Question

The de - Broglie wavelength associated with the electron in the $n=4$ level is :

• A. half of the de-Broglie wavelength of the electron in the ground state
• B. four times the de-Broglie wavelength of the electron in the ground state
• C. $1/4^{th}$ of the de-Broglie wavelength of the electron in the ground state
• D. two times the de-Broglie wavelength of the electron in the ground state

Answer 1

The correct option is B four times the de-Broglie wavelength of the electron in the ground state

De-Broglie wavelength is given as

$\lambda =\frac{h}{mv}$...(i) where h is planck constant

The velocity of electron in Bohr model is given as

$v=\frac{h}{2\pi m_e{a}_{\circ }n}$...(i)
in the ground state n=1

$v_{n=1}=\frac{h}{2\pi m_e{a}_{\circ }}$

$v_{n=4}=\frac{h}{2\pi m_e{a}_{\circ }4}$

$v_{n=4}=\frac{v_{n=1}}{4}$...(iii)

from equation (i)

${\lambda }_{n=4}=\frac{h}{m_e{v}_{n=4}}$

substituting value of $v_{n=4}$ from equation (iii)

${\lambda }_{n=4}=\frac{h}{m_e\frac{v_{n=1}}{4}}$

${\lambda }_{n=4}=4{\lambda }_{n=1}$

Hence correct answer will be option B.