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### College Physics (Urone)

Book edition 1st Edition
Author(s) Paul Peter Urone
Pages 1272 pages
ISBN 9781938168000

# Violet light of wavelength $${\rm{400 nm}}$$ ejects electrons with a maximum kinetic energy of $${\rm{0}}{\rm{.860 eV}}$$ from sodium metal. What is the binding energy of electrons to sodium metal?

The binding energy of electrons to sodium metal $$2.24\,{\rm{eV}}$$

See the step by step solution

## Step 1: Given data

Given,

Wavelength is, $$\lambda = 400\,{\rm{nm}} = 400 \times {10^{ - 9}}{\rm{m}}$$.

Kinetic energy is, $${\rm{KE}} = 0.860\,{\rm{eV}}$$.

We also know that: Planks constant $$h = 4.13 \times {10^{ - 15}}\,{\rm{eV}}{\rm{.s}}$$

Speed of light $$c = 3 \times {10^8}\,{\rm{m/s}}$$

## Step 2: The longest-wavelength EM radiation can eject an electron

The kinetic energy of the electron is given by

$$K{E_e} = hf - BE$$ ...(1)

Here $$K{E_e}$$ is the kinetic energy, $$h$$ is the plank constant, $$f$$is the frequency of the EM radiation and $$BE$$ is the binding energy.

Now we know that the wavelength of EM radiation is given by

$$\lambda = \frac{c}{f}$$ ...(2)

Where $$c$$ is the speed of light.

So equation becomes,

$$K{E_e} = \frac{{hc}}{\lambda } - BE$$ ...(3)

## Step 3: Calculate the binding energy of electrons to sodium metal

Hence the binding energy is expressed as,

$$BE = \frac{{hc}}{\lambda } - KE$$

Substitute all the value in the above equation

\begin{aligned}{}BE &= \frac{{\left( {4.13 \times {{10}^{ - 15}}\,{\rm{eV}}{\rm{.s}}} \right)\left( {3.00 \times {{10}^8}\,{\rm{m}}{{\rm{s}}^{{\rm{ - 1}}}}} \right)}}{{400 \times {{10}^{ - 9}}\,{\rm{m}}}} - (0.860\,{\rm{eV}})\\ &= 2.24\,{\rm{eV}}\end{aligned}

Therefore, the binding energy of electrons to sodium metal $$2.24\,{\rm{eV}}$$