MHT-CET Physics · Structure of Atoms and Nuclei

The Hydrogen Spectrum and its Series

Every line of the hydrogen spectrum is a jump down to a fixed lower level, with 1/λ = RZ²(1/n_f² − 1/n_i²); the lower level names the series — Lyman, Balmer, Paschen, Brackett — and within a series the first line is the longest wavelength and the series limit the shortest.

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Why this matters

28 PYQs, 7 of them HARD. Sixteen are about the series themselves: which one is visible, the frequency or wave number of a series limit, and relations between limits and first lines. Twelve ask for the ratio of two named lines, sometimes in a hydrogen-like ion, or for the level a photon came from. Two cards.

Concept 1 of 2: The Series, Their First Lines and Their Limits

Fix the lower level n_f and let the upper level run: n_f = 1 gives the Lyman series (ultraviolet), 2 the Balmer (visible), 3 the Paschen, 4 the Brackett and 5 the Pfund (all infrared). The first line comes from the nearest level, n_f + 1, and has the smallest energy, so the LONGEST wavelength. The series limit comes from n = ∞, 1/λ = R/n_f², and has the SHORTEST wavelength. Frequency is c/λ and wave number is 1/λ, so the Balmer limit has wave number R/4 and frequency Rc/4. Because limits and first lines are differences of the same terms, they combine: the Lyman limit minus the first Lyman line is the Balmer limit. A source moving away stretches every line by Δλ/λ = v/c.

Definition

Lyman (n_f = 1, UV), Balmer (2, visible), Paschen (3), Brackett (4), Pfund (5) — infrared.
First line $(n_f + 1 \to n_f)$ : longest λ. Series limit $(\infty \to n_f)$ : shortest λ, $\dfrac{1}{\lambda} = \dfrac{R}{n_f^2}$ .
Balmer limit: wave number $\dfrac{R}{4}$ , frequency $\dfrac{Rc}{4}$ . Lyman max/min λ = 4 : 3.
Limits in a ratio: $\dfrac{\lambda_{\text{Lyman}}}{\lambda_{\text{Balmer}}} = \dfrac{1}{4}$ ; Paschen/Balmer = 9/4.
Combining terms: $\dfrac{1}{\lambda_{L,\infty}} - \dfrac{1}{\lambda_{L,1}} = \dfrac{1}{\lambda_{B,\infty}}$ ; $\nu_{B,\infty} - \nu_{B,1} = \nu_{P,\infty}$ .
Photon from an energy gap: $\lambda = \dfrac{hc}{\Delta E} = \dfrac{1240}{\Delta E\,(\text{eV})}$ nm.
Receding source: $\dfrac{\Delta\lambda}{\lambda} = \dfrac{v}{c}$ .

Series limit

\frac{1}{\lambda_\infty} = \frac{R}{n_f^2}, \qquad \nu_\infty = \frac{Rc}{n_f^2}

Worked example

With R = 1.097 × 10⁷ m⁻¹, what is the wavelength of the Balmer series limit?

Practice this conceptself-check · 2 quick reps

The same idea in a real exam question:

MHT-CET · 2024 · 12th May Shift 1 · Q16Moderate

Example 1 · Structure of Atoms and Nuclei · Hydrogen Spectrum and Spectral Series

In Lyman series, series limit of wavelength is

\lambda_{1}

. The wavelength of first line of Lyman series is

\lambda_{2}

and in Balmer series, the series limit of wavelength is

\lambda_{3}

. Then the relation between

\lambda_{1}

\lambda_{2}

and

\lambda_{3}

Calling the series limit the longest wavelength

The limit comes from n = ∞, the biggest jump into that level, so it has the most energy and the SHORTEST wavelength. The first line is the longest.

Using n_f = 1 for every series

Only Lyman ends on n = 1. The Balmer limit is R/4, not R; Paschen's is R/9. Set the lower level from the series name before substituting.

Drill 15 more on the series, their first lines and their limits

Concept 2 of 2: Ratios of Two Lines

Write 1/λ = RZ²(1/n_f² − 1/n_i²) for each line and divide; R cancels, and so does Z if both lines are in one atom. The wavelength ratio is the INVERSE of the ratio of the brackets, and the frequency ratio is the ratio of the brackets itself. The same transition in two hydrogen-like ions gives λ ∝ 1/Z². To find the level a photon came from, solve the formula for n_i. The highest-frequency emission is the one with the biggest bracket, which is usually a jump into n = 1: 2 → 1 beats 5 → 3 and 6 → 2. A jump UP is absorption, not emission.

Definition

$\dfrac{1}{\lambda} = RZ^2\left(\dfrac{1}{n_f^2} - \dfrac{1}{n_i^2}\right)$ ; $\dfrac{\lambda_1}{\lambda_2}$ = inverse ratio of the brackets.
Common brackets: $2\to1: \tfrac{3}{4}$ ; $3\to1: \tfrac{8}{9}$ ; $3\to2: \tfrac{5}{36}$ ; $4\to2: \tfrac{3}{16}$ ; $4\to3: \tfrac{7}{144}$ ; $5\to4: \tfrac{9}{400}$ .
First Paschen to first Lyman: $\dfrac{\lambda_P}{\lambda_L} = \dfrac{3/4}{7/144} = \dfrac{108}{7}$ .
Same transition, two ions: $\lambda \propto \dfrac{1}{Z^2}$ (He⁺⁺ : Li⁺⁺⁺ ⇒ 9 : 4).
Level of origin, photon λ into n = 1: $n = \sqrt{\dfrac{\lambda R}{\lambda R - 1}}$ .

Rydberg formula

\frac{1}{\lambda} = RZ^2\left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)

Worked example

With R = 1.097 × 10⁷ m⁻¹, wavelength of the first Balmer line (3 → 2)?

Practice this conceptself-check · 2 quick reps

The same idea in a real exam question:

MHT-CET · 2023 · 3rd May Shift 1 · Q49Hard

Example 2 · Structure of Atoms and Nuclei · Hydrogen Spectrum and Spectral Series

If '

\lambda_1

' and '

\lambda_2

' are the wavelengths of the first line of the Lyman and Paschen series respectively, then

\lambda_2:\lambda_1

Dividing wavelengths the way the brackets divide

The bracket is 1/λ. A larger bracket means a SHORTER wavelength, so the wavelength ratio is the brackets' ratio turned upside down.

Counting an upward jump as emission

n = 1 to n = 2 absorbs a photon. 'Which transition emits the highest frequency' excludes every jump upward, however large.

Drill 11 more on ratios of two lines

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

The Series, Their First Lines and Their Limits
Series limit
$\frac{1}{\lambda_\infty} = \frac{R}{n_f^2}, \qquad \nu_\infty = \frac{Rc}{n_f^2}$
Ratios of Two Lines
Rydberg formula
$\frac{1}{\lambda} = RZ^2\left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)$

Watch out for (4)

Calling the series limit the longest wavelength
→ The Series, Their First Lines and Their Limits
Using n_f = 1 for every series
→ The Series, Their First Lines and Their Limits
Dividing wavelengths the way the brackets divide
→ Ratios of Two Lines
Counting an upward jump as emission
→ Ratios of Two Lines

Test yourself on Structure of Atoms and Nuclei

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