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

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1. WHAT is adsorption? (set the stage)

WHY a surface adsorbs at all: A surface atom has unsatisfied bonds — it has fewer neighbours than a bulk atom, so it carries excess energy (surface free energy). Sticking a molecule onto it lowers that energy. Hence adsorption is always exothermic ($\Delta H_{ads}<0$).

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2. The Langmuir isotherm — DERIVE it from scratch

Assumptions (steel-frame the model):

1. Surface has a fixed number of identical sites.
2. Monolayer only — one molecule per site, no stacking.
3. No interaction between adsorbed molecules.
4. Dynamic equilibrium: rate of adsorption = rate of desorption.

Let $\theta$ = fraction of sites occupied (coverage), so $(1-\theta)$ = fraction empty, $p$ = gas pressure.

Rate of adsorption ∝ (gas hitting) × (empty sites): $r_{ads} = k_a\, p\,(1-\theta)$ Why? A molecule must (a) arrive — frequency ∝ $p$ — and (b) find an empty site — probability ∝ $(1-\theta)$.

Rate of desorption ∝ (occupied sites): $r_{des} = k_d\,\theta$ Why? Only filled sites can release a molecule.

At equilibrium set them equal: $k_a\,p\,(1-\theta) = k_d\,\theta$ Let $K = k_a/k_d$ (the adsorption equilibrium constant). Solve for $\theta$: $Kp(1-\theta)=\theta \;\Rightarrow\; Kp = \theta(1+Kp)$

Linearized form (so you can extract $K$ from data). With $\theta = V/V_m$ (volume adsorbed / monolayer volume):

\;\Rightarrow\; \boxed{\frac{p}{V}=\frac{1}{KV_m}+\frac{p}{V_m}}$$ Plot $p/V$ vs $p$ → straight line, slope $=1/V_m$, intercept $=1/(KV_m)$. > [!example] Worked: find $\theta$ > $K = 2.0\ \text{atm}^{-1}$, $p = 0.50$ atm. > $\theta = \dfrac{2.0\times0.50}{1+2.0\times0.50} = \dfrac{1.0}{2.0}=0.50$. > **Why this step?** Plug into the boxed formula; coverage of 50% means half the sites are filled at this pressure. Doubling $p$ to $1.0$ gives $\theta = 2/3$, NOT $1.0$ — saturation is gradual, that's the point. ![[5.1.06-Surface-chemistry-—-adsorption-isotherms-(Langmuir,-Freundlich,-BET),-catalysis-on-.png]] --- ## 3. The Freundlich isotherm — empirical, for messy real surfaces Real surfaces are heterogeneous (sites of different strength). Freundlich is a **fit**, not a derivation: > [!formula] Freundlich > $$\frac{x}{m}=k\,p^{1/n}\qquad (n>1)$$ > $x/m$ = mass adsorbed per gram adsorbent. Linearize by taking logs: > $$\log\frac{x}{m}=\log k+\frac{1}{n}\log p$$ > Plot $\log(x/m)$ vs $\log p$ → slope $1/n$, intercept $\log k$. > [!intuition] WHY a power law? Because the strongest sites fill first, then weaker ones. The "effective $K$" keeps dropping as $p$ rises, which a single power $p^{1/n}$ with $1/n<1$ mimics. It **fails at high $p$** (no saturation built in) — that's its honest limitation. > [!example] Worked: Freundlich slope > Data: at $p=1$, $x/m = 4$; at $p=100$, $x/m = 40$. > $\log 40 - \log 4 = \frac1n(\log100-\log1)$ → $1 = \frac1n(2)$ → $1/n = 0.5$, so $n=2$. > **Why this step?** Slope on log–log plot directly *is* $1/n$. Then $\log k = \log4 - 0.5\log1 = \log4$, so $k=4$. --- ## 4. BET isotherm — multilayer (where Langmuir gives up) **WHY BET:** Physisorption stacks layers. Brunauer–Emmett–Teller extend Langmuir by letting molecules adsorb on top of already-adsorbed ones (using the liquefaction energy for layers 2,3,…). > [!formula] BET equation > $$\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; $c=\exp\!\big[(E_1-E_L)/RT\big]$ where $E_1$=heat of 1st-layer adsorption, $E_L$=heat of liquefaction. Plot $\dfrac{p}{V(p_0-p)}$ vs $p/p_0$ → straight line. Slope $s=\frac{c-1}{V_mc}$, intercept $i=\frac{1}{V_mc}$. Then $V_m = \dfrac{1}{s+i}$ and $c = \dfrac{s}{i}+1$. **This $V_m$ → surface area** (count monolayer molecules × area each). BET surface area is *the* industrial standard. > [!intuition] Langmuir is the special case $p_0\to\infty$ (no liquefaction, no layers) → BET collapses back to one layer. Always good to see the simpler model living inside the harder one. --- ## 5. Heterogeneous catalysis on surfaces > [!intuition] The catalyst is a *matchmaker*. It adsorbs reactants (chemisorption), **stretches/weakens their bonds**, holds them close in the right orientation, they react, then products desorb to free the site. Lower activation energy ⇒ faster reaction, catalyst regenerated. **Mechanism stages:** diffusion to surface → adsorption → surface reaction → desorption → diffusion away. **Langmuir–Hinshelwood** (both reactants adsorbed): rate $\propto \theta_A\theta_B$. **Eley–Rideal** (one adsorbed, one from gas): rate $\propto \theta_A\,p_B$. > [!intuition] Sabatier principle — the "Goldilocks" rule. Binding too weak → reactant won't stick → slow. Binding too strong → product won't leave (site poisoned) → slow. **Best catalyst binds intermediate**: this gives a "volcano plot" of activity vs binding strength. > [!example] Why a single-substrate catalysed rate plateaus > For $A \xrightarrow{cat} P$ with $r=k\theta_A$ and $\theta_A=\frac{Kp_A}{1+Kp_A}$: > Low $p_A$: $r\approx kKp_A$ → **first order**. High $p_A$: $\theta_A\to1$, $r\to k$ → **zero order** (surface saturated). One model explains a changing reaction order — that's the power of adsorption thinking. --- ## 6. Common mistakes (Steel-man + fix) > [!mistake] "Langmuir allows multilayers." > *Why it feels right:* more pressure should pile up more gas, surely it keeps stacking. *The flaw:* Langmuir assumes **one molecule per site** — once a site is full it's done. Stacking → that's **BET**. Fix: monolayer ⇒ plateau at $\theta=1$; multilayer ⇒ BET diverges as $p\to p_0$. > [!mistake] "Higher temperature → more adsorption." > *Why it feels right:* higher $T$ usually speeds things up. *The flaw:* adsorption is **exothermic**; by Le Chatelier heat is a product, so raising $T$ shifts equilibrium *back* to gas. Physisorption falls with $T$. (Chemisorption can first *rise* — it needs activation energy to start — then fall: a maximum.) Fix: remember $\Delta H_{ads}<0$. > [!mistake] "Freundlich and Langmuir are the same shape." > *Why it feels right:* both increase with $p$. *The flaw:* Freundlich ($p^{1/n}$) **never saturates**; Langmuir plateaus. Fix: at high $p$ Freundlich keeps climbing (its known failure region). > [!mistake] "A strong-binding catalyst is the best." > *Why it feels right:* strong binding = activates the molecule strongly. *The flaw:* too strong and the product/intermediate never leaves → poisoning. Fix: **Sabatier / volcano** — intermediate binding wins. --- ## 7. Active-recall flashcards #flashcards/chemistry Why is adsorption always exothermic? ::: Adsorption lowers molecular freedom so $\Delta S<0$; for $\Delta G<0$ with $-T\Delta S>0$, we need $\Delta H<0$. State the Langmuir isotherm. ::: $\theta = Kp/(1+Kp)$, $K=k_a/k_d$. Langmuir derivation key equation? ::: Equate $r_{ads}=k_a p(1-\theta)$ and $r_{des}=k_d\theta$ at equilibrium. Low-pressure limit of Langmuir? ::: $\theta\approx Kp$ (linear, Henry-like). High-pressure limit of Langmuir? ::: $\theta\to1$ (monolayer saturation). Linear Langmuir plot? ::: $p/V$ vs $p$: slope $1/V_m$, intercept $1/(KV_m)$. Freundlich isotherm and its log form? ::: $x/m=k p^{1/n}$; $\log(x/m)=\log k+(1/n)\log p$. Main weakness of Freundlich? ::: No saturation — fails at high pressure. What does BET add to Langmuir? ::: Multilayer adsorption using liquefaction energy for upper layers. BET parameter $c$ meaning? ::: $c=\exp[(E_1-E_L)/RT]$; ratio of first-layer to liquefaction binding. What is BET $V_m$ used for? ::: Calculating surface area (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? ::: $r\propto\theta_A\theta_B$ (both adsorbed). Eley–Rideal rate law form? ::: $r\propto\theta_A\,p_B$ (one adsorbed, one gas-phase). Sabatier principle? ::: Best catalyst binds intermediates moderately — gives a volcano plot of activity vs binding strength. Why does a surface-catalysed reaction shift from 1st to 0 order? ::: $r=k\theta$; $\theta\propto p$ at low $p$ (1st order), $\theta\to1$ at high $p$ (0 order). --- > [!recall]- Feynman: explain to a 12-year-old > Imagine a parking lot (the surface) with a fixed number of spots (sites). Cars (gas molecules) drive in and park, and some drive out again. At "rush hour" (high pressure) the lot fills up — once it's full, no more cars fit (that's Langmuir's plateau). If cars start stacking on roofs of parked cars, that's BET (multilayer). A catalyst is like a friendly parking attendant: cars stay just long enough to swap passengers (react) and then leave fast, so traffic flows quicker. If the attendant clamps the cars too hard, nobody leaves and the lot jams — that's why the *best* catalyst holds molecules with a medium grip. > [!mnemonic] Remember the three isotherms > **"L-F-B = Lid, Forever-up, Bunkbeds."** > **L**angmuir = a **Lid** (monolayer, saturates). **F**reundlich = **Forever-up** (power law, never saturates). **B**ET = **Bunkbeds** (multilayers stacking). > And catalysts: **"Stick, Stretch, Snap, Split"** = adsorb, weaken bond, react, desorb. > [!intuition] Connections > - [[Chemical Kinetics — Rate Laws]] (reaction order, activation energy lowered by catalyst) > - [[Chemical Equilibrium & Le Chatelier]] (T-dependence of exothermic adsorption) > - [[Thermodynamics — Gibbs Free Energy]] (why $\Delta H_{ads}<0$ is forced) > - [[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 ``` ## 🔊 Hinglish (regional understanding) > [!intuition]- Hinglish mein samjho > Dekho, surface chemistry ka core idea simple hai: kisi solid ki surface pe atoms ke paas adhoore (unsatisfied) bonds hote hain, isliye woh gas molecules ko apni taraf "chipka" leti hai — isko **adsorption** kehte hain (bulk ke andar ghusna **absorption** hota hai). Yeh process hamesha exothermic hota hai, kyunki molecule apni freedom khota hai ($\Delta S<0$), aur $\Delta G<0$ tabhi aayega jab $\Delta H<0$ ho. Yeh thermodynamics ka pakka rule hai. > > **Langmuir isotherm** ka derivation rato mat — samjho. Adsorption ka rate $= k_a p(1-\theta)$ (molecule aana chahiye = $p$, aur khaali site milni chahiye = $1-\theta$), aur desorption rate $=k_d\theta$. Equilibrium pe dono barabar, solve karo, mil jaata hai $\theta = Kp/(1+Kp)$. Low pressure pe yeh linear ($\theta\approx Kp$), high pressure pe surface bhar jaati hai ($\theta\to1$). Ek hi monolayer banti hai — yahi Langmuir ki assumption hai. > > **Freundlich** ($x/m=kp^{1/n}$) sirf ek empirical fit hai messy surfaces ke liye — log-log plot se $1/n$ aur $k$ nikaalte hain, par yeh saturate nahi karta (high $p$ pe fail). **BET** tab kaam aata hai jab molecules ek doosre ke upar layers banayein (multilayer) — isi se industry **surface area** measure karti hai. > > **Catalysis on surface**: catalyst ek matchmaker hai — reactant ko chipka ke uske bond ko weak karta hai, reaction fast hoti hai, phir product nikal jaata hai. **Sabatier principle** yaad rakho: binding na bahut weak ho na bahut strong — *medium* grip best hai (volcano plot). Bahut strong binding = product nikalta hi nahi = catalyst poison. Bas yahi sab exam aur real life dono mein kaam aata hai! ![[audio/5.1.06-Surface-chemistry-—-adsorption-isotherms-(Langmuir,-Freundlich,-BET),-catalysis-on-.mp3]]