Answer:
Thus, the different spectral
lines in the spectra of atoms correspond to different transitions of electrons
from higher energy levels to lower energy levels.
Let \[{{n}_{1}}\] and \[{{n}_{2}}\]
be the two energy shells in the hydrogen atom \[({{n}_{1}}<{{n}_{2}})\]
According Bohr atomic model:
Energy associated with \[{{n}_{1}}\]
shell \[=-\frac{2{{\pi }^{2}}m{{e}^{4}}{{k}^{2}}}{n_{1}^{2}{{h}^{2}}}\]
Energy associated with \[{{n}_{2}}\]
shell
\[=-\frac{2{{\pi
}^{2}}m{{e}^{4}}{{k}^{2}}}{n_{2}^{2}{{h}^{2}}}\]
\[{{E}_{{{n}_{2}}}}-{{E}_{{{n}_{1}}}}=-\frac{2{{\pi
}^{2}}m{{e}^{4}}{{k}^{2}}}{n_{2}^{2}{{h}^{2}}}-\left[ -\frac{2{{\pi
}^{2}}m{{e}^{4}}{{k}^{2}}}{n_{1}^{2}{{h}^{2}}} \right]\]
\[\Delta
E=\frac{2{{\pi }^{2}}m{{e}^{4}}{{k}^{2}}}{{{h}^{2}}}\left[
\frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}} \right]\]
\[\Delta
E=hv=h\cdot \frac{c}{\lambda }\]
\[h\cdot
\frac{c}{\lambda }=\frac{2{{\pi }^{2}}m{{e}^{4}}{{k}^{2}}}{{{h}^{2}}}\left[
\frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}} \right]\]
or \[\frac{1}{\lambda
}=\frac{2{{\pi }^{2}}m{{e}^{4}}{{k}^{2}}}{{{h}^{3}}c}\left[
\frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}} \right]\]
\[\frac{1}{\lambda
}=\overset{\_}{\mathop{v}}\,={{R}_{H}}\left[ \frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}}
\right]\]
here \[\overset{\_}{\mathop{v}}\,\]
= wave number
or \[v={{R}_{H}}\times
c\left[ \frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}} \right]\]
where \[{{R}_{H}}\] is Rydberg
constant for hydrogen. Its value is\[109677.76c{{m}^{-1}}\]
You need to login to perform this action.
You will be redirected in
3 sec
