Liquid state — vapour pressure, viscosity, surface tension

1. Vapour Pressure

WHAT is happening (Feynman-level): In a closed vessel, fast surface molecules escape (evaporate). As vapour builds up, some molecules crash back (condense). When rate of evaporation = rate of condensation, the vapour pressure stops changing — that steady value is the vapour pressure.

WHY does it depend on temperature? The fraction of molecules with enough energy to escape is set by the Boltzmann distribution. The number that can beat the escape barrier grows exponentially with $T$ . Deriving this:

The equilibrium liquid⇌vapour has $\Delta G = 0$ , so vapour pressure $p$ obeys the Clausius–Clapeyron relation. Starting from Clapeyron: $\frac{dp}{dT} = \frac{\Delta H_{vap}}{T\,\Delta V}$

Why this step? At phase equilibrium the molar Gibbs energies of both phases stay equal along the coexistence line, and equating their changes gives this slope.

Assume vapour is ideal and $V_{gas}\gg V_{liq}$ , so $\Delta V \approx V_{gas} = RT/p$ : $\frac{dp}{dT} = \frac{\Delta H_{vap}\,p}{RT^2} \;\Rightarrow\; \frac{d\ln p}{dT} = \frac{\Delta H_{vap}}{RT^2}$

Why this step? Substituting $\Delta V$ and dividing by $p$ turns the LHS into $d\ln p/dT$ — a clean, integrable form.

Integrating (assuming $\Delta H_{vap}$ constant):

Worked example Finding

$\Delta H_{vap}$ of water Water: $p_1 = 1.00$ atm at $T_1 = 373$ K; $p_2 = 2.00$ atm at $T_2 = 393$ K. Find $\Delta H_{vap}$ . $\ln\frac{2}{1} = -\frac{\Delta H_{vap}}{8.314}\left(\frac{1}{393}-\frac{1}{373}\right)$ Why this step? Plug measured pressures/temperatures directly into the integrated law. Inner bracket $= 2.544\times10^{-3}-2.681\times10^{-3} = -1.365\times10^{-4}\,\text{K}^{-1}$ . $0.693 = -\frac{\Delta H_{vap}}{8.314}(-1.365\times10^{-4}) \Rightarrow \Delta H_{vap} = \frac{0.693\times8.314}{1.365\times10^{-4}} \approx 42.2\text{ kJ/mol}$ Why sensible? Real water is ~40.7 kJ/mol — our constant- $\Delta H$ assumption gives a close value. ✓

Trends: Higher IMF ⇒ lower vapour pressure ⇒ higher boiling point. More volatile liquid = higher vapour pressure.

2. Surface Tension

HOW the two definitions are the same (derivation): Slide a wire of length $\ell$ by distance $dx$ against the surface, doing work $dW = F\,dx$ . This creates new area $dA = \ell\,dx$ (or $2\ell\,dx$ for a film with two faces). Energy per area: $\gamma = \frac{dW}{dA} = \frac{F\,dx}{\ell\,dx} = \frac{F}{\ell}$ Why this step? Force-per-length and energy-per-area are literally the same ratio — units confirm ( $\text{N/m}=\text{J/m}^2$ ).

Capillary rise (derivation from force balance): Weight of raised column = upward pull of surface tension around the circumference. $\underbrace{\rho g h \,\pi r^2}_{\text{weight}} = \underbrace{2\pi r\,\gamma\cos\theta}_{\text{upward force}}$ $\boxed{h = \frac{2\gamma\cos\theta}{\rho g r}}$ Why this step? Vertical surface-tension force acts along the wetted circumference ( $2\pi r$ ), tilted by contact angle $\theta$ ; balancing it against liquid weight gives the height.

Trends: ↑IMF ⇒ ↑surface tension. ↑Temperature ⇒ ↓surface tension (thermal motion weakens the inward pull).

Figure — Liquid state — vapour pressure, viscosity, surface tension

3. Viscosity

HOW it's defined (Newton's law of viscosity): Force to slide a layer is proportional to its area $A$ and to the velocity gradient $du/dz$ perpendicular to flow: $F = \eta\,A\,\frac{du}{dz}$ Why this step? Experiment shows drag grows with contact area and with how quickly speed changes between layers; $\eta$ is the proportionality constant defining that resistance.

Temperature dependence (Arrhenius-type — unlike gases!): $\eta = A\,e^{E_a/RT}$ Why this step? Flow needs molecules to hop over an energy barrier $E_a$ to slip past neighbours. More hopping-capable molecules at high $T$ ⇒ lower viscosity. Note the + sign: viscosity decreases as $T$ rises (honey flows better when warm).

Trends: ↑IMF ⇒ ↑viscosity. ↑Molecular size / H-bonding (glycerol, glucose) ⇒ ↑viscosity. ↑Temperature ⇒ ↓viscosity.

Common mistake Steel-manning the classic errors

(a) "Vapour pressure depends on the container's volume / surface area." Why it feels right: bigger surface → faster evaporation rate. Fix: rate is faster, but condensation also speeds up equally; equilibrium pressure depends only on $T$ and the liquid's IMF, not on how much liquid or area exists.

(b) "Higher temperature ⇒ higher viscosity (like it raises vapour pressure)." Why it feels right: many properties increase with $T$ . Fix: heat gives molecules energy to escape IMF traps and slip past each other — viscosity falls. Note the sign in $\eta = Ae^{+E_a/RT}$ .

(c) Confusing surface tension units. Why it feels right: "tension" sounds like force → students write N. Fix: it's force per length, N/m — equal to energy per area J/m².

(d) "Mercury rises in a capillary like water." Why it feels right: both are liquids in tubes. Fix: mercury has $\theta > 90°$ so $\cos\theta<0$ ⇒ $h<0$ ⇒ it depresses (non-wetting).

Recall Feynman: explain to a 12-year-old

Think of molecules as sticky little magnets.

Vapour pressure: how many jumpy magnets fly off the top of a puddle. Hotter = more jumpers = higher vapour pressure. Stronger stickiness = fewer jumpers.
Surface tension: the top magnets get pulled down and sideways by friends below but have no friends above, so the surface stretches tight like a trampoline skin — that's why a water strider walks on water.
Viscosity: how hard the magnets grip each other when you try to pour. Honey grips hard (thick), water grips loosely (thin). Warm it up and the grip loosens.

Connections

Intermolecular Forces — the common root cause of all three properties
Clausius-Clapeyron Equation — thermodynamic derivation of vapour pressure
Boiling Point and Phase Diagrams — boiling = vapour pressure meets external pressure
Gibbs Free Energy — $\Delta G=0$ at phase equilibrium underlies the derivation
Boltzmann Distribution — why escaping/hopping fractions grow exponentially with $T$
Capillary Action and Contact Angle — application of surface tension

Flashcards

Define vapour pressure.

Pressure of vapour in dynamic equilibrium with its liquid at a given temperature (closed container).

Does vapour pressure depend on surface area or liquid amount?

No — only on temperature and the liquid's intermolecular forces.

State the Clausius–Clapeyron equation.

\ln(p_2/p_1) = -\frac{\Delta H_{vap}}{R}(1/T_2 - 1/T_1)

What assumption lets us replace

\Delta V

with

RT/p

in the Clapeyron derivation?

Vapour is ideal and its volume far exceeds the liquid's.

When does a liquid boil?

When its vapour pressure equals the external pressure (normal BP at 1 atm).

Give the two equivalent definitions of surface tension.

Force per unit length (N/m) and energy per unit area (J/m²).

Why are small liquid drops spherical?

A sphere has minimum surface area per volume, minimizing surface energy.

Derive/state the capillary rise formula.

h = 2\gamma\cos\theta/(\rho g r)

from balancing column weight against surface-tension pull.

Why does mercury depress in a capillary?

Contact angle >90° so cosθ<0, giving negative h (non-wetting).

Define viscosity and give Newton's law.

Resistance to flow;

F = \eta A (du/dz)

How does viscosity depend on temperature?

Decreases with T:

\eta = A e^{+E_a/RT}

(molecules more easily hop past neighbours).

Effect of stronger IMF on the three properties?

↓vapour pressure, ↑surface tension, ↑viscosity.

Units of viscosity?

Poise (CGS) or Pa·s (SI); 1 Pa·s = 10 poise.

How does surface tension change with temperature?

Decreases (thermal motion weakens inward pull).

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, liquid ke teen properties — vapour pressure, surface tension, aur viscosity — sabki jadd (root) ek hi cheez hai: intermolecular forces (IMF) vs thermal energy ki ladai. Jab molecules ek doosre ko zyada strongly pakadte hain, toh unse bahar nikalna mushkil (kam vapour pressure), surface ka skin tight (high surface tension), aur ek doosre ke upar se slide karna hard (high viscosity). Yeh ek golden rule yaad rakho: strong forces ⇒ vapour pressure LOW, baaki dono HIGH.

Vapour pressure matlab — closed dabbe mein liquid se molecules udd ke vapour banate hain, aur kuch wapas girte hain; jab dono rate barabar ho jaate hain (dynamic equilibrium) tab jo pressure banta hai wahi vapour pressure hai. Temperature badhao toh jyada molecules escape kar paate hain, isliye vapour pressure exponentially badhta hai — Clausius–Clapeyron equation isko batati hai. Boiling tab hoti hai jab vapour pressure external pressure ke barabar ho jaye.

Surface tension — surface ke molecules ko neeche wale kheechte hain par upar koi nahi, isliye surface ek stretched trampoline jaisa tight ho jaata hai; isiliye water drop gol banti hai (least area). Capillary rise ka formula $h = 2\gamma\cos\theta/(\rho g r)$ force balance se aata hai — patli tube mein pani zyada upar chadhta hai. Mercury ka contact angle 90° se zyada hai, isliye wo neeche jaata hai (depression).

Viscosity ek important trap hai! Bahut students sochte hain temperature badhne se viscosity badhti hai — GALAT. Honey ko garam karo toh patli ho jaati hai na? Heat molecules ko energy deta hai barrier cross karne ke liye, isliye viscosity KAM hoti hai: $\eta = A e^{+E_a/RT}$ . Bas yeh sign yaad rakhna — "heat frees everything."

Liquid state — vapour pressure, viscosity, surface tension

1. Vapour Pressure

2. Surface Tension

3. Viscosity

Connections

Flashcards

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — States of Matter (Quantitative)

Connections