Question

How fast must an electron move to have a kinetic energy equal to the photon energy of sodium light at wavelength 590 nm?

Short answer

The speed of the electron is $8.6\times {10}^5\mathrm{m}/\mathrm{s}$.

Step-by-step solution

Describe the expression for the kinetic energy of the electron and energy of the photon

The expression for kinetic energy of the electron is given by,

$K=\frac{1}{2}{\mathrm{mv}}^2$

Here, m is the mass of an electron, and v is the speed of the electron.

Combine the above two equations.

$E=\frac{hc}{\lambda }$

Here, h is the Planck’s constant, c is the speed of the light, and $\lambda$is the wavelength.

Determine the speed of the electron

The energy of the photon is equal to the kinetic energy of the electron.

…… (1)

Substitute below values in equation 1.

$$\begin{gathered} \mathrm{m}=9.1\times {10}^{-3}\mathrm{Kg} \\ \mathrm{\lambda }=590\mathrm{nm} \\ \mathrm{h}=6.626\times {10}^{-34}\mathrm{J}/\mathrm{s} \\ \mathrm{c}=3\times {10}^8\mathrm{m}/\mathrm{s} \end{gathered}$$

Therefore, the speed of the electron is$8.6\times {10}^5\mathrm{m}/\mathrm{s}$