 ## Nuclei

• α-decay can be written as, ZPA  →  Z−2DA− 4 + 2He4
• Ex: 92U238  →  90Th234 + 2He4
• β – decay can be written in the form, ZPAZ+1DA + −1e0
• Ex: 90Th234  →  91Pa234 + −1e0
• 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 = N0e−λt]
• A = λN = λN0e−λt = A0e−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 × 1010 disintegrations / sec
• 1 Becquerel = 1 disintegration per second.
• The time taken by the atoms to decrease from N0 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 (t1/2) 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 2t1/2
• t87.5% (or) t7/8 = 3t1/2
t90% = \tt \frac{10}{3} t_{1/2}
\tt t_{\frac{15}{16}} or t93.75% = 4t1/2
• t99% = \tt \frac{20}{3} t_{1/2}
• t99.9% = 10t1/2
• t29.3% = \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 (T1/2): 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 = T1/2
Hence form N = N0e−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}