Electron exhibiting a particle wavelength of 6.81 x 10-9 are emitted from hydrogen atoms after interaction with high energy photons. Assuming that the electrons in the hydrogen atom were in their ground state prior to interaction, determine the wavelength of the photon which generated the free electrons

To determine the wavelength of the photon that generated the free electrons, we can use the concept of conservation of energy.

The energy of the photon is equal to the difference in energy between the initial and final states of the electron. In the ground state of a hydrogen atom, the electron is in the lowest energy level, also known as the n = 1 level.

The energy of the electron in the nth energy level of a hydrogen atom can be given by the formula:

E = -13.6 eV / n²

where E is the energy in electron volts (eV) and n is the principal quantum number.

Since the electron is emitted after interaction with a high-energy photon, its energy increases to become free. Therefore, we can say that the energy of the electron after interaction is zero, as it is no longer bound to the atom.

Setting the initial and final energies equal to each other, we can solve for the principal quantum number, n:

0 eV = -13.6 eV / n²

Simplifying this equation, we get:

n² = -13.6 eV / 0 eV

Since we cannot divide by zero, we cannot solve this equation directly. However, we know that the electron is in the ground state prior to interaction, which means n = 1.

Now, we can find the wavelength (λ) of the photon that generated the free electron using the equation:

λ = c / ν

Where λ is the wavelength, c is the speed of light (2.998 x 10^8 m/s), and ν is the frequency.

To find the frequency, we can use the formula:

ν = E / h

Where ν is the frequency, E is the energy, and h is Planck's constant (6.626 x 10^-34 J·s).

Converting the energy from electron volts (eV) to joules (J):

E = -13.6 eV

E = -13.6 eV x 1.6 x 10^-19 J/eV

E = -2.176 x 10^-18 J

Now, we can calculate the frequency (ν):

ν = (-2.176 x 10^-18 J) / (6.626 x 10^-34 J·s)

ν = 3.286 x 10^15 Hz

Finally, we can calculate the wavelength (λ):

λ = (2.998 x 10^8 m/s) / (3.286 x 10^15 Hz)

λ ≈ 9.13 x 10^-8 m

Therefore, the wavelength of the photon that generated the free electrons is approximately 9.13 x 10^-8 meters.