Surface chemistry — adsorption isotherms (Langmuir, Freundlich, BET), catalysis on surfaces

5.1.6 · Chemistry › Physical Chemistry (Advanced)

1. Adsorption KYA hai? (stage set karo)

Surface kyun adsorb karta hai: Surface ka atom ke paas unsatisfied bonds hote hain — uske bulk atom se kam neighbours hote hain, isliye usmein excess energy (surface free energy) hoti hai. Uspar ek molecule chipkaane se woh energy kam hoti hai. Isliye adsorption hamesha exothermic hota hai ( $Δ H_{a d s} < 0$ ).

2. Langmuir isotherm — scratch se DERIVE karo

Assumptions (model ka steel-frame):

Surface par identical sites ki fixed number hoti hai.
Monolayer hi — ek site par ek hi molecule, stacking nahi.
Adsorbed molecules ke beech koi interaction nahi.
Dynamic equilibrium: adsorption ki rate = desorption ki rate.

Maano $θ$ = sites ka occupied fraction (coverage), toh $(1 - θ)$ = empty fraction, $p$ = gas pressure.

Adsorption ki rate ∝ (gas ka aaana) × (empty sites): $r_{a d s} = k_{a} p (1 - θ)$ Kyun? Molecule ko (a) aana padega — frequency ∝ $p$ — aur (b) ek empty site milni chahiye — probability ∝ $(1 - θ)$ .

Desorption ki rate ∝ (occupied sites): $r_{d es} = k_{d} θ$ Kyun? Sirf bhari hui sites hi molecule release kar sakti hain.

Equilibrium par dono ko equal rakho: $k_{a} p (1 - θ) = k_{d} θ$ Maano $K = k_{a} / k_{d}$ (adsorption equilibrium constant). $θ$ ke liye solve karo: $K p (1 - θ) = θ \Rightarrow K p = θ (1 + K p)$

Linearized form (taaki data se $K$ nikaala ja sake). $θ = V / V_{m}$ (adsorbed volume / monolayer volume) leke:

\;\Rightarrow\; \boxed{\frac{p}{V}=\frac{1}{KV_m}+\frac{p}{V_m}}$$ $p/V$ vs $p$ plot karo → straight line, slope $=1/V_m$, intercept $=1/(KV_m)$. > [!example] Worked: $\theta$ find karo > $K = 2.0\ \text{atm}^{-1}$, $p = 0.50$ atm. > $\theta = \dfrac{2.0\times0.50}{1+2.0\times0.50} = \dfrac{1.0}{2.0}=0.50$. > **Yeh step kyun?** Boxed formula mein plug karo; 50% coverage matlab is pressure par aadhi sites bhari hain. $p$ ko double karke $1.0$ karo toh $\theta = 2/3$ milta hai, $1.0$ nahi — saturation dheere dheere hota hai, yahi toh baat hai. ![[5.1.06-Surface-chemistry-—-adsorption-isotherms-(Langmuir,-Freundlich,-BET),-catalysis-on-.png]] --- ## 3. Freundlich isotherm — empirical, messy real surfaces ke liye Real surfaces heterogeneous hoti hain (alag alag strength ke sites). Freundlich ek **fit** hai, derivation nahi: > [!formula] Freundlich > $$\frac{x}{m}=k\,p^{1/n}\qquad (n>1)$$ > $x/m$ = adsorbent ke per gram adsorbed mass. Log lekar linearize karo: > $$\log\frac{x}{m}=\log k+\frac{1}{n}\log p$$ > $\log(x/m)$ vs $\log p$ plot karo → slope $1/n$, intercept $\log k$. > [!intuition] Power law kyun? Kyunki sabse strong sites pehle bharti hain, phir weak wali. Jaise jaise $p$ badhta hai "effective $K$" girta rehta hai, jise $1/n<1$ wala single power $p^{1/n}$ mimic karta hai. Yeh **high $p$ par fail** karta hai (ismein saturation built-in nahi hai) — yahi iska honest limitation hai. > [!example] Worked: Freundlich slope > Data: $p=1$ par, $x/m = 4$; $p=100$ par, $x/m = 40$. > $\log 40 - \log 4 = \frac1n(\log100-\log1)$ → $1 = \frac1n(2)$ → $1/n = 0.5$, toh $n=2$. > **Yeh step kyun?** Log–log plot par slope directly $1/n$ *hota hai*. Phir $\log k = \log4 - 0.5\log1 = \log4$, toh $k=4$. --- ## 4. BET isotherm — multilayer (jahan Langmuir haar maanta hai) **BET kyun:** Physisorption layers stack karti hai. Brunauer–Emmett–Teller, Langmuir ko extend karte hain taaki molecules already-adsorbed molecules ke upar bhi adsorb ho sakein (layer 2, 3, … ke liye liquefaction energy use karte hue). > [!formula] BET equation > $$\frac{p}{V(p_0-p)}=\frac{1}{V_m c}+\frac{c-1}{V_m c}\cdot\frac{p}{p_0}$$ > $p_0$ = saturation vapour pressure; $c=\exp\!\big[(E_1-E_L)/RT\big]$ jahan $E_1$=1st layer adsorption ki heat, $E_L$=liquefaction ki heat. $\dfrac{p}{V(p_0-p)}$ vs $p/p_0$ plot karo → straight line. Slope $s=\frac{c-1}{V_mc}$, intercept $i=\frac{1}{V_mc}$. Phir $V_m = \dfrac{1}{s+i}$ aur $c = \dfrac{s}{i}+1$. **Yeh $V_m$ → surface area** (monolayer molecules ko count karo × har molecule ka area). BET surface area industrial standard hai. > [!intuition] Langmuir special case hai jab $p_0\to\infty$ (koi liquefaction nahi, koi layer nahi) → BET wapas ek layer par aa jaata hai. Hamesha accha lagta hai jab simple model, bade model ke andar zinda dikhe. --- ## 5. Surfaces par Heterogeneous catalysis > [!intuition] Catalyst ek *matchmaker* hai. Yeh reactants ko adsorb karta hai (chemisorption), **unke bonds ko stretch/kamzor karta hai**, unhe sahi orientation mein paas laata hai, woh react karte hain, phir products desorb hoke site free kar dete hain. Lower activation energy ⇒ tez reaction, catalyst regenerate. **Mechanism ke stages:** surface tak diffusion → adsorption → surface reaction → desorption → wapas diffusion. **Langmuir–Hinshelwood** (dono reactants adsorbed): rate $\propto \theta_A\theta_B$. **Eley–Rideal** (ek adsorbed, ek gas se): rate $\propto \theta_A\,p_B$. > [!intuition] Sabatier principle — "Goldilocks" rule. Binding bahut weak → reactant chipkega nahi → slow. Binding bahut strong → product niklega nahi (site poisoned) → slow. **Best catalyst intermediate bind karta hai**: yahi "volcano plot" deta hai activity vs binding strength ka. > [!example] Single-substrate catalysed rate kyun plateau karta hai > $A \xrightarrow{cat} P$ ke liye $r=k\theta_A$ aur $\theta_A=\frac{Kp_A}{1+Kp_A}$: > Low $p_A$: $r\approx kKp_A$ → **first order**. High $p_A$: $\theta_A\to1$, $r\to k$ → **zero order** (surface saturated). Ek hi model changing reaction order explain karta hai — yahi adsorption thinking ki power hai. --- ## 6. Common mistakes (Steel-man + fix) > [!mistake] "Langmuir multilayers allow karta hai." > *Kyun sahi lagta hai:* zyada pressure mein zyada gas hogi, toh zaroor stack hogi. *Galati:* Langmuir maanta hai **ek site par ek molecule** — site bhar gayi toh ho gaya. Stacking → woh **BET** hai. Fix: monolayer ⇒ $\theta=1$ par plateau; multilayer ⇒ BET diverge karta hai jab $p\to p_0$. > [!mistake] "Zyada temperature → zyada adsorption." > *Kyun sahi lagta hai:* high $T$ zyada cheezein tez karti hai. *Galati:* adsorption **exothermic** hai; Le Chatelier ke hisaab se heat ek product hai, toh $T$ badhane se equilibrium *wapas* gas ki taraf shift karta hai. Physisorption $T$ ke saath girta hai. (Chemisorption pehle *badh* sakta hai — shuru mein activation energy chahiye — phir girta hai: ek maximum hota hai.) Fix: yaad rakho $\Delta H_{ads}<0$. > [!mistake] "Freundlich aur Langmuir ka shape same hai." > *Kyun sahi lagta hai:* dono $p$ ke saath badhte hain. *Galati:* Freundlich ($p^{1/n}$) **kabhi saturate nahi karta**; Langmuir plateau karta hai. Fix: high $p$ par Freundlich badhta rehta hai (yahi iska jaana-maana failure region hai). > [!mistake] "Strong-binding catalyst sabse accha hota hai." > *Kyun sahi lagta hai:* strong binding = molecule ko strong activate karta hai. *Galati:* bahut strong binding mein product/intermediate kabhi niklega nahi → poisoning. Fix: **Sabatier / volcano** — intermediate binding jeetती hai. --- ## 7. Active-recall flashcards #flashcards/chemistry Adsorption hamesha exothermic kyun hota hai? ::: Adsorption molecular freedom ghata deta hai isliye $\Delta S<0$; $-T\Delta S>0$ ke saath $\Delta G<0$ ke liye, humein $\Delta H<0$ chahiye. Langmuir isotherm batao. ::: $\theta = Kp/(1+Kp)$, $K=k_a/k_d$. Langmuir derivation ki key equation? ::: Equilibrium par $r_{ads}=k_a p(1-\theta)$ aur $r_{des}=k_d\theta$ ko equal karo. Langmuir ka low-pressure limit? ::: $\theta\approx Kp$ (linear, Henry-jaisa). Langmuir ka high-pressure limit? ::: $\theta\to1$ (monolayer saturation). Linear Langmuir plot? ::: $p/V$ vs $p$: slope $1/V_m$, intercept $1/(KV_m)$. Freundlich isotherm aur uska log form? ::: $x/m=k p^{1/n}$; $\log(x/m)=\log k+(1/n)\log p$. Freundlich ki main weakness? ::: Saturation nahi — high pressure par fail hota hai. BET Langmuir mein kya add karta hai? ::: Upper layers ke liye liquefaction energy use karke multilayer adsorption. BET parameter $c$ ka matlab? ::: $c=\exp[(E_1-E_L)/RT]$; first-layer aur liquefaction binding ka ratio. BET $V_m$ kis kaam aata hai? ::: Surface area calculate karne mein (monolayer capacity × molecular area). Physisorption vs chemisorption energy? ::: Physisorption ~20 kJ/mol (vdW, multilayer); chemisorption ~80–400 kJ/mol (bond, monolayer). Langmuir–Hinshelwood rate law form? ::: $r\propto\theta_A\theta_B$ (dono adsorbed). Eley–Rideal rate law form? ::: $r\propto\theta_A\,p_B$ (ek adsorbed, ek gas-phase). Sabatier principle? ::: Best catalyst intermediates ko moderately bind karta hai — activity vs binding strength ka volcano plot deta hai. Surface-catalysed reaction 1st se 0 order mein kyun shift karta hai? ::: $r=k\theta$; low $p$ par $\theta\propto p$ (1st order), high $p$ par $\theta\to1$ (0 order). --- > [!recall]- Feynman: 12-saal ke bacche ko explain karo > Socho ek parking lot (surface) hai jisme fixed number of spots (sites) hain. Cars (gas molecules) aakar park karti hain, aur kuch nikal bhi jaati hain. "Rush hour" (high pressure) mein lot bhar jaata hai — ek baar bhar gaya toh aur cars nahi aayengi (yahi Langmuir ka plateau hai). Agar cars parked cars ki chhat par stack hone lagein, woh BET hai (multilayer). Catalyst ek dost parking attendant ki tarah hai: cars sirf itni der rukti hain jitni der mein passengers swap ho jaayein (react ho jaayein) aur phir jaldi nikal jaati hain, toh traffic tez chalta hai. Agar attendant cars ko bahut tight clamp kare, koi nahi niklega aur lot jam ho jaayega — isliye *best* catalyst molecules ko medium grip se pakaDta hai. > [!mnemonic] Teen isotherms yaad karo > **"L-F-B = Lid, Forever-up, Bunkbeds."** > **L**angmuir = ek **Lid** (monolayer, saturate hota hai). **F**reundlich = **Forever-up** (power law, kabhi saturate nahi hota). **B**ET = **Bunkbeds** (multilayers stack karti hain). > Aur catalysts ke liye: **"Stick, Stretch, Snap, Split"** = adsorb, bond kamzor karo, react karo, desorb karo. > [!intuition] Connections > - [[Chemical Kinetics — Rate Laws]] (reaction order, catalyst se activation energy kam hona) > - [[Chemical Equilibrium & Le Chatelier]] (exothermic adsorption ki T-dependence) > - [[Thermodynamics — Gibbs Free Energy]] (kyun $\Delta H_{ads}<0$ forced hai) > - [[Intermolecular Forces]] (van der Waals = physisorption; covalent = chemisorption) > - [[Henry's Law]] (low-pressure Langmuir limit) > - [[Colloids & Surface Tension]] (surface free energy) ## 🖼️ Concept Map ```mermaid flowchart TD SURF[Solid surface with sticky sites] ADS[Adsorption at interface] EXO[Always exothermic dH<0] ENTROPY[Lost translational freedom dS<0] PHYS[Physisorption weak vdW multilayer] CHEM[Chemisorption real bond monolayer] LANG[Langmuir isotherm] ASSUME[Monolayer fixed sites no interaction] EQUIL[Rate ads = rate des] THETA[Coverage theta = Kp / 1+Kp] LINEAR[Linearized p/V form] CAT[Surface catalysis] SURF -->|unsatisfied bonds| ADS ADS -->|dS<0 forces sign| EXO ENTROPY -->|dG<0 needs dH<0| EXO ADS -->|weak low T| PHYS ADS -->|strong Ea| CHEM ASSUME -->|steel-frame| LANG EQUIL -->|solve for theta| THETA LANG -->|assumes| EQUIL THETA -->|is the| LANG THETA -->|rearranged to extract K| LINEAR CHEM -->|weakens bonds| CAT ADS -->|enables| CAT ```