2})} \Rightarrow T_{1/2} = \frac{\log_{e}2}{\lambda} = \frac{0.693}{\lambda}

3. Mean (orl average) life (τ) : The time for which a radioactive material remains active is defined as mean (average) life of that material.

i.e., τ = \tt \frac{Sum \ of \ the \ lives \ of \ all \ the \ atoms}{Total \ number \ of \ atoms} = \frac{1}{\lambda}