Energy at the atomic scale comes in packets, not a smooth ramp.
Light acts like a wave (it has a wavelength and frequency) and also like a stream of particles called photons, each carrying a fixed packet of energy, E = hν. Electrons in atoms can only sit at specific energies, so they only absorb or emit photons whose energy exactly matches a gap. That's why each element has its own line spectrum, like a barcode.
Speed of light
Photon energy
Takeaway
Shorter λ → higher ν → more energy per photonSlide across the electromagnetic spectrum
Bohr's energy levels and the line spectrum
For hydrogen, En = −2.18 × 10⁻¹⁸ J (1/n²). An electron moving from ni to nf emits or absorbs a photon with |ΔE| = 2.18 × 10⁻¹⁸ J |1/nf² − 1/ni²|. Drops to n = 2 land in the visible range (the Balmer series).
Visible hydrogen emission lines (Balmer series). The highlighted one is your transition, if it's visible.
Brightness doesn't help if the color is wrong.
Shine light on a metal and electrons pop off only if each photon has at least the metal's work function (Φ). A brighter red light just means more photons that are each too weak. This is the experiment that proved light comes in packets. Einstein won his Nobel Prize for it.
Everything moving has a wavelength: λ = h / mv
de Broglie flipped the idea around: if light can act like particles, particles can act like waves. The wavelength only matters when it's comparable to the space the particle moves through. For an electron in an atom, it is. For a baseball, it's absurdly small.
Is this electron address allowed?
Rules
n = 1, 2, 3, … (shell)ℓ = 0 to n − 1 (s=0, p=1, d=2, f=3)
mℓ = −ℓ to +ℓ (which orbital)
mₛ = +½ or −½ (spin)
Counting
Orbitals in a subshell = 2ℓ + 1Orbitals in shell n = n²
Electrons in shell n = 2n²
n = 3 holds 9 orbitals and 18 electrons.
sphere, 1 orbital
dumbbell, 3 orbitals (x, y, z)
cloverleaf (mostly), 5 orbitals
Light and photon problems
Hydrogen transitions and quantum numbers
Germicidal UV is photon-energy math.
A 254 nm UV-C photon carries 4.9 eV, or about 471 kJ/mol. That's more than the energy of a C–C bond (~348 kJ/mol), and it's right around where DNA bases absorb most strongly. Adjacent thymines fuse into dimers, and the organism can't replicate. Visible light at 600 nm carries only ~200 kJ/mol, which isn't enough to break those bonds. The packet size matters, not the brightness, the same lesson as the photoelectric effect.
The same physics runs other plant instruments. IR thermometers read the infrared photons every warm surface emits. Colorimeters and spectrophotometers measure which wavelengths a sample absorbs.
Common mistakes
- Leaving λ in nm in c = λν. Convert to meters first (1 nm = 10⁻⁹ m).
- Thinking a brighter light means higher-energy photons. Brightness is the number of photons. Energy per photon depends only on λ.
- Sign confusion in Bohr problems. Emission (high n → low n) releases energy, so ΔE of the atom is negative. The photon's energy is always positive.
- Allowing ℓ = n. For n = 2, ℓ can only be 0 or 1. There's no 2d.
- Forgetting that J per photon × 6.022 × 10²³ gives J per mole, the unit that compares to bond energies.