Q55E Verify that equation (4-19) foll... [FREE SOLUTION]

Chapter 1: Q55E (page 1)

Verify that equation (4-19) follows from (4-16) and (4-18).

Short Answer

Expert verified

The proof of the equation $r = \frac{(4 蟺蔚_{0}) {鈩�}^{2}}{m e^{2}} n^{2}$ is stated below.

Step by step solution

Velocity in terms of charge and permittivity of free space.

According to Bohr model, the expression for the velocity of an electron, rotating around a proton, in terms of charge and permittivity of free space is given by,

$v^{2} = \frac{e^{2}}{4 {蟺蔚}_{o} mr} 路路路路路路路路路路路路路路路路路路路路路路路路 (1)$

According to the Bohr model, only those orbits are available for the revolution of electrons, for which the angular momentum is an integral multiple of $鈩�$ . Where,

and $h$ is Planck鈥檚 constant $(h = 6.626 脳 10^{- 34} \frac{{kgm}^{2}}{s})$ .

Thus, for an electron of mass m moving in an orbit of radius r and moving with velocity v, the angular momentum L is given as-

$L = mvr L = n 鈩� v = \frac{n 鈩�}{mr} 路路路路路路路路路路路路路路路路路路路路路路路路路 (2)$

Proof the equation r=4πε0ℏ2me2n2 with the help of equation (1)

Consider an electron, having charge $e$ , rotating around a proton in a circular orbit of radius $r$ , for which the electron of mass $m$ has angular momentum $L$ and is moving with velocity $v$ .

Using equations (1) and (2), the following relation can be obtained-

$v^{2} = \frac{e^{2}}{4 蟺蔚_{0} m r}$

$(\frac{n^{2} {鈩�}^{2}}{m^{2} r^{2}}) = \frac{e^{2}}{4 蟺蔚_{0} m r}$