Question
Using Bohr’s postulates of the atomic model, derive the expression for radius of $n^{th}$ electron orbit. Hence obtain the expression for Bohr’s radius.
✓
Answer
For the electron, we have
Bohr’s Postulate $(\text{mvr} = \frac{\text{nh}}{2\pi})$
$\frac{\text{mv}^{2}}{\text{r}} = \frac{1}{4\pi\in_{o}}\frac{\text{ze}^{2}}{\text{r}^{2}}$
and mvr $ = \frac{\text{nh}}{2\pi}$
$\therefore\text{m}^{2}\text{v}^{2}\text{r}^{2} = \frac{\text{n}^{2}\text{h}^{2}}{4\pi^{2}}$
and $mv^2r = \frac{1}{4\pi\in_{o}}\text{ze}^{2}$
$\therefore\text{r} = \frac{\in_{o}\text{n}^{2}\text{h}^{2}}{\pi\text{ze}^{2}\text{m}}$
Bohr’s radius $($for $n = 1) = \in_{o } \text{h}^{2}/\pi\text{ze}^{2}\text{m}.$
Bohr’s Postulate $(\text{mvr} = \frac{\text{nh}}{2\pi})$
$\frac{\text{mv}^{2}}{\text{r}} = \frac{1}{4\pi\in_{o}}\frac{\text{ze}^{2}}{\text{r}^{2}}$
and mvr $ = \frac{\text{nh}}{2\pi}$
$\therefore\text{m}^{2}\text{v}^{2}\text{r}^{2} = \frac{\text{n}^{2}\text{h}^{2}}{4\pi^{2}}$
and $mv^2r = \frac{1}{4\pi\in_{o}}\text{ze}^{2}$
$\therefore\text{r} = \frac{\in_{o}\text{n}^{2}\text{h}^{2}}{\pi\text{ze}^{2}\text{m}}$
Bohr’s radius $($for $n = 1) = \in_{o } \text{h}^{2}/\pi\text{ze}^{2}\text{m}.$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
The potential difference across the terminals of a battery of emf 12V and internal resistance $2\Omega$ drops to 10V when it is connected to a silver voltameter. Find the silver deposited at the cathode in half an hour. Atomic weight of silver is 107.9g/mole.
→Water level is maintained in a cylindrical vessel upto a fixed height H. The vessel is kept on a horizontal plane. At what height above the bottom should a hole be made in the vessel so that the water stream coming out of the hole strikes the horizontal plane at the greatest distance from the vessel.
Define refractive index of a transparent medium.
A ray of light passes through a triangular prism. Plot a graph showing the variation of the angle of deviation with the angle of incidence.
A ray of light passes through a triangular prism. Plot a graph showing the variation of the angle of deviation with the angle of incidence.
Draw a diagram of magnetic field lines due to any current carrying circular loop.
The circuit arrangement given below shows that when an a.c. passes through the coil A, the current starts flowing in the coil B.
- State the underlying principle involved.
- Mention two factors on which the current produced in the coil B depends.
It is heard sometime that mercury is used in defining the temperature scale because it expands uniformly with temperature. If the temperature scale is not yet defined, is it logical to say that a substance expands uniformly with temperature?
Write applications of the resonance phenomenon in the $L - C - R$ series circuit.
→A tightly-wound, long solenoid has n turns per unit length, a radius r and carries a current i. A particle having charge q and mass m is projected from a point on the axis in a direction perpendicular to the axis. What can be the maximum speed for which the particle does not strike the solenoid ?
In Young’s double $-$ slit experiment using monochromatic light of wavelength $\lambda, $ the intensities of two sources is $I.$ What is the intensity of light at a point where path difference between wave front is $\lambda / 4$ ?
→An electroplating unit plates $3.0g$ of silver on a brass plate in $3.0$ minutes. Find the current used by the unit. The electrochemical equivalent of silver is $1.12 \times 10^{-6}\ kg\ C^{-1.}$
→