Atomic Structure Result Question 7-1
7. If the de-Broglie wavelength of the electron in $n^{\text {th }}$ Bohr orbit in a hydrogenic atom is equal to $1.5 \pi a _0$ ( $a _0$ is Bohr radius), then the value of $n / Z$ is
(2019 Main, 12 Jan II)
(a) 1.0
(b) 0.75
(c) 0.40
(d) 1.50
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Answer:
Correct Answer: 7. ( b )
Solution:
- $ \text { Number of waves }=\dfrac{\text { Circumference }}{\text { Wavelength }} \Rightarrow n=\dfrac{2 \pi r}{\lambda} $
$ \therefore \quad 2 \pi r=n \lambda \hspace{15mm} …(i)$
Also, we know that radius $(r)$ of an atom is given by
$r=\dfrac{a_0 n^2}{Z}$
Thus, Eq. (i) becomes
$\begin{gathered} 2 \pi a_0 \dfrac{n^2}{Z}=n \lambda \hspace {15mm}…(ii) \\ \therefore \quad 2 \pi a_0 \dfrac{n^2}{Z}=n\left(1.5 \pi a_0\right)\left[\text { Given, } \lambda=1.5 \pi a_0\right] \\ \dfrac{n}{Z}=\dfrac{1.5 \pi a_0}{2 \pi a_0}=\dfrac{1.5}{2}=0.75 \end{gathered}$
BETA
