Write and derive the law of radioactivity decay.
Write and derive the law of radioactivity decay.
"In any radioactive sample, which undergoes $\alpha, \beta$ or $\gamma$-decay, it is found that the number of nuclei undergoing the decay per unit time is proportional to the total number of nuclei in the sample".
If $\mathrm{N}$ is the number of nuclei in the sample and $\Delta \mathrm{N}$ undergo decay in time $\Delta t$ then $\frac{\Delta \mathrm{N}}{\Delta t} \propto \mathrm{N}$
The number $\Delta \mathrm{N}$ of the decaying nucleus is always positive, $\therefore \frac{\Delta \mathrm{N}}{\Delta t}=\lambda \mathrm{N}$
where $\lambda$ is called decay constant or disintegration constant. If the period $\Delta t$ corresponds to zero,
$\lim _{\Delta t \rightarrow 0} \frac{\Delta \mathrm{N}}{\Delta t}=-\lambda \mathrm{N}$
$\therefore-\frac{d \mathrm{~N}}{d t}=\lambda \mathrm{N}$$...(1)$
where $d \mathrm{~N}$ is the change in $\mathrm{N}$, which may be positive or negative. Here it is negative because as time goes by the number of nucleus remaining will decreases.
Writing equation $(1)$ as follows,
$\frac{d \mathrm{~N}}{\mathrm{~N}}=-\lambda d t$
“In any radioactive sample, which undergoes $\alpha, \beta$ or $\gamma$-decay, it is found that the number of nuclei undergoing the decay per unit time is proportional to the total number of nuclei in the sample".
If $\mathrm{N}$ is the number of nuclei in the sample and $\Delta \mathrm{N}$ undergo decay in time $\Delta t$ then $\frac{\Delta \mathrm{N}}{\Delta t} \propto \mathrm{N}$
The number $\Delta \mathrm{N}$ of the decaying nucleus is always positive,
$\therefore \frac{\Delta \mathrm{N}}{\Delta t}=\lambda \mathrm{N}$
where $\lambda$ is called decay constant or disintegration constant.
If the period $\Delta t$ corresponds to zero,
$\lim _{\Delta t \rightarrow 0} \frac{\Delta \mathrm{N}}{\Delta t}=-\lambda \mathrm{N}$
$\therefore-\frac{d \mathrm{~N}}{d t}=\lambda \mathrm{N}$
where $d \mathrm{~N}$ is the change in $\mathrm{N}$, which may be positive or negative. Here it is negative because as time goes by the number of nucleus remaining will decreases.
Writing equation $(1)$ as follows,
$\frac{d \mathrm{~N}}{\mathrm{~N}}=-\lambda d t$
Similar Questions
Two radioactive isotopes $P$ and $Q$ have half Jives $10$ minutes and $15$ minutes respectively. Freshly prepared samples of each isotope initially gontain the same number of atoms. After $30$ minutes, the ratio $\frac{\text { number of atoms of } P}{\text { number of atoms of } Q}$ will be
Two radioactive isotopes $P$ and $Q$ have half Jives $10$ minutes and $15$ minutes respectively. Freshly prepared samples of each isotope initially gontain the same number of atoms. After $30$ minutes, the ratio $\frac{\text { number of atoms of } P}{\text { number of atoms of } Q}$ will be
Deuteron is a bound state of a neutron and a proton with a binding energy $B = 2.2\, MeV$. A $\gamma $ -ray of energy $E$ is aimed at a deuteron nucleus to try to break it into a (neutron + proton) such that the $n$ and $p$ move in the direction of the incident $\gamma $ -ray. If $E = B$, show that this cannot happen. Hence calculate how much bigger than $B$ must $E$ be for such a process to happen.
Deuteron is a bound state of a neutron and a proton with a binding energy $B = 2.2\, MeV$. A $\gamma $ -ray of energy $E$ is aimed at a deuteron nucleus to try to break it into a (neutron + proton) such that the $n$ and $p$ move in the direction of the incident $\gamma $ -ray. If $E = B$, show that this cannot happen. Hence calculate how much bigger than $B$ must $E$ be for such a process to happen.
The radioactivity of a given sample of whisky due to tritium (half life $12.3$ years) was found to be only $3\%$ of that measured in a recently purchased bottle marked $"7$ years old". The sample must have been prepared about
The radioactivity of a given sample of whisky due to tritium (half life $12.3$ years) was found to be only $3\%$ of that measured in a recently purchased bottle marked $"7$ years old". The sample must have been prepared about
After absorbing a slowly moving neutron of mass $m_N$ $(momentum $~ $0)$ a nucleus of mass $M$ breaks into two nuclei of masses $m_1$ and $3m_1$ $(4m_1 = M + m_N)$, respectively. If the de Broglie wavelength of the nucleus with mass $m_1$ is $\lambda$, then de Broglie wavelength of the other nucleus will be
After absorbing a slowly moving neutron of mass $m_N$ $(momentum $~ $0)$ a nucleus of mass $M$ breaks into two nuclei of masses $m_1$ and $3m_1$ $(4m_1 = M + m_N)$, respectively. If the de Broglie wavelength of the nucleus with mass $m_1$ is $\lambda$, then de Broglie wavelength of the other nucleus will be
Write the definition of half life of radioactive substance and obtain its relation to decay constant.
Write the definition of half life of radioactive substance and obtain its relation to decay constant.