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Bohr Model Of The Hydrogen Atom - Practice Questions & MCQ

Edited By admin | Updated on Sep 25, 2023 25:23 PM | #NEET

Syllabus

Quick Facts

Radius of orbit and velocity of electron is considered one the most difficult concept.
Bohr's Model of hydrogen atom is considered one of the most asked concept.
61 Questions around this concept.

Solve by difficulty

EASY

In which of the following systems will the radius of the first orbit $(n=1)$ be minimum?

Hydrogen atom

Doubly ionized lithium

Singly ionized helium

Deuterium atom

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Concepts Covered - 2

Bohr's Model of hydrogen atom

Bohr's Model of hydrogen atom:

Bohr proposed a model for hydrogen atom which is also applicable for some lighter atoms in which a single electron revolves around a stationary nucleus of positive charge $Z_e$ (called hydrogen like atom)
Bohr's model is based on the following postulates-

(1). An atom consists of positively charged nucleus responsible for almost the entrie mass of the assumption is retention
of Rutherford model.
(2). The electrons revive around the nucleus in cerrain permitted circular of an infinite radii.
(3). The permitted orbits are those for which the angular momentum of an electron is an intergral muttiple of $\frac{h}{2\pi}$ where h is the
Planck's constant If m is the mass and v is the veloctron in a permitted ortit of radius r, then
$L=m v r=\frac{n h}{2 \Pi} ; n=1,2,3...... \infty$

(4). An electron can transition from a non-radiating orbit to another of a lower energy level. In doing so, a photon is emitted whose energy is equal to the energy difference between the two states. Hence, the frequency of the emitted photon is:

$h_v = E_i - E_f$

Ei is the energy of the initial state and Ef is the energy of the final state. Also, Ei > Ef.

Radius of orbit and velocity of electron

Radius of orbit and velocity of electron

Radius of orbit : For an electron around a stationary nucleus the electrostatics force of attraction provides the neccesary centripetal force.

$\begin{array}{l}{\text { ie. } \frac{1}{4 \pi \varepsilon_{0}} \frac{(Z e) e}{r^{2}}=\frac{m v^{2}}{r} \quad \cdots \text { (i) }} \\ \\ {\text { also } m v r=\frac{n h}{2 \pi}}\end{array}$

From equation (i) and (ii) radius of r orbit

$\begin{array}{l}{r_{n}=\frac{n^{2} h^{2}}{4 \pi^{2} k Z m e^{2}}=\frac{n^{2} h^{2} \varepsilon_{0}}{m n Z e^{2}}=0.53 \frac{n^{2}}{Z} A \quad\left(k=\frac{1}{4 \pi \varepsilon_{0}}\right)} \\ \\ {\Rightarrow r_{n} \propto \frac{n^{2}}{Z}}\end{array}\\ \implies r_n=0.53 \frac{n^2}{Z} A^0$

Speed of electron:

From the above relations, speed of electron in $n^{th}$ orbit can be calculated as
$v_{n}=\frac{2 \pi k Z e^{2}}{n h}=\frac{Z e^{2}}{2 \varepsilon_{0} n h}=\left(\frac{c}{137}\right) \frac{Z}{n}=2.2 \times 10^{6} \frac{Z}{n} m / \text {sec}$