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Question: Answered & Verified by Expert
In hydrogen atom, the de-Broglie wavelength of an electron in the second Bohr orbit is [Given that, Bohr radius, $\mathrm{a}_0=52.9 \mathrm{pm}$ ]
ChemistryStructure of AtomNEETNEET 2019 (Odisha)
Options:
  • A $211.6 \mathrm{pm}$
  • B $211.6 \pi \mathrm{pm}$
  • C $52.9 \pi \mathrm{pm}$
  • D $105.8 \mathrm{pm}$
Solution:
1316 Upvotes Verified Answer
The correct answer is: $211.6 \pi \mathrm{pm}$
According to Bohr,
$$
\begin{aligned}
& \mathrm{mvr}=\frac{\mathrm{nh}}{2 \pi} \\
& 2 \pi \mathrm{r}=\frac{\mathrm{nh}}{\mathrm{mv}}=\mathrm{n} \lambda
\end{aligned}
$$
..(i) $\left[\because \lambda=\frac{\mathrm{h}}{\mathrm{mv}}\right]$


where, $\mathrm{r}=$ radius, $\lambda=$ wavelength
$\mathrm{n}=$ number of orbit
Also,
$$
r=\frac{a_0 n^2}{\mathrm{Z}}
$$
where, $\mathrm{a}_0=$ Bohr radius $=52.9 \mathrm{pm}$
$\mathrm{Z}=$ atomic number
On substituting the value of ' $r$ ' from Eq. (ii) to Eq. (i), we get
$$
\begin{aligned}
\mathrm{n} \lambda & =\frac{2 \pi \mathrm{n}^2 \mathrm{a}_0}{\mathrm{Z}} \Rightarrow \lambda=\frac{2 \pi \mathrm{na}_0}{\mathrm{Z}} \\
\lambda & =2 \pi \times 2 \times 52.9 \quad[\because \mathrm{n}=2, \mathrm{Z}=1] \\
& =211.6 \pi \mathrm{pm}
\end{aligned}
$$

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