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

1.2.9 · Chemistry › Atomic Structure (Classical)

Hydrogen line spectrum kyun deta hai?

KYA: Ek line (discrete) spectrum ka matlab hai ki sirf kuch specific wavelengths appear hoti hain.

KYUN: Atom ke andar electron energies quantized hoti hain (Bohr). Energy fixed steps mein aati hai, isliye emitted photons sirf fixed energy differences carry karte hain, $Δ E = E_{n_{2}} - E_{n_{1}}$ .

KAISE: Photon energy wavelength fix karti hai: $Δ E = h ν = \frac{h c}{λ}$ . Fixed $Δ E$ ⇒ fixed $λ$ ⇒ ek sharp line.

Formula scratch se derive karna

Step 1 — Emission ek fall hai. Electron $n_{2} \to n_{1}$ mein drop karta hai jahan $n_{2} > n_{1}$ . Released energy: $Δ E = E_{n_{2}} - E_{n_{1}} = - R_{H} h c (\frac{1}{n _{2}^{2}} - \frac{1}{n _{1}^{2}}) = R_{H} h c (\frac{1}{n _{1}^{2}} - \frac{1}{n _{2}^{2}})$

Yeh step kyun? Hum final − initial energies subtract karte hain; kyunki $n_{1} < n_{2}$ hai, bracket positive hai isliye $Δ E > 0$ (energy genuinely ek photon ke roop mein release hoti hai).

Step 2 — Energy ko wavelength mein convert karo. Ek photon $Δ E = \frac{h c}{λ}$ carry karta hai, toh $\frac{h c}{λ} = R_{H} h c (\frac{1}{n _{1}^{2}} - \frac{1}{n _{2}^{2}})$

Yeh step kyun? Conservation of energy: lost electronic energy exactly ek photon ban jaati hai.

Step 3 — $h c$ cancel karo aur Rydberg formula paao: $\frac{1}{λ} = \overset{ν}{ˉ} = R_{H} (\frac{1}{n _{1}^{2}} - \frac{1}{n _{2}^{2}})$

Paanch named series (har ek $n_{1}$ fix karti hai)

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

Ek series ke andar: pehli line ( $n_{2} = n_{1} + 1$ ) sabse lamba $λ$ hai (sabse chota jump). Series limit ( $n_{2} \to \infty$ ) us series ka sabse chota $λ$ hai: $\frac{1}{λ _{limit}} = \frac{R _{H}}{n _{1}^{2}}$

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

Worked examples

Common mistakes (steel-manned)

Active recall

Recall Flip me: series kya determine karta hai?

Lower level $n_{1}$ . Lyman→1, Balmer→2, Paschen→3, Brackett→4, Pfund→5.

Recall Flip me: Rydberg formula & constant

$\frac{1}{λ} = R_{H} (\frac{1}{n _{1}^{2}} - \frac{1}{n _{2}^{2}})$ , $R_{H} = 1.097 \times 1 0^{7} m^{- 1}$ .

Recall Feynman (ek 12-saal ke bacche ko explain karo)

Socho atom ek staircase hai aur electron ek ball hai. Har step ki ek set height hoti hai. Jab ball neeche girti hai, toh ek light flash karti hai. Bade drops bluish-violet/UV light flash karte hain (bahut energy), chote drops red/infrared light flash karte hain (thodi energy). Agar sab balls usi ek bottom step par land karein, toh woh group of flashes ek "family" hai (ek series). Lyman = sab ground floor par land karte hain; Balmer = sab step 2 par land karte hain; aur aise hi aage. Kyunki steps fixed hain, flashes hamesha same exact colours ki hoti hain — isliye hydrogen sharp lines deta hai, smear nahi.

Connections

Bohr Model of the Atom — $E_{n} = - 13.6/ n^{2}$ eV deta hai jo formula derive karne mein use hota hai.
Quantization of Energy — yahi reason hai ki lines discrete hain.
Photon Energy and Planck Relation — $E = h c / λ$ jump ko colour se link karta hai.
Rydberg Constant and Spectra of Hydrogen-like Ions — $Z^{2}$ ke saath generalize karta hai.
Ionization Energy of Hydrogen — Lyman series limit ke barabar hai, 13.6 eV.
Electromagnetic Spectrum — har series ki UV/visible/IR placement.

Konsa quantum number ek spectral series define karta hai?

Lower level

n_{1}

(electron ke fall ki destination).

Rydberg formula state karo.

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

with

n_{2} > n_{1}

Rydberg constant

R_{H}

ki value?

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

Konsi series visible region mein hoti hai?

Balmer (

n_{1} = 2

Konsi series mein overall sabse short wavelengths hoti hain?

Lyman (

n_{1} = 1

, UV) — sabse bade energy drops.

Lyman, Balmer, Paschen, Brackett, Pfund ke liye

n_{1}

ki values?

1, 2, 3, 4, 5.

Diye gaye

n_{1}

ke liye series limit ki wavelength?

λ_{limit} = n_{1}^{2} / R_{H}

(kyunki

n_{2} \to \infty

Spectrum discrete kyun hai, continuous kyun nahi?

Kyunki electron energies quantized hain, isliye sirf fixed

Δ E

(hence fixed

λ

) wale photons emit hote hain.

Ek series ke andar jab

n_{2}

badhta hai, toh

λ

badhta hai ya ghatta hai?

Series limit ki taraf ghatta hai (jumps bade hote jaate hain).

H-α ki wavelength (Balmer first line,

3 \to 2

Approximately 656 nm (red).