Question: Q. 16. (i) If electron in the atom is replaced by a particle (muon) having the same charge but mass about 200 times as that of the electron to form amuonic atom, how would : (i) the radius and (ii) the ground state energy of this be affected?
(ii) Calculate the wavelength of the first spectral line in the corresponding Lyman series of this atom.
A [Delhi Comptt. I, II, III 2012]
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Solution:
Ans. (i) Charged muon of mass about $200 m_{e}$, revolves around a proton.
In Bohr’s atomic model
$$ \begin{aligned} & & & \propto \frac{1}{m} \ & \text { So, } & \frac{r_{\mu}}{r_{e}} & =\frac{m_{e}}{m_{\mu}} \ & \text { or } & \frac{r_{\mu}}{r_{e}} & =\frac{1}{200} \ & \text { or } & 200 r_{\mu} & =r_{e} \ & \text { or } & r_{\mu} & =\frac{r_{e}}{200} \end{aligned} $$
i.e., the radius reduces to $\frac{1}{200}$ times.
Again, in Bohr’s atomic model
So
$$ \frac{E_{\mu}}{E_{e}}=\frac{m_{\mu}}{m_{e}}=\frac{200}{1} $$
$E_{\mu}=200 E_{e}$
i.e. the ground state energy increases by 200
times.
(ii) Eor first spectral line of Lyman series,
$$ n_{1}=1, n_{2}=2 $$
Difference of energy of electron in hydrogen atom when $n_{1}=1$ & $n_{2}=2$
$$ (\text { Lyman Series })=10.2 \mathrm{eV} $$
Hence, for muonic atom this energy
$$ \begin{aligned} & =10.2 \times 200 \mathrm{eV} \ \mathrm{E} & =\frac{12375}{\lambda(\text { in } \AA)}(\text { in eV }) \ \text { So, } \quad \lambda & =\frac{12375}{10.2 \times 200 \mathrm{eV}} \AA \ & =6.1 \AA \ & =0.61 \mathrm{~nm} \end{aligned} $$
