Spectral series — Lyman, Balmer, Paschen: Question bank

True or false — justify

Each answer explains why, not just the verdict. Cover the reasoning before you reveal.

The Balmer series is the lowest-energy series in hydrogen.

False — Balmer lands on

n_{1} = 2

, but Paschen, Brackett, Pfund land higher and have smaller gaps, so they emit lower-energy (longer-wavelength) photons. Balmer is only famous because it happens to be visible.

Every line of the Lyman series has a shorter wavelength than every line of the Balmer series.

True — the smallest Lyman gap (

n_{2} = 2 \to n_{1} = 1

, 122 nm) is still bigger than the largest Balmer gap (

3 \to 2

, 656 nm), because landing on

n_{1} = 1

makes every jump deeper than any Balmer jump.

The series limit of a series is its longest-wavelength line.

False — the limit is

n_{2} \to \infty

, which is the biggest energy gap and therefore the shortest wavelength. The longest wavelength is the first line,

n_{2} = n_{1} + 1

An electron falling from

n = 5

n = 3

produces a Paschen line.

True — the series is named by the final (landing) level, here

n_{1} = 3

, which is the Paschen landing strip; the starting level

n_{2} = 5

just picks which Paschen line.

The Rydberg formula with

R_{H}

works for singly-ionized helium (He

^{+}

False as written — He

^{+}

is hydrogen-like with

Z = 2

, so you must use

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

; the plain

R_{H}

form is only for

Z = 1

. See Hydrogen-like Ions and Z dependence.

Absorption of light by cold hydrogen shows the same wavelengths as emission.

True — the energy gaps are the same whether the electron jumps up (absorbs) or down (emits), so the wavelengths coincide; only the direction of the transition differs. See Emission vs Absorption Spectra.

The gaps between hydrogen energy levels get bigger as

n

increases.

False — since

E_{n} = - 13.6/ n^{2}

, the levels crowd together at the top; gaps shrink toward the ionization edge, which is why high-

n

transitions give long-wavelength (low-energy) light.

A photon with energy exactly

13.6

eV can excite a ground-state hydrogen atom to

n = 2

False —

13.6

eV is the ionization energy (from

n = 1

to free). The

1 \to 2

gap is

13.6 (1 - \frac{1}{4}) = 10.2

eV; a

13.6

eV photon would ionize, not neatly land on

n = 2

. See Ionization Energy.

Spot the error

Each prompt contains a mistake; the reveal names it and fixes it.

"For emission,

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

with

n_{2} > n_{1}

The bracket is backwards — with the higher level

n_{2} > n_{1}

this is negative, giving a nonsense negative

1/ λ

. Put the lower (landing) level first:

\frac{1}{n _{1}^{2}} - \frac{1}{n _{2}^{2}} > 0

"The Lyman series lies in the infrared because

n_{1} = 1

gives the smallest jumps."

Two errors: landing on

n_{1} = 1

gives the largest jumps (deepest landing), and those large gaps mean ultraviolet, not infrared.

R_{H} \approx 1.097 \times 1 0^{7}

has units of metres."

Wrong units —

R_{H}

is a wavenumber, so its units are

m^{- 1}

(lines per metre). It equals

1/ λ

scale, the reciprocal of a length.

"Since

n = \infty

is the biggest quantum number, the series limit gives the longest wavelength."

The confusion is size-of-

n

versus size-of-gap.

n_{2} \to \infty

maximizes the energy gap, so it gives the shortest wavelength of the series, not the longest.

"H-alpha (656 nm) is a Lyman line because it's the brightest visible red."

H-alpha is a Balmer line (

3 \to 2

, landing on

n_{1} = 2

). Lyman lines land on

n_{1} = 1

and are all ultraviolet — invisible to the eye.

"To convert a

1.89

eV gap to wavelength, use

λ (nm) = 1240 \times 1.89

The relation is a division:

λ (nm) = 1240/Δ E (eV)

, because

λ

and

E

are inversely related (

E = h c / λ

). Multiplying inflates the answer wildly.

"An electron can emit a photon while sitting in the ground state

n = 1

There is no lower level to fall to, so a ground-state electron cannot emit — it can only absorb to jump up. Ground state is the floor of the staircase. See Bohr Model of the Atom.

Why questions

The reveal is the causal chain, not a restatement.

Why is the sign of the released energy positive when the electron drops?

With

n_{2}

the initial (higher) level and

n_{1}

the final (lower) level, the lower level is more negative (

E_{n} = - 13.6/ n^{2}

); the emitted energy is

E_{n_{2}} - E_{n_{1}}

, and since

E_{n_{2}} > E_{n_{1}}

this difference is positive and carried off as a photon.

Why does the Rydberg formula use

1/ λ

rather than

λ

directly?

Photon energy is

E = h c / λ

, so energy is linear in

1/ λ

; since the energy is a clean difference of

1/ n^{2}

terms, writing it in

1/ λ

keeps the formula a simple subtraction. See Photon Energy and Planck's Relation.

Why do all lines of one series share the same landing level rather than the same starting level?

Grouping by the final level

n_{1}

fixes the "floor" the photons build up from; every allowed higher floor

n_{2}

contributes a line, so the series is a fan of transitions sharing one destination.

Why can hydrogen only emit at discrete wavelengths instead of a continuous rainbow?

Because the energy levels are quantized (angular momentum is quantized,

m v r = n ℏ

), only fixed energy gaps exist, so only fixed photon energies — hence fixed wavelengths — are possible. See Quantization of Angular Momentum.

Why does the ionization energy of hydrogen equal

13.6

eV exactly?

It is the energy to lift the electron from

n = 1

(

E_{1} = - 13.6

eV) all the way to free (

E_{\infty} = 0

); the gap is

0 - (- 13.6) = 13.6

eV. See Ionization Energy.

Why do the spectral lines within a series bunch up toward the series limit?

Because

1/ n_{2}^{2}

shrinks fast as the starting level

n_{2}

grows, successive lines' wavenumbers get closer together, piling up at the limit value

R_{H} / n_{1}^{2}

where the electron is barely bound.

Edge cases

Boundary and degenerate scenarios the topic invites.

What transition does

n_{2} = n_{1}

correspond to physically?

None — the "gap" is zero, so

1/ λ = 0

and no photon is emitted; the electron isn't moving between levels, so there is no line.

For the series limit (

n_{2} \to \infty

), what happens to the electron just above that energy?

It is no longer bound — energies above

0

form a continuum, so instead of a sharp line you get continuous emission/absorption; the limit marks the edge of the bound spectrum.

Does the "longest wavelength = first line" rule survive if we skip

n_{1} + 1

The actual longest line always comes from the smallest possible gap,

n_{2} = n_{1} + 1

; any larger starting level

n_{2}

gives a bigger gap and shorter wavelength, so the rule holds by definition of "smallest jump."

For a hydrogen-like ion with

Z = 2

, does the series structure (Lyman/Balmer/Paschen landing levels) change?

No — the landing levels $n_{1}$ stay the same; only the wavelengths scale by

1/ Z^{2}

(shorter), because

\frac{1}{λ}

multiplies by

Z^{2}

. The names/structure are unchanged. See Hydrogen-like Ions and Z dependence.

Is there an upper limit to how many lines a single series can have?

No — the starting level

n_{2}

can be any integer above the landing level

n_{1}

, so infinitely many lines exist, crowding ever closer to the series limit but never quite reaching it.

Can two different transitions ever give the exact same wavelength in hydrogen?

Not within the pure Rydberg (Bohr) spectrum — each pair

(n_{1}, n_{2})

gives a distinct

\frac{1}{n _{1}^{2}} - \frac{1}{n _{2}^{2}}

. (In real atoms, once you add fine-structure/relativistic corrections or move to multi-electron atoms, accidental near-overlaps can occur; the "never" applies only to the idealized formula.)

Connections

Bohr Model of the Atom — the source of the quantized energy levels these traps rely on.
Photon Energy and Planck's Relation — why energy is linear in $1/ λ$ .
Emission vs Absorption Spectra — same wavelengths, opposite direction.
Ionization Energy — the $13.6$ eV and series-limit edge cases.
Hydrogen-like Ions and Z dependence — the $Z^{2}$ scaling traps.
Quantization of Angular Momentum — why the spectrum is discrete at all.