treat a hydrogen atom as a one dimensional box of length 1.50pm (the approximate diameter of an atom), containing an electron and predict the wavelength of the radiation emitted when the electron falls to the lower energy level from the next highest energy level. use the simplified form of particles in the box
The energy levels of a particle in a one-dimensional box are given by the equation:
E_n = (n^2 * h^2) / (8mL^2)
where E_n is the energy level, n is the quantum number (n=1, 2, 3,...), h is the Planck constant, m is the mass of the electron, and L is the length of the box.
For a hydrogen atom in a 1D box with L = 1.50 pm, the energy levels are:
E_1 = 3.63 x 10^-18 J
E_2 = 1.45 x 10^-17 J
E_3 = 3.27 x 10^-17 J
When the electron falls from the n=2 energy level to the n=1 energy level, the energy emitted as radiation is equal to the difference in energy levels:
ΔE = E_2 - E_1 = 1.09 x 10^-17 J
Using the equation for the energy of a photon:
E = hc/λ
where E is the energy of the photon, h is the Planck constant, c is the speed of light, and λ is the wavelength of the radiation emitted.
Substituting in the value for ΔE, we can solve for the wavelength of the radiation emitted:
1.09 x 10^-17 J = (6.63 x 10^-34 J s * 3.00 x 10^8 m/s) / λ
λ = 1.83 x 10^-6 m
Therefore, the wavelength of the radiation emitted when the electron falls from the n=2 energy level to the n=1 energy level in a hydrogen atom in a 1D box with a length of 1.50 pm is 1.83 μm.