Hydrogen emission spectrum — Lyman, Balmer, Paschen, Brackett, Pfund: Foundations

This page assumes nothing. We build every letter the parent note fires at you — $n$ , $E_{n}$ , $Δ E$ , $λ$ , $ν$ , $\overset{ν}{ˉ}$ , $h$ , $c$ , $R_{H}$ — one at a time, each anchored to a picture, each earned before the next.

0. The staircase picture (hold this in your head)

Figure — Hydrogen emission spectrum — Lyman, Balmer, Paschen, Brackett, Pfund

Look at the figure. Each horizontal line is an allowed energy the electron is permitted to have — nothing in between is allowed. That "only certain heights allowed" fact is the single strangest thing about atoms, and everything below is just a way of labelling or measuring this staircase.

The rungs are labelled by a counting number (we'll call it $n$ in §1).
The height of each rung is an energy (we'll call it $E_{n}$ in §2).
A fall from a high rung to a low one is a drop in energy (we'll name it $Δ E$ in §3).
The flash of light released has a colour set by its wavelength (named $λ$ in §4).

To keep the promise "no symbol before it is defined," the figure above uses only plain words — "rung," "height," "fall," "flash." Starting in §1 we attach one letter at a time. Nothing else is bookkeeping — let's earn each symbol.

1. $n$ — the level number (which rung)

The picture: in the staircase figure, $n$ is just the label on the rung — rung 1, rung 2, rung 3. It is never a fraction, never negative, never zero. It is the electron's "address."

Why the topic needs it: every spectral line is described by two addresses — where the electron started ( $n_{2}$ ) and where it landed ( $n_{1}$ ). Without a way to name the rungs, we couldn't say which jump made which colour.

2. $E_{n}$ — the energy of rung $n$ (how high)

Reading the symbols:

$eV$ (electron-volt) is simply a small unit of energy, handy for atoms — like using millimetres instead of metres. We will need its SI value later: $1 eV = 1.602 \times 1 0^{- 19} J$ (see §5, where energies must be in joules).
The minus sign is the important part. It means the electron is trapped (bound) inside the atom, like a ball sitting inside a well: you'd have to add energy to lift it out.
$n^{2}$ in the bottom means: as $n$ grows, $E_{n}$ gets closer to zero (less negative), i.e. higher.

The picture: look at the well figure. Zero energy (a free, escaped electron) is the top rim. All the rungs sit below the rim, so all $E_{n}$ are negative. The rungs bunch together near the top because of the $1/ n^{2}$ — big jumps at the bottom, tiny jumps near the rim.

3. $Δ E$ — the size of the fall (how big the drop)

The picture: in the staircase figure, $Δ E$ is the vertical gap between the two rungs — the height the ball falls. A tall gap = big $Δ E$ ; a short gap = small $Δ E$ .

Why $n_{1} < n_{2}$ and why the order matters: emission is a fall, so the electron ends up lower than it started. If we always keep the smaller address as $n_{1}$ , the released energy comes out positive (energy genuinely leaves the atom). This is exactly why the parent note writes the bracket as $(\frac{1}{n _{1}^{2}} - \frac{1}{n _{2}^{2}})$ with the smaller $n$ first — it guarantees a positive answer.

4. Light as a wave: $λ$ , $ν$ , and $c$ (what colour comes out)

Before the flash of light can have "a colour," we need the language of light itself.

The picture: imagine the wave in the figure sliding past you at speed $c$ . If each ripple is long ( $λ$ big), fewer ripples pass per second ( $ν$ small). If ripples are short, many pass per second. So $λ$ and $ν$ are opposites — big one, small other — locked by the constant $c$ . This inverse relationship is the seed of the topic's most common mistake ("bigger jump ⇒ bigger wavelength" — it's the reverse).

The full ordering of colours by wavelength lives in the Electromagnetic Spectrum — UV (short), visible (middle), IR (long).

5. $h$ and the Planck relation — from energy to colour (mind the units!)

We have "size of the fall" ( $Δ E$ ) and "colour of light" ( $λ$ ). We need the bridge between them.

Why this tool and not another? We specifically want to turn an energy drop inside the atom into a wavelength we can see or measure. Frequency alone won't do it — we need a rule that says "this much energy ↔ this exact colour." The Planck relation is that rule, and $h$ is its conversion factor.

The chain, in one breath (all energies in joules): $fall Δ E = conservation E_{photon} = \frac{h c}{λ} ⟹ λ = \frac{h c}{Δ E}$

Read the last arrow: $λ \propto 1/Δ E$ . Big fall ⇒ short wavelength. This single inverse is why Lyman (biggest falls, landing on rung 1) is UV, and Paschen/Brackett/Pfund (gentle falls) are infrared.

6. $\overset{ν}{ˉ}$ — the wavenumber (a convenient handle)

Why bother? Because $\overset{ν}{ˉ}$ is proportional to $Δ E$ , comparing energies of two spectral lines becomes as easy as comparing two plain numbers — no dividing needed. That's exactly what the parent note's worked example 3 exploits (" $0.75 > 0.1389 \Rightarrow$ Lyman is more energetic").

7. $R_{H}$ — the Rydberg constant (the one measured number)

Where it comes from — and how $13.6$ eV and $R_{H}$ are the same fact. The Bohr energy can be written two equivalent ways: $E_{n} = - \frac{R _{H} h c}{n ^{2}} and E_{n} = - \frac{13.6 eV}{n ^{2}} .$ These agree only if $R_{H} h c = 13.6$ eV. Let's check the units convert correctly: $R_{H} h c = (1.097 \times 1 0^{7} m^{- 1}) (6.626 \times 1 0^{- 34} J\cdots) (3.00 \times 1 0^{8} m/s) = 2.18 \times 1 0^{- 18} J .$ Now convert to eV by dividing by $1.602 \times 1 0^{- 19} J/eV$ : $\frac{2.18 \times 1 0 ^{- 18} J}{1.602 \times 1 0 ^{- 19} J/eV} = 13.6 eV . ✓$ So " $- 13.6$ eV/ $n^{2}$ " and " $- R_{H} h c / n^{2}$ " are not two rules but one — the eV version is just the SI version passed through the eV↔J conversion.

Feeding this into the chain of §§3–6 collapses all constants into the single $R_{H}$ , giving the topic's headline formula: $\frac{1}{λ} = \overset{ν}{ˉ} = R_{H} (\frac{1}{n _{1}^{2}} - \frac{1}{n _{2}^{2}})$

Now every symbol in that boxed formula has been earned: $λ$ (§4), $\overset{ν}{ˉ}$ (§6), $R_{H}$ (§7), $n_{1}, n_{2}$ (§1). Nothing in the parent note is a mystery anymore.

How the foundations feed the topic

The picture below (built by the figure script, not raw code) shows how each symbol we defined feeds forward into the Rydberg formula — read it left-to-right, bottom-to-top.

Equipment checklist

Test yourself — you are ready for the main note only if each reveal matches your own answer.

What does the symbol

n

stand for, and what values can it take?

The principal quantum number — which rung the electron is on; whole numbers

1, 2, 3, \dots

only.

Why is every

E_{n}

negative?

The electron is bound (trapped in the well); zero energy means a free, escaped electron, and all rungs sit below that.

Which is higher in energy,

E_{1}

E_{2}

E_{2} = - 3.4

eV is higher than

E_{1} = - 13.6

eV — less negative means higher.

What does

Δ E

mean and why keep

n_{1} < n_{2}

"Change in energy" = size of the fall; keeping the smaller address as

n_{1}

makes emission energy positive.

Before using

λ = h c /Δ E

, what must you do to

Δ E

given in eV?

Convert to joules: multiply by

1.602 \times 1 0^{- 19}

J/eV, so

h

(J·s) and

c

(m/s) are unit-consistent.

What is the relationship between

λ

and

ν

Inverse:

c = λ ν

, so a long wavelength means a low frequency.

Write the Planck relation linking energy and wavelength.

E_{photon} = h ν = \frac{h c}{λ}

Δ E

doubles, what happens to

λ

It halves —

λ \propto 1/Δ E

Show that

R_{H} h c

equals

13.6

eV.

R_{H} h c = 2.18 \times 1 0^{- 18}

J; dividing by

1.602 \times 1 0^{- 19}

J/eV gives

13.6

eV.

What is the wavenumber

\overset{ν}{ˉ}

and why is it convenient?

\overset{ν}{ˉ} = 1/ λ

; it is directly proportional to photon energy, so energy comparisons become simple number comparisons.

State the value and role of

R_{H}

1.097 \times 1 0^{7} m^{- 1}

; it sets the overall scale of every hydrogen wavelength in the Rydberg formula.

Assemble the Rydberg formula from these pieces.

\frac{1}{λ} = R_{H} (\frac{1}{n _{1}^{2}} - \frac{1}{n _{2}^{2}})

with

n_{2} > n_{1}

Connections

Bohr Model of the Atom — source of $E_{n} = - 13.6/ n^{2}$ eV, the staircase heights.
Quantization of Energy — why $n$ takes whole values only.
Photon Energy and Planck Relation — the $E = h c / λ$ bridge built in §5.
Electromagnetic Spectrum — where UV, visible, IR sit by wavelength.
Ionization Energy of Hydrogen — the well depth, $13.6$ eV.
Rydberg Constant and Spectra of Hydrogen-like Ions — where $R_{H}$ comes from and how it generalises.