Question

Calculate the wavelengths of the photon having an energy of 1 electron volt in Angstroms.

Answer 1

Hint: Wavelength is given as the distance between two corresponding points on the adjacent waves. Photons are tiny particles that do not have any charge and travel with the speed of light.

Complete step by step answer:
Wavelength is given by the distance between two corresponding points on adjacent waves. It is denoted by the Greek letter ‘lambda’$\lambda $ .
It can be measured in centimeters, meters or nanometers.
Wavelength is related to energy and frequency by:
$E = h\upsilon $ …1
where $ E = $ energy ,$h = $Planck’s constant $\upsilon = $velocity .
But $\upsilon = \dfrac{c}{\lambda }$
Substituting this value in equation 1 we get,
$E = \dfrac{{hc}}{\lambda }$ ….2
where E$ = $ energy ,
$h = $ Planck’s constant
$c = $ speed of light
$\lambda = $ wavelength
So we will use this formula to calculate the wavelength of photons having energy $1eV$ in Angstrom.
Photons do not have mass but have energy. The energy of the photon is inversely proportional to the wavelength.
Angstrom: it is the unit of length that is used to measure the size of atoms, molecules and electromagnetic wavelengths.
$1{A^ \circ } = 1 \times {10^{ - 10}}m$ .
$1eV$is defined as the energy gained by an electron having the potential difference of $1V$ .
The value of $1eV = 1.602 \times {10^{ - 19}}J$ .
Now we will calculate the wavelength of the photon having energy $1eV$ using the formula from equation 2.
Given data: $1eV = 1.602 \times {10^{ - 19}}J$ , $c = 3 \times {10^8}m/s$, $h = 6.62 \times {10^{ - 34}}Js$
$E = \dfrac{{hc}}{\lambda }$
rearranging the equation we get,
$\lambda = \dfrac{{hc}}{E}$
Substituting the values we get,
$\lambda = \dfrac{{6.62 \times {{10}^{ - 34}} \times 3 \times {{10}^8}}}{{1.602 \times {{10}^{ - 19}}}}$
$\lambda = \dfrac{{19.86 \times {{10}^{ - 26}}}}{{1.602 \times {{10}^{ - 19}}}}$
$\lambda = 12.397 \times {10^{ - 7}}m$
But, $1{A^ \circ } = 1 \times {10^{ - 10}}m$
Therefore, $\lambda = \dfrac{{12.397 \times {{10}^{ - 7}}}}{{1 \times {{10}^{ - 10}}}}$
$\lambda = 12.397 \times {10^3}{A^ \circ }$

So the wavelength of photon having energy $1eV$ is $12.397 \times {10^3}{A^ \circ }$ .

Note: In order to convert wavelengths from meter to centimeters or from meter to nanometers, the conversions are given below as follows:
$1m = {10^2}cm$
$1m = {10^9}nm$
Wavelength is inversely proportional to frequency which means that longer the wavelength than frequency will be lower.
Wavelengths are widely used in spectroscopy.