Rydberg formula 1 - λ = R(1 - n₁² − 1 - n₂²)

1.2.10 · Chemistry › Atomic Structure (Classical)

YEH HAI KYA?

YEH $1/ n^{2}$ JAISA KYU DIKHTA HAI? — Derivation shuru se

Hum isse Bohr model se derive karenge taaki koi bhi step magical na lage.

Step 1 — Bohr level ki energy. Bohr ka quantisation hydrogen mein level $n$ ki energy deta hai. Kyunki electron aur proton dono apne common centre of mass ke around orbit karte hain, isliye sahi mass reduced mass $μ = \frac{m _{e} m _{p}}{m _{e} + m _{p}}$ hai, na ki sirf electron mass $m_{e}$ : $E_{n} = - \frac{13.6 eV}{n ^{2}} = - \frac{μ e ^{4}}{8 ε _{0}^{2} h ^{2}} \cdot \frac{1}{n ^{2}}$ Yeh step kyun? Electron bound hai (negative energy), aur $1/ n^{2}$ Coulomb attraction ko quantised angular momentum $L = n ℏ$ ke saath balance karne se aata hai. Energy levels $n$ badhne ke saath ek doosre ke kareeb aate jaate hain. $m_{e}$ ki jagah $μ$ use karna finite (infinite nahi) proton mass ka account karta hai.

Step 2 — Emitted photon ki energy. Jab electron $n_{2} \to n_{1}$ se girta hai, energy conserve hoti hai: atomic energy ka jo loss hota hai woh ek photon ban jaata hai. $Δ E = E_{n_{2}} - E_{n_{1}} = - \frac{μ e ^{4}}{8 ε _{0}^{2} h ^{2}} (\frac{1}{n _{2}^{2}} - \frac{1}{n _{1}^{2}})$ Photon energy woh amount released hai $= E_{n_{2}} - E_{n_{1}}$ lekin positive liya gaya: $E_{photon} = E_{n_{2}} - E_{n_{1}} = \frac{μ e ^{4}}{8 ε _{0}^{2} h ^{2}} (\frac{1}{n _{1}^{2}} - \frac{1}{n _{2}^{2}})$ Yeh step kyun? Kyunki $n_{2} > n_{1}$ , isliye $\frac{1}{n _{1}^{2}} > \frac{1}{n _{2}^{2}}$ , toh bracket positive hai → photon energy positive hai. Sahi hai.

Step 3 — Photon energy ko wavelength mein convert karo. Photon ke liye $E_{photon} = h ν = \frac{h c}{λ}$ hota hai। $\frac{h c}{λ} = \frac{μ e ^{4}}{8 ε _{0}^{2} h ^{2}} (\frac{1}{n _{1}^{2}} - \frac{1}{n _{2}^{2}})$ Yeh step kyun? Hum $λ$ mein formula chahte hain, isliye $E$ ko $h c / λ$ se replace karo.

Step 4 — $1/ λ$ ke liye solve karo. Dono sides ko $h c$ se divide karo: $\frac{1}{λ} = R_{H} \frac{μ e ^{4}}{8 ε _{0}^{2} h ^{3} c} (\frac{1}{n _{1}^{2}} - \frac{1}{n _{2}^{2}})$

Spectral series (crowding staircase)

$n_{1}$ series fix karta hai (electron kahan land karta hai); $n_{2}$ us series ke andar line fix karta hai.

Series	$n_{1}$	Region
Lyman	1	Ultraviolet
Balmer	2	Visible
Paschen	3	Infrared
Brackett	4	Infrared
Pfund	5	Far Infrared

Figure — Rydberg formula 1 - λ = R(1 - n₁² − 1 - n₂²)

Worked Examples

Common Mistakes (Steel-manned)

Active Recall

Recall Forecast-then-Verify: kaun si series visible hai, aur kyun?

Pehle forecast karo, phir open karo. Answer: Balmer series ( $n_{1} = 2$ ). Sirf uski lines 380–700 nm visible window mein aati hain kyunki $n = 2$ pe land karne wale energy gaps visible photon energies se match karte hain. Lyman ( $n_{1} = 1$ ) ke gaps bade hain → UV; Paschen aur uske aage chhote gaps hain → IR.

Recall 12-saal ke bachche ko explain karo (Feynman)

Socho atom ke andar ek staircase hai jisme electron baitha hai. Steps jitni upar jaati hain, utni ek doosre ke paas aati jaati hain. Jab electron neeche kisi lower step pe slide karta hai, toh usse apni energy phenkhni padti hai — aur woh use coloured light ki ek flash ke roop mein bahar phenkta hai. Bada drop = badi flash = bluer light; chota drop = choti flash = redder light. Rydberg formula woh rulebook hai jo batata hai ki staircase pe har slide ke liye exactly kaun sa colour nikalta hai, sirf start aur end step numbers use karke.

Flashcards

Rydberg formula mein

n_{1}

kya represent karta hai?

Lower (final, smaller) energy level ka quantum number; series terminus.

n_{2} > n_{1}

kyun hona chahiye?

Taaki bracket

(1/ n_{1}^{2} - 1/ n_{2}^{2})

positive rahe, jisse positive wavenumber aur real, positive wavelength milti hai.

R_{H}

ki value aur units kya hain?

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

(hydrogen ke liye Rydberg constant).

True Rydberg constant mein kaun sa mass aata hai, aur kyun?

Reduced mass

μ = m_{e} m_{p} / (m_{e} + m_{p})

, kyunki electron aur proton dono apne common centre of mass ke around orbit karte hain.

R_{\infty}

aur

R_{H}

mein kya fark hai?

R_{\infty} = m_{e} e^{4} / (8 ε_{0}^{2} h^{3} c)

infinite-nuclear-mass idealisation hai;

R_{H} = R_{\infty} μ / m_{e}

real hydrogen ke liye thoda chota hai.

Kaun si spectral series visible region mein hai aur

n_{1}

kya hai?

Balmer series,

n_{1} = 2

1/ λ

ko kya kehte hain aur yeh kiske proportional hai?

Wavenumber

\overset{ν}{ˉ}

; photon energy

E = h c / λ

ke proportional.

Charge Z wale hydrogen-like ion ke liye Rydberg formula kaise modify hota hai?

1/ λ = R Z^{2} (1/ n_{1}^{2} - 1/ n_{2}^{2})

(apne reduced mass ke saath).

Series limit (sabse chota λ) kaun sa transition deta hai?

n_{2} \to \infty

, jisse

1/ λ = R_{H} / n_{1}^{2}

milta hai.

Bohr energies se

R_{H}

symbolically derive karo.

R_{H} = μ e^{4} / (8 ε_{0}^{2} h^{3} c)

h c / λ

ko Bohr energy gap ke barabar rakhne se.

Mnemonic

Connections

Bohr Model of the Atom — $E_{n} = - 13.6/ n^{2}$ eV deta hai jis par poori derivation based hai.
Reduced Mass in Two-Body Systems — kyun $μ$ real Rydberg constant mein $m_{e}$ ki jagah aata hai.
Energy of a Photon E = hc over lambda — energy gap se wavelength tak ka bridge.
Hydrogen Spectrum and Spectral Series — experimental lines jo yeh formula predict karta hai.
Quantisation of Angular Momentum — $1/ n^{2}$ energy dependence ka origin.
Ionisation Energy — series limit ( $n_{2} \to \infty$ ) ground state ko ionise karne ke barabar hai.
Hydrogen-like Ions and Z-scaling — $Z^{2}$ factor ke saath generalisation.