Bohr’s quantum conditions explain why atoms emit and absorb only light of particular wavelengths.
Basic Knowledge
Bohr’s model of the hydrogen atom has two assumptions:
① Quantum condition: Electrons occupy only circular orbits, and the angular momentum of an orbit is an integer multiple of h/(2π):mvr = nh/(2π) = nℏ（n = 1, 2, 3, ...: principal quantum number）
② Frequency condition: Photons are emitted or absorbed during transitions between energy levels:hf = E_m − E_n（E_m > E_n）
Energy levels of the hydrogen atom:E_n = −13.6/n² eV（n = 1: ground state −13.6 eV）
When an electron transitions from n = m to n = n（m > n）, the frequency of the emitted light is:hf = E_m − E_n = 13.6(1/n² − 1/m²) eV
Hydrogen spectral series: Lyman series (ultraviolet), Balmer series (visible), Paschen series (infrared)
Energy level (energy level)（a discrete value of energy that an electron in an atom can have）
Ground state and excited state（the lowest-energy state / a higher-energy state）
Ionization energy（the energy required to remove an electron from n = 1 to infinity. Hydrogen = 13.6 eV）
Spectral series（classified by the value of the destination n. The Balmer series, n = 2, is visible light）
ℏ (h-bar)（ℏ = h/(2π): reduced Planck constant）
Further Exploration (Background and Significance)
E_n = −13.6/n² can be derived by combining the mechanics of circular orbits (Coulomb force = centripetal force) with the quantum condition. The electron’s total energy (kinetic + potential) is negative, and E_n approaches 0 as n increases (ionization occurs at n → ∞).
Bohr’s model completely explained the experimental spectrum of hydrogen, but it cannot be applied to atoms with many electrons. The complete solution is given by the Schrödinger equation (quantum mechanics).
A laser (LASER) amplifies light of the same phase and frequency by causing atoms in excited states to undergo stimulated emission. Understanding spectra (differences between energy levels) is fundamental to this process.
- E_n = −13.6/n² eV
- n = 1 is the ground state (most stable)
- Photon emission: higher level → lower level
- hf = E_m − E_n
- Balmer series (transitions to n = 2) is visible light
- Ionization energy: 13.6 eV
- Quantum condition: mvr = nℏ
Notes (Easily Confused Points)
① E_n is negative. As n increases, E_n becomes larger (closer to 0). ② Photons are emitted in transitions from higher to lower levels. ③ Absorption is the reverse (low → high). ④ Ionization means moving to n → ∞.
Practice
- Find the photon energy in eV when a hydrogen atom transitions from n = 3 to n = 2.
- What is the ionization energy of hydrogen in eV?
- Explain what kind of transitions produce the Balmer series.