Search any question & find its solution
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, $\mathrm{a}_0=52.9 \mathrm{pm}$ ]
Options:
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
$$
\begin{aligned}
\mathrm{n} \lambda & =\frac{2 \pi \mathrm{n}^2 \mathrm{a}_0}{\mathrm{Z}} \Rightarrow \lambda=\frac{2 \pi \mathrm{na}_0}{\mathrm{Z}} \\
\lambda & =2 \pi \times 2 \times 52.9 \quad[\because \mathrm{n}=2, \mathrm{Z}=1] \\
& =211.6 \pi \mathrm{pm}
\end{aligned}
$$
$$
\begin{aligned}
& \mathrm{mvr}=\frac{\mathrm{nh}}{2 \pi} \\
& 2 \pi \mathrm{r}=\frac{\mathrm{nh}}{\mathrm{mv}}=\mathrm{n} \lambda
\end{aligned}
$$
..(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
$$
\begin{aligned}
\mathrm{n} \lambda & =\frac{2 \pi \mathrm{n}^2 \mathrm{a}_0}{\mathrm{Z}} \Rightarrow \lambda=\frac{2 \pi \mathrm{na}_0}{\mathrm{Z}} \\
\lambda & =2 \pi \times 2 \times 52.9 \quad[\because \mathrm{n}=2, \mathrm{Z}=1] \\
& =211.6 \pi \mathrm{pm}
\end{aligned}
$$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.