Atomic Spectra

Chem 3240 · Lecture 1.5

Davit Potoyan

Spectroscopy: Light as a Fingerprint

Spectroscopy: interaction of matter and light
Heated atoms emit at characteristic frequencies
Spectrum is unique per element: an atomic fingerprint
Reveals structure and composition

Discrete Lines, Not a Continuum

Solar spectrum shows dark and bright lines
Lines identify elements in the Sun’s atmosphere
Discrete lines are impossible in classical mechanics
A puzzle awaiting a new physics

The Rydberg Formula

Balmer (1885) fit part of the hydrogen spectrum. Rydberg generalized it:

\tilde{\nu} = R_H\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)

R_H = 1.097 \times 10^7 \ \text{m}^{-1} (Rydberg constant)
n_1 = 1,2,3,... and n_2 = n_1+1, n_1+2,...
Fits the data beautifully, but offers no physics
Why should integers govern atomic light?

Spectral Series

Each series = all transitions to one lower level
Lyman: n_1=1
Balmer: n_1=2
Paschen: n_1=3
Named after their discoverers

Bohr’s Model (1913)

Electron in circular orbits around a fixed proton
New quantization rule stops the spiral into the nucleus
Orbit must hold an integer number of standing waves, n=1,2,3,...
Yields discrete energy levels labeled by n

Quantized Angular Momentum

Fit an integer number of de Broglie waves around the orbit:

2\pi r = n \lambda_e, \qquad \lambda_e = \frac{h}{m_e v}

Substituting gives the quantization condition:

m_e v r = \frac{n h}{2\pi} = n \hbar

m_e v r is the angular momentum
Bohr: angular momentum is quantized in units of \hbar

Circular Motion: What Actually Balances

\vec{v} always tangent, \vec{a} always inward
Magnitudes fixed, only directions turn
Lab frame: Coulomb pull is unbalanced, it supplies a = v^2/r
Rotating frame: centrifugal cancels it
Each component is simple harmonic motion

From Orbits to Energy Levels

Coulomb pull balances centrifugal force, \dfrac{e^2}{4\pi\varepsilon_0 r^2} = \dfrac{m_e v^2}{r}, combined with m_e v r = n\hbar:

r_n = n^2 a_0

a_0 = \frac{4\pi \varepsilon_0 \hbar^2}{m_e e^2} \approx 0.529 \,\text{Å}

E_n = -\frac{m_e e^4}{8 \varepsilon_0^2 h^2}\cdot\frac{1}{n^2} = -\frac{13.6\ \text{eV}}{n^2}

Bohr radius a_0: the length scale of atoms
Negative energy: the electron is bound; ionization from n=1 costs 13.6 eV
Orbits grow as n^2, energies close up as 1/n^2

Rydberg Constant from First Principles

Photon energy of a transition, \tilde{\nu} = \nu/c:

\tilde{\nu} = R_H \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)

Now R_H is derived, not fitted:

R_H = \frac{m_e e^4}{8 \varepsilon_0^2 c h^3}

Bohr’s model explains the empirical Rydberg formula
The mysterious integers are quantum numbers

Lines are Jumps Between Levels

A photon carries away exactly E_{n_2} - E_{n_1} = h\nu
Each series shares its lower level
Lyman ends on n=1 (UV), Balmer on n=2 (visible), Paschen on n=3 (IR)
Lines pile up at the series limit, the ionization edge

Explore the Series

Slide the lower level: the whole series moves between UV, visible and IR.

viewof n1 = Inputs.range([1, 4], {step: 1, value: 2, label: "lower level n₁"})

lamColor = function (w) {
  if (w < 380 || w > 750) return "#8c8c8c";          // invisible: grey
  if (w < 440) return "#7b2fbe";
  if (w < 490) return "#2b6fd6";
  if (w < 510) return "#12b5a6";
  if (w < 580) return "#3aa93a";
  if (w < 645) return "#e0a521";
  return "#C8102E";
}

{
  const lines = d3.range(n1 + 1, n1 + 9).map(n2 => ({
    n2,
    lam: 1 / (1.097e-2 * (1 / (n1 * n1) - 1 / (n2 * n2)))   // nm
  }));
  return Plot.plot({
    width: 1120, height: 380, marginLeft: 40, marginBottom: 55,
    x: {type: "log", label: "wavelength (nm)  →", domain: [80, 5000],
        ticks: [100, 200, 500, 1000, 2000, 5000], tickFormat: d => d},
    y: {axis: null, domain: [0, 1]},
    marks: [
      Plot.rect([{}], {x1: 380, x2: 750, y1: 0, y2: 1, fill: "#ffd700", fillOpacity: 0.2}),
      Plot.text([{}], {x: 534, y: 0.04, text: ["visible"], fill: "#8a6d00", fontSize: 13}),
      Plot.ruleX(lines, {x: "lam", stroke: d => lamColor(d.lam), strokeWidth: 3}),
      Plot.text(lines.filter(d => d.n2 <= n1 + 2),
                {x: "lam", y: 0.97, text: d => `n₂=${d.n2}`, fill: "#333",
                 fontSize: 15, textAnchor: "start", dx: 6})
    ]
  });
}

md`**${["", "Lyman (UV)", "Balmer (visible)", "Paschen (IR)", "Brackett (IR)"][n1]}** &nbsp;·&nbsp;
first line **${(1/(1.097e-2*(1/(n1*n1) - 1/((n1+1)*(n1+1))))).toFixed(0)} nm** &nbsp;·&nbsp;
series limit **${(1/(1.097e-2/(n1*n1))).toFixed(0)} nm** (lines pile up here)`

Hydrogen-like Atoms

For one-electron ions (He^+, Li^{2+}), add nuclear charge Z:

E_n = -13.6\,\frac{Z^2}{n^2}\,\,\,[\text{eV}]

Z=1 for H, Z=2 for He^+, Z=3 for Li^{2+}
Energies scale as Z^2: He^+ ionization is 54.4 eV
Orbits shrink as r_n = n^2 a_0 / Z

What Bohr Gets Wrong

Two electrons break it: no helium, no periodic table
No intensities, no selection rules: which lines are bright?
No fine structure: one quantum number n is not enough
An orbit is not allowed: definite r and v defy uncertainty

Keep the physics, drop the picture: quantized energies, integer quantum numbers, light from jumps between levels

Takeaway

Atomic spectra are discrete because energy is quantized: Bohr’s standing-wave condition fixes the orbits and yields E_n = -13.6\,/\,n^2 eV, deriving the empirical Rydberg formula from first principles.