Single Choice

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

Ahalf of the de-Broglie wavelength of the electron in the ground state
Bfour times the de-Broglie wavelength of the electron in the ground state
Correct Answer
C$$1/4^{th}$$ of the de-Broglie wavelength of the electron in the ground state
Dtwo times the de-Broglie wavelength of the electron in the ground state

Solution

De-Broglie wavelength is given as

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

The velocity of electron in Bohr model is given as

$$v=\dfrac{h}{2\pi m_e a_\circ{} n}$$...(i)

in the ground state n=1

$$v_{n=1}=\dfrac{h}{2\pi m_e a_\circ{} }$$

$$v_{n=4}=\dfrac{h}{2\pi m_e a_\circ{} 4}$$

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

from equation (i)

$$\lambda_{n=4}=\dfrac{h}{m_e v_{n=4} }$$

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

$$\lambda_{n=4}=\dfrac{h}{m_e \dfrac{ v_{n=1}}{4} }$$

$$\lambda_{n=4}=4 \lambda_{n=1}$$

Hence correct answer will be option B.

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