Will a photon of light of wavelength 480 nm excite an electron in the hydrogen atom from the n = 1 level to the n = 2 level?
How would I figure this out?
Energy needed to excite an electron from the n = 1 to the n = 2 level is
E = 2.18E-18 J(1/1^2 - 1/2^2)
Compare that energy needed with what you have in a 480 nm beam of light. E = hc/wavelength
Post your work if you get stuck.
To determine whether a photon of light with a wavelength of 480 nm can excite an electron in the hydrogen atom from the n = 1 level to the n = 2 level, you need to compare the energy of the photon with the energy difference between these two levels.
The energy of a photon is given by the equation E = hc/λ, where E is the energy, h is Planck's constant (6.626 x 10^-34 J·s), c is the speed of light (approximately 3.0 x 10^8 m/s), and λ is the wavelength of the photon.
To calculate the energy of the photon with a wavelength of 480 nm, you can plug the values into the equation:
E = (6.626 x 10^-34 J·s) x (3.0 x 10^8 m/s) / (480 x 10^-9 m)
E ≈ 4.13 x 10^-19 J
Now, you need to calculate the energy difference between the n = 1 and n = 2 levels in the hydrogen atom. The formula to calculate the energy difference between two energy levels in a hydrogen atom is given by:
ΔE = -13.6 eV x (1/n2^2 - 1/n1^2)
Where ΔE is the energy difference, -13.6 eV is the ionization energy of hydrogen, and n1 and n2 are the quantum numbers representing the initial and final energy levels, respectively. Note that 1 eV = 1.6 x 10^-19 J.
Plugging in the values for n1 = 1 and n2 = 2:
ΔE = -13.6 eV x (1/2^2 - 1/1^2)
ΔE = -13.6 eV x (1/4 - 1/1)
ΔE = -13.6 eV x (1/4 - 1)
ΔE = -13.6 eV x (-3/4)
ΔE ≈ 10.2 eV
To compare the energy of the photon with the energy difference between the levels, convert the energy of the photon to eV:
Energy of the photon ≈ 4.13 x 10^-19 J / (1.6 x 10^-19 J/eV) ≈ 2.58 eV
Comparing the energy of the photon (2.58 eV) with the energy difference between the n = 1 and n = 2 levels (10.2 eV), you can see that the energy of the photon is smaller. Therefore, a photon with a wavelength of 480 nm does not have enough energy to excite an electron in the hydrogen atom from the n = 1 level to the n = 2 level.