Question

(a) State Bohr’s postulate to define stable orbits in hydrogen atom. How does de Broglie’s hypothesis explain the stability of these orbits ?
(b) A hydrogen atom initially in the ground state absorbs a photon which excites it to the n = 4 level. Estimate the frequency of the photon.

Answer 1

Answer:

Part a)

As per Bohr's postulate of stationary orbit we know that

$mvR = \frac{nh}{2\pi}$

Part b)

Frequency of photon which is absorbed by the electron is

$\nu = 3.1 \times 10^{15} Hz$

Explanation:

Part a)

As per Bohr's postulate of stationary orbit we know that if an electron is in stationary orbit then the angular momentum of the electron in that stationary orbit must be integral multiple of $\frac{h}{2\pi}$

so we have

$mvR = \frac{nh}{2\pi}$

Now as per de-Broglie hypothesis we know that de-broglie wavelength of an electron is given as

$\lambda = \frac{h}{mv}$

now if wave associated with electron forms stationary waves in its orbit then there is no energy loss

so we have

$2\pi r = n\lambda$

$2\pi r = n\frac{h}{mv}$

so we have

$mvr = \frac{nh}{2\pi}$

Part b)

As hydrogen atom absorbs photon in ground state and reaches to n = 4

so here we have

$\Delta E = -\frac{13.6}{4^2} + \frac{13.6}{1^2}$

$\Delta E = 13.6(1 - \frac{1}{16})$

$\Delta E = 12.75 eV$

Now this is energy of photon which is absorbed by an electron

so we will have

$h\nu = 12.75$

$\nu = \frac{12.75 \times (1.6 \times 10^{-19})}{6.6 \times 10^{-34}}$

$\nu = 3.1 \times 10^{15} Hz$