Ejection of Electrons from Hydrogen by Incident Photons Light of wavelength 80 nm is incident on a sample of hydrogen gas, resulting in electrons being ejected from the atoms. N.B. The energy of these photons is much smaller than the rest energy of the electron, so you don't need relativistic expressions. Assume that the photon is destroyed in this process, leaving only the electron moving away from a stationary hydrogen ion.
a. With the gas in thermal equilibrium at room temperature, what would be the maximum kinetic energy of ejected electrons?
b. When an intense light source of the same wavelength is shone on the gas sample, a small number of electrons with kinetic energy as much as 10.2 eV greater than you calculated in part a. Explain how this can be.

a) [tex]K_{max}[/tex] = 1.9 eV = 3.04 10⁻¹⁹ J,b ) This means that some electrons are at the first excited level of the hydrogen atom, which is highly likely as the temperature rises.

Explanation:

a) To calculate the maximum kinetic energy of the expelled electrons let's use the relationships of the photoelectric effect

[tex]K_{max}[/tex]= h f - Φ

Where K is the kinetic energy, h the Planck constant that is worth 6.63 10⁻³⁴ Js, f the frequency and Φ the work function

The speed of light is related to wavelength and frequency

c = λ f

Let's analyze the work function, it is the energy needed to start an electron from a metal, in this case to start an electron from a hydrogen atom its fundamental energy is needed, so

Φ= E₀ = 13.6 eV

let's replace and calculate the energy of the incident photon

E = h c / λ

E = 6.63 10⁻³⁴ 3 10⁸/80 10⁻⁹

E = 2,486 10⁻¹⁸ J

Let's reduce to eV

E = 2,486 10⁻¹⁸ (1 eV / 1.6 10⁻¹⁹)

E = 15.5 eV

Now we can calculate the kinetic energy

[tex]K_{max}[/tex]= h c / f - fi

[tex]K_{max}[/tex] = 15.5 -13.6

[tex]K_{max}[/tex] = 1.9 eV

b) Extra energy = 10.2 eV

The total kinetic energy of electrons is

Total kinetic energy = 1.9 +10.2 = 12.1 eV

For the calculation we are assuming that all the electors are in the hydrogen base state, but for temperatures greater than 0K some electors may be in some excited state, so less energy is needed to tear them out of hydrogen atom.

Let's analyze this possibility

ΔE = E photon - Total kinetic energy electron

ΔE = 15.5 - 12.1

ΔE = 3.4 eV

If we use the Bohr ratio for the hydrogen atom

[tex]E_{n}[/tex] = 13.606 / n2

n = √ 13.606 / En

n = √ (13606 / 3.4)

n = 2

This means that some electrons are at the first excited level of the hydrogen atom, which is highly likely as the temperature rises.