a beam of electron 13.0 ev is used to bombard gaseous hydrogen. what series of wavelengths will be emitted?
lecture
what was the solution
To determine the series of wavelengths emitted when a beam of electrons with an energy of 13.0 eV bombards gaseous hydrogen, we need to understand the concept of atomic spectra and the energy levels of hydrogen atoms.
1. Atomic Spectra:
When atoms or molecules are excited, they can emit light or electromagnetic radiation. This emission occurs at specific wavelengths, which create distinct lines in the electromagnetic spectrum. These lines correspond to the transitions between different energy levels within the atom or molecule.
2. Energy Levels of Hydrogen:
The energy levels of a hydrogen atom are described by the Rydberg equation:
1/λ = R_H * (1/n_final^2 - 1/n_initial^2)
Where:
- λ represents the wavelength of light emitted or absorbed.
- R_H is the Rydberg constant for hydrogen (approximately 1.097 × 10^7 m^-1).
- n_initial and n_final are the principal quantum numbers representing the initial and final energy levels, respectively.
3. Energy of an Electron:
The energy of an electron can be converted to wavelength using the equation:
E = hc/λ
Where:
- E is the energy of the electron (in eV).
- h is Planck's constant (approximately 4.136 × 10^-15 eV s).
- c is the speed of light (approximately 3.0 × 10^8 m/s).
Now, let's calculate the series of wavelengths emitted when the electron beam with an energy of 13.0 eV bombards gaseous hydrogen:
Step 1: Determine the wavelength corresponding to the energy of the electron.
Given E = 13.0 eV, h, and c, we can calculate λ.
E = hc/λ
13.0 = (4.136 × 10^-15) × (3.0 × 10^8) / λ
Solving for λ, we find:
λ ≈ 9.563 × 10^-8 m
Step 2: Calculate the corresponding wavelengths for different series.
The wavelengths emitted in the hydrogen spectrum fall into several different series: Lyman, Balmer, Paschen, Brackett, and Pfund. Each series corresponds to different energy level transitions.
By substituting the calculated wavelength (9.563 × 10^-8 m) into the Rydberg equation, we can find the corresponding series:
1/λ = 1.097 × 10^7 * (1/n_final^2 - 1/n_initial^2)
- For Lyman Series (Ultraviolet): n_initial = 1
- n_final = 2, 3, 4, ...
- For Balmer Series (Visible): n_initial = 2
- n_final = 3, 4, 5, ...
- For Paschen Series (Infrared): n_initial = 3
- n_final = 4, 5, 6, ...
- For Brackett Series (Infrared): n_initial = 4
- n_final = 5, 6, 7, ...
- For Pfund Series (Infrared): n_initial = 5
- n_final = 6, 7, 8, ...
Substitute the initial energy level (n_initial) values and solve the equation to find the wavelengths for each series.
Please note that this is a simplified explanation, and the actual configuration of energy levels and transitions is more complex. Advanced models take into account factors like fine structure and quantum mechanics to provide more accurate predictions.