Question 14

The wavelength of light from the spectral emission line of sodium is 589 nm. Find the kinetic energy at which

(a) an electron, and

(b) a neutron, would have the same de Broglie wavelength.

Answer

Wavelength of light of a sodium line, λ = 589 nm = 589 × 10^{−9 }m

Mass of an electron, m_{e}= 9.1 × 10^{−31} kg

Mass of a neutron, m_{n}= 1.66 × 10^{−27} kg

Planck’s constant, h = 6.6 × 10^{−34} Js

**(a)** For the kinetic energy K, of an electron accelerating with a velocity v, we have the relation:

*K = 1/2 m _{e}v^{2 } ..... (1)*

We have the relation for de Broglie wavelength as:

Substituting equation (2) in equation (1), we get the relation:

Hence, the kinetic energy of the electron is 6.9 × 10^{−25} J or 4.31 μeV.

**(b)** Using equation (3), we can write the relation for the kinetic energy of the neutron as:

Hence, the kinetic energy of the neutron is 3.78 × 10^{−28} J or 2.36 neV.

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