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

WHY does hydrogen give a line spectrum?

WHAT: A line (discrete) spectrum means only certain wavelengths appear.

WHY: Electron energies inside an atom are quantized (Bohr). Energy comes in fixed steps, so emitted photons carry only fixed energy differences, $\Delta E = E_{n_2}-E_{n_1}$ .

HOW: Photon energy fixes wavelength: $\Delta E = h\nu = \dfrac{hc}{\lambda}$ . Fixed $\Delta E$ ⇒ fixed $\lambda$ ⇒ a sharp line.

Deriving the formula from scratch

Step 1 — Emission is a fall. Electron drops $n_2 \to n_1$ with $n_2 > n_1$ . Energy released: $\Delta E = E_{n_2}-E_{n_1} = -R_Hhc\left(\frac{1}{n_2^2}-\frac{1}{n_1^2}\right)= R_Hhc\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)$

Why this step? We subtract final − initial energies; because $n_1<n_2$ , the bracket is positive so $\Delta E>0$ (energy is genuinely released as a photon).

Step 2 — Turn energy into wavelength. A photon carries $\Delta E = \dfrac{hc}{\lambda}$ , so $\frac{hc}{\lambda} = R_Hhc\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)$

Why this step? Conservation of energy: the lost electronic energy becomes exactly one photon.

Step 3 — Cancel $hc$ to get the Rydberg formula: $\boxed{\ \frac{1}{\lambda} = \bar{\nu} = R_H\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)\ }$

The five named series (each fixes $n_1$ )

Series	$n_1$ (lands on)	$n_2$	Region
Lyman	1	2,3,4,…	Ultraviolet
Balmer	2	3,4,5,…	Visible
Paschen	3	4,5,6,…	Infrared
Brackett	4	5,6,7,…	Infrared
Pfund	5	6,7,8,…	Infrared

Within one series: the first line ( $n_2=n_1+1$ ) is the longest $\lambda$ (smallest jump). The series limit ( $n_2\to\infty$ ) is the shortest $\lambda$ of that series: $\frac{1}{\lambda_{\text{limit}}} = \frac{R_H}{n_1^2}$

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

Worked examples

Worked example 1) Balmer red line H-α (

$n_2=3\to n_1=2$ ) $\frac1\lambda = 1.097\times10^7\left(\frac1{2^2}-\frac1{3^2}\right)=1.097\times10^7\left(\frac14-\frac19\right)$ Why this step? Balmer ⇒ $n_1=2$ ; first line ⇒ $n_2=3$ . $\frac14-\frac19 = \frac{9-4}{36}=\frac{5}{36}=0.1389$ . $\frac1\lambda = 1.097\times10^7\times0.1389 = 1.524\times10^6\ \text{m}^{-1}$ . $\lambda = 6.56\times10^{-7}\text{ m} = 656\text{ nm}$ → red, matches the visible line. ✓

Common mistakes (steel-manned)

Active recall

Recall Flip me: what determines the series?

The lower level $n_1$ . Lyman→1, Balmer→2, Paschen→3, Brackett→4, Pfund→5.

Recall Flip me: Rydberg formula & constant

$\dfrac1\lambda = R_H\left(\dfrac1{n_1^2}-\dfrac1{n_2^2}\right)$ , $R_H=1.097\times10^7\ \text{m}^{-1}$ .

Recall Feynman (explain to a 12-year-old)

Imagine the atom is a staircase and the electron is a ball. Each step has a set height. When the ball tumbles down, it flashes a light. Big drops flash bluish-violet/UV light (lots of energy), small drops flash red/infrared light (little energy). If all the balls land on the same bottom step, that group of flashes is one "family" (a series). Lyman = they all land on the ground floor; Balmer = they all land on step 2; and so on. Because the steps are fixed, the flashes are always the same exact colours — that's why hydrogen gives sharp lines, not a smear.

Connections

Bohr Model of the Atom — supplies $E_n=-13.6/n^2$ eV used to derive the formula.
Quantization of Energy — the reason lines are discrete.
Photon Energy and Planck Relation — $E=hc/\lambda$ links jump to colour.
Rydberg Constant and Spectra of Hydrogen-like Ions — generalises with $Z^2$ .
Ionization Energy of Hydrogen — equals Lyman series limit, 13.6 eV.
Electromagnetic Spectrum — UV/visible/IR placement of each series.

Which quantum number defines a spectral series?

The lower level

n_1

(the destination of the electron's fall).

State the Rydberg formula.

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

with

n_2>n_1

Value of the Rydberg constant

R_H

1.097\times10^{7}\ \text{m}^{-1}

Which series lies in the visible region?

Balmer (

n_1=2

Which series has the shortest wavelengths overall?

Lyman (

n_1=1

, UV) — biggest energy drops.

n_1

values for Lyman, Balmer, Paschen, Brackett, Pfund?

1, 2, 3, 4, 5.

Wavelength of the series limit for a given

n_1

\lambda_{\text{limit}}=n_1^2/R_H

(since

n_2\to\infty

Why is the spectrum discrete not continuous?

Because electron energies are quantized, so only fixed

\Delta E

(hence fixed

\lambda

) photons are emitted.

n_2

increases within a series, does

\lambda

increase or decrease?

Decreases toward the series limit (jumps get bigger).

Wavelength of H-α (Balmer first line,

3\to2

About 656 nm (red).

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, hydrogen ke atom mein electron sirf fixed "orbits" (energy levels) pe reh sakta hai — beech mein kahin nahi. Jab electron upar wale level ( $n_2$ ) se neeche wale level ( $n_1$ ) pe girta hai, to ek photon (light ka packet) release hota hai jiski energy exactly do levels ke difference ke barabar hoti hai. Kyunki energy fixed steps mein hai, isliye sirf fixed colours (wavelengths) hi nikalte hain — yeh "line spectrum" kehlata hai, ek smooth rainbow nahi.

Ab series ka funda simple hai: series ka naam decide hota hai us niche wale level se jahan electron aakar girta hai. Lyman = sab $n=1$ pe aate hain (UV, sabse zyada energy, sabse chhoti wavelength). Balmer = $n=2$ pe (yeh visible light hai, isliye humein dikhta hai). Paschen ( $n=3$ ), Brackett ( $n=4$ ), Pfund ( $n=5$ ) — ye sab infrared mein hote hain.

Formula ek hi hai, Rydberg formula: $\frac{1}{\lambda}=R_H\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)$ , jahan $R_H=1.097\times10^7$ m⁻¹. Yaad rakho: chhota $n$ hamesha pehle (positive term), warna answer negative aa jayega. Ek important trap: bada jump matlab bada energy, aur bada energy matlab chhoti wavelength — energy aur wavelength ulta-ulta chalte hain. Series limit ( $n_2\to\infty$ ) pe us series ki sabse chhoti wavelength milti hai, aur Lyman ki limit hi hydrogen ki ionization energy (13.6 eV) deti hai. Bas itna clear ho gaya to poora topic aapka.

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

WHY does hydrogen give a line spectrum?

Deriving the formula from scratch

The five named series (each fixes $n_1$ )

Worked examples

Common mistakes (steel-manned)

Active recall

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Atomic Structure (Classical)

Connections