A photon interacts with a ground state electron in a hydrogen atom and is absorbed. The electron is ejected from the atom and exhibits a de Broglie wavelength of 5.908×10−10 m. Determine the frequency (in hz) of the interacting photon.
To determine the frequency of the interacting photon, we can use the equation that relates the energy of a photon to its frequency:
E = hf
where E is the energy of the photon, h is the Planck's constant (6.626 x 10^-34 J s), and f is the frequency of the photon.
Since the photon is absorbed by the electron in the hydrogen atom, its energy is used to eject the electron from the ground state. Therefore, the energy of the photon is equal to the ionization energy of the hydrogen atom.
The energy of a photon can also be related to its wavelength using the equation:
E = hc/λ
where c is the speed of light (3 x 10^8 m/s) and λ is the wavelength of the photon.
We are given the de Broglie wavelength (λ) of the ejected electron, which is 5.908 x 10^-10 m. The de Broglie wavelength is calculated using the equation:
λ = h/mv
where h is Planck's constant, m is the mass of the electron, and v is the velocity of the electron. Since the electron is ejected, its velocity is determined by the energy of the photon.
To find the velocity of the electron, we can use the equation for the energy of a photon:
E = (1/2)mv^2
where E is the energy of the photon, m is the mass of the electron, and v is the velocity of the electron.
From the given wavelength (λ), we can solve the de Broglie equation for the velocity (v):
v = h/(mλ)
Now we have the velocity of the electron, we can substitute it back into the equation for the energy of a photon:
E = (1/2)m(h/(mλ))^2
Simplifying the equation, we get:
E = (h^2)/(2mλ^2)
Since the energy of the photon is equal to the ionization energy of the hydrogen atom, we can equate the two equations:
hc/λ = (h^2)/(2mλ^2)
Simplifying the equation, we find:
c = (h/(2mλ))
Now we can substitute the given values:
c = (6.626 x 10^-34 J s)/((2 x 9.1 x 10^-31 kg)(5.908 x 10^-10 m)^2)
Now we can solve for c to find the speed of light.
c ≈ 3 x 10^8 m/s
Finally, we can use the speed of light and the wavelength (λ) to calculate the frequency (f) of the photon:
f = c/λ
Substituting the values:
f = (3 x 10^8 m/s)/(5.908 x 10^-10 m)
f ≈ 5.08 x 10^17 Hz
Therefore, the frequency of the interacting photon is approximately 5.08 x 10^17 Hz.