Spectral series — Lyman, Balmer, Paschen — Hinglish notes

2.3.15 · Physics › Modern Physics

Hum kya explain kar rahe hain

KYA: Hydrogen sirf kuch specific discrete wavelengths par light emit/absorb karta hai, jo named series mein organized hain.
KYUN: Energy levels quantized hain; unke beech ke transitions quantized hain; isliye wavelengths bhi quantized hain.
KAISE: Energy levels derive karo, difference lo, wavelength mein convert karo.

Bilkul scratch se Derivation (Bohr model)

$13.6$ eV kahaan se aata hai? Bohr ne Coulomb attraction ko centripetal need ke saath balance kiya, phir angular momentum $m v r = n ℏ$ ko quantize kiya. Result hai $E_{n} = - \frac{m e ^{4}}{8 ε _{0}^{2} h ^{2}} \cdot \frac{1}{n ^{2}} .$ Constants daalne par prefactor $13.6$ eV milta hai (yahi Rydberg energy hai). Tumhe constants ka pura pile yaad karne ki zaroorat nahi — bas yaad rakho ki yeh $- 13.6/ n^{2}$ hai.

Ab photon ki baat. Electron level $n_{2}$ (high) se $n_{1}$ (low) par girta hai, $n_{2} > n_{1}$ :

= 13.6\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)\ \text{eV}.$$ Yeh **positive** hai (energy photon ke roop mein release hoti hai). **Yeh step kyun?** Lower level zyada negative hoti hai, isliye neeche jaane par magnitude badh jaata hai → energy bahar aati hai. Ek photon $E_\text{photon} = h\nu = \dfrac{hc}{\lambda}$ carry karta hai. Dono ko equal karo: $$\frac{hc}{\lambda} = 13.6\,\text{eV}\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right).$$ > [!formula] Rydberg formula (master equation) > $$\boxed{\ \frac{1}{\lambda} = R_H\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)\ }\qquad n_2>n_1$$ > jahan **Rydberg constant** hai > $$R_H = \frac{13.6\ \text{eV}}{hc} \approx 1.097\times10^{7}\ \text{m}^{-1}.$$ > $1/\lambda$ **wavenumber** hai (lines per metre). Bada gap ⇒ bada $1/\lambda$ ⇒ **choti** wavelength. **$1/\lambda$ kyun use karte hain?** Kyunki energy $1/\lambda$ mein *linear* hai, formula clean rehta hai. --- ## Named series (landing level $n_1$ ke hisaab se grouped) | Series | $n_1$ (final) | $n_2$ (from) | Region | Woh region kyun | |--------|------|-----|--------|-----------------| | ==Lyman== | 1 | 2,3,4,… | ==Ultraviolet== | Sabse gehra landing ⇒ sabse bade gaps ⇒ sabse choti $\lambda$ | | ==Balmer== | 2 | 3,4,5,… | ==Visible== | Medium gaps ⇒ visible light (jo hum *dekhte* hain) | | ==Paschen== | 3 | 4,5,6,… | Infrared | Shallow gaps ⇒ lambi $\lambda$ | | Brackett | 4 | 5,6,… | Infrared | Aur bhi shallow | | Pfund | 5 | 6,7,… | Far infrared | Sabse shallow | ![[2.3.15-Spectral-series-—-Lyman,-Balmer,-Paschen.png]] > [!intuition] Series limit aur first line > - Kisi bhi series ki **first line** (sabse chota gap) $n_2 = n_1+1$ use karti hai → us series mein **sabse lambi** $\lambda$. > - **Series limit** $n_2 \to \infty$ use karta hai → $\frac{1}{\lambda}=R_H/n_1^2$ → us series ki **sabse choti** $\lambda$ (sabse zyada energy). Iske baad electron free (ionized) ho jaata hai. --- ## Worked examples > [!example] 1 — Lyman series ki sabse lambi wavelength > Lyman ⇒ $n_1=1$. Sabse lambi $\lambda$ ⇒ sabse chota gap ⇒ $n_2=2$. > $$\frac{1}{\lambda}=R_H\left(\frac{1}{1^2}-\frac{1}{2^2}\right)=1.097\times10^7\left(1-\tfrac14\right) > =1.097\times10^7\cdot\tfrac34=8.23\times10^6\ \text{m}^{-1}.$$ > $$\lambda = \frac{1}{8.23\times10^6}=1.215\times10^{-7}\ \text{m}=121.5\ \text{nm (UV)}.$$ > **$n_2=2$ kyun?** Sabse lambi wavelength = sabse kam energy = sabse chota jump = nearest higher level. > [!example] 2 — Red H-alpha line (Balmer, $3\to2$) > $$\frac{1}{\lambda}=R_H\left(\frac{1}{2^2}-\frac{1}{3^2}\right) > =1.097\times10^7\left(\tfrac14-\tfrac19\right)=1.097\times10^7\cdot\tfrac{5}{36}=1.524\times10^6\ \text{m}^{-1}.$$ > $$\lambda = 6.56\times10^{-7}\ \text{m}=656\ \text{nm}\ \text{(red, visible)}.$$ > **Yeh kyun important hai:** Yeh wahi famous deep-red line hai jo tum hydrogen lamp mein actually dekhte ho. > [!example] 3 — Paschen series ka series limit > Paschen ⇒ $n_1=3$, limit ⇒ $n_2\to\infty$ isliye $1/n_2^2\to0$: > $$\frac{1}{\lambda}=R_H\cdot\frac{1}{9}=1.219\times10^6\ \text{m}^{-1}\Rightarrow \lambda = 820\ \text{nm (infrared)}.$$ > **Yeh step kyun?** Series limit = electron bilkul free hone ki edge se aa raha hai; us series ka sabse zyada energy wala photon deta hai. > [!example] 4 — Balmer photon ki energy eV mein > $\Delta E = 13.6(\tfrac1{4}-\tfrac1{9}) = 13.6\cdot\tfrac{5}{36}=1.89\ \text{eV}$ use karo. > Check karo: $\lambda=\frac{1240\ \text{eV·nm}}{1.89\ \text{eV}}=656\ \text{nm}$. ✓ Example 2 se match karta hai. > **Dono methods kyun agree karte hain:** Dono ek hi energy gap use karte hain; eV-route aur $R_H$-route same physics hai. --- ## Forecast-then-Verify > [!recall]- Compute karne se pehle Forecast karo > Q: Bina compute kiye, kaunsi wavelength **choti** hogi — Lyman $2\to1$ ya Balmer $3\to2$? > > Forecast: Lyman jumps zyada gehra land karte hain ⇒ bada energy gap ⇒ choti $\lambda$. > Verify: Lyman = 121.5 nm, Balmer = 656 nm. **Lyman chota hai.** ✓ --- ## Common mistakes (Steel-man + fix) > [!mistake] "$n_1$ aur $n_2$ swap karna" — negative $1/\lambda$ milna > **Kyun sahi lagta hai:** Log reflex mein formula likhte hain aur from-level pehle daal dete hain. > **Trap:** $\frac{1}{n_2^2}-\frac{1}{n_1^2}$ emission ke liye negative hota hai. > **Fix:** **Emission** wavelengths ke liye, hamesha **lower level pehle** likho: $\frac{1}{n_1^2}-\frac{1}{n_2^2}>0$. > [!mistake] "Balmer pehli/sabse lowest series hai" > **Kyun sahi lagta hai:** Balmer historically pehle padhaya jaata hai kyunki yeh *visible* hai. > **Fix:** Lyman ($n_1=1$) sabse lowest, sabse zyada energetic series hai. Balmer bas woh hai jo humari aankhein pakad leti hain. > [!mistake] "Sabse lambi wavelength $n_2=\infty$ use karti hai" > **Kyun sahi lagta hai:** $\infty$ "sabse bada" answer lagta hai. > **Fix:** $n_2=\infty$ **sabse badi energy** deta hai = **sabse choti** $\lambda$ (series *limit*). Sabse lambi $\lambda$ = sabse chota jump = $n_2=n_1+1$. > [!mistake] "Rydberg formula kisi bhi atom ke liye kaam karta hai" > **Fix:** Yeh $R_H$ form sirf **hydrogen (ek electron)** ke liye hai. Charge $Z$ wale hydrogen-like ion ke liye $Z^2$ se multiply karo: $\frac1\lambda=R_H Z^2(\frac1{n_1^2}-\frac1{n_2^2})$. --- ## Active recall #flashcards/physics Hydrogen energy level formula ::: $E_n=-13.6/n^2$ eV Wavenumber ke liye Rydberg formula ::: $1/\lambda=R_H(1/n_1^2-1/n_2^2)$, $n_2>n_1$ Rydberg constant $R_H$ ki value ::: $1.097\times10^7\ \text{m}^{-1}$ Lyman series ka final level aur region ::: $n_1=1$, ultraviolet Balmer series ka final level aur region ::: $n_1=2$, visible Paschen series ka final level aur region ::: $n_1=3$, infrared Kaunsa transition kisi series mein SABSE LAMBI wavelength deta hai ::: $n_2=n_1+1$ (sabse chota gap) Series limit transition ::: $n_2\to\infty$, series ki sabse choti wavelength deta hai Lyman first line ki wavelength ::: 121.5 nm (UV) H-alpha ki wavelength (Balmer 3→2) ::: 656 nm (red) Emission mein sign positive kyun hota hai ::: Lower level zyada negative hoti hai; girne par positive energy release hoti hai eV gap ko nm mein jaldi convert kaise karein ::: $\lambda(\text{nm})=1240/\Delta E(\text{eV})$ Charge Z wale hydrogen-like ion ka formula ::: $1/\lambda=R_H Z^2(1/n_1^2-1/n_2^2)$ --- > [!recall]- Feynman: 12-saal ke bachche ko explain karo > Socho ek seedhi (staircase) jisme neeche ke steps bahut door-door hain aur upar ke steps ekdum paas-paas. Ek ball (electron) kisi step par baitha hai. Jab woh **neeche jump** karta hai, toh ek light ki flash nikalta hai. Neeche ke bade jumps **bright, high-energy** flashes dete hain (Lyman). Jo jumps 2nd step par land karte hain woh **visible** colours dete hain jo hum dekh sakte hain (Balmer). 3rd step par land karne wale jumps **gentle** hote hain, invisible heat-light dete hain (Paschen, infrared). Saari flashes ko **kis step par land karte hain** ke hisaab se group karo, aur tumhe spectral series mil jaayegi. Colour fix hota hai kyunki step heights fix hain — isliye hydrogen hamesha same colours mein glow karta hai! > [!mnemonic] Series order aur landing level > **"L**arry **B**uys **P**ie **B**efore **P**arty**"** → **L**yman(1), **B**almer(2), **P**aschen(3), **B**rackett(4), **P**fund(5). > Aur: **L**yman→**UV**, **B**almer→**V**isible, **P**aschen→**IR** = "UV-Vis-IR seedhi chadhte waqt." --- ## Connections - [[Bohr Model of the Atom]] — quantized energy levels provide karta hai. - [[Photon Energy and Planck's Relation]] — $E=hc/\lambda$ gap ko wavelength se jodta hai. - [[Hydrogen-like Ions and Z dependence]] — $R_H \to R_H Z^2$ generalize karta hai. - [[Ionization Energy]] — series limit = level $n_1$ se electron ko free karne ki energy. - [[Emission vs Absorption Spectra]] — same lines, jump ka opposite direction. - [[Quantization of Angular Momentum]] — woh assumption jo levels ko discrete banata hai. ## 🖼️ Concept Map ```mermaid flowchart TD Q[Quantized energy levels] -->|En equals -13.6 over n squared| E[Energy of level n] B[Bohr model] -->|derived from| Q E -->|take difference n2 to n1| D[Energy gap Delta E] D -->|released as| P[Photon] P -->|E equals hc over lambda| R[Rydberg formula] D -->|set equal to hc over lambda| R R -->|1 over lambda equals RH times gap| W[Wavenumber and wavelength] R -->|group by final level n1| S[Spectral series] S -->|n1 equals 1| L[Lyman ultraviolet] S -->|n1 equals 2| BA[Balmer visible] S -->|n1 equals 3| PA[Paschen infrared] L -->|biggest gaps give| SW[Shorter wavelength] PA -->|shallow gaps give| LW[Longer wavelength] ```