Chapter 5 — States Of Matter Gases And Liquids

Fig 1 — Educational Diagram: Kinetic Molecular Theory of Gases

What it is
The Kinetic Molecular Theory (KMT) of gases explains macroscopic gas behaviour (pressure, temperature, diffusion) using the motion and collisions of a large number of tiny particles (molecules or atoms). It links temperature to the average kinetic energy of gas particles.

Basic assumptions

• Gases consist of a large number of identical particles moving in random straight-line motion.
• The volume of the individual gas particles is negligible compared with the volume of the container (particles are point-like).
• There are no long-range attractive or repulsive forces between particles except during collisions (ideal gas behaviour).
• Collisions between gas particles and with the walls of the container are perfectly elastic (no net loss of kinetic energy).
• The time spent in collisions is negligible compared with the time between collisions.
• The average kinetic energy of the particles is proportional to the absolute temperature: temperature measures the average translational kinetic energy.

Derivation sketch: pressure from particle motion
Consider N molecules each of mass m in a cubic container of side L (volume V = L³). For one molecule with velocity component v_x, colliding elastically with a wall changes its momentum by Δp = 2m v_x. Time between successive collisions with that wall is Δt = 2L / |v_x|. So average force by that molecule on the wall is F = Δp / Δt = m v_x² / L. Summing over all molecules and using isotropy (〈v_x²〉 = 〈v_y²〉 = 〈v_z²〉 = (1/3)〈v²〉) gives:

P = (1/3) (N/V) m <v²>

where <v²> is the mean square speed. Using kinetic energy of a particle (½ m <v²>),

P V = (2/3) N × (average translational kinetic energy per molecule)

Comparing with the ideal gas equation PV = NkT leads to

average translational kinetic energy per molecule = (3/2) k T

and for one mole (N = N_A), total translational kinetic energy per mole = (3/2) R T.

Key physical consequences

• Temperature is a measure of the average translational kinetic energy of gas molecules: higher T → faster molecules.
• Pressure arises from collisions of molecules with the container walls; more/stronger collisions → higher pressure.
• At the same temperature, lighter molecules move faster than heavier ones (important for diffusion/effusion).

Limits and real gases
KMT describes ideal gases. Real gases deviate at high pressure and low temperature where molecular volume and intermolecular forces become significant. Corrections (e.g., Van der Waals equation) account for these effects.

Fig 2 — Educational Diagram: Gas Laws and Ideal Gas Equation

Overview
Gas laws are empirical relationships that describe how pressure (P), volume (V), temperature (T) and amount (n, moles) of a gas are interrelated. The ideal gas equation unifies these relationships into one formula valid for ideal gases and is a good approximation for many real gases at moderate pressures and temperatures.

Key empirical laws

• Boyle's law (constant T, n): P ∝ 1/V or PV = constant. If temperature and amount are fixed, pressure increases as volume decreases.
• Charles's law (constant P, n): V ∝ T. For fixed pressure and amount, volume is directly proportional to absolute temperature (T in K). Extrapolating V to zero gives absolute zero (~0 K).
• Gay-Lussac's law (constant V, n): P ∝ T. At fixed volume and amount, pressure is proportional to absolute temperature.
• Avogadro's law (constant T, P): V ∝ n. Equal volumes of gases, at the same T and P, contain equal numbers of molecules.
• Dalton's law of partial pressures: For a mixture of non-reacting gases, total pressure Ptot = Σ Pi, where Pi = partial pressure of component i. Pi = Xi Ptot, where Xi is mole fraction.

Combined gas law
For a fixed amount of gas (n constant): (P1 V1)/T1 = (P2 V2)/T2. This combines Boyle's, Charles's and Gay-Lussac's laws.

Ideal gas equation
PV = nRT, where R is the universal gas constant. Common values: R = 0.082057 L·atm·K-1·mol-1 = 8.314 J·K-1·mol-1. Use consistent units: P in Pa (or atm), V in m3 (or L), T in K, n in mol.

Derived relations

• Density: rho = m/V = PM/(RT), where M is molar mass (kg mol-1 or g mol-1 with consistent units).
• Moles from conditions: n = PV/(RT).
• Molar volume at STP (273 K, 1 atm): Vm ≈ 22.414 L mol-1.

Kinetic Molecular Theory (KMT) and molecular-level interpretation
Assumptions: gases consist of many small particles in random motion; particle volume is negligible compared to container volume; no intermolecular forces except elastic collisions; collisions are elastic. Consequences: average kinetic energy ∝ T (in K). Root-mean-square speed u_rms = sqrt(3RT/M) where M is molar mass in kg mol-1 and R = 8.314 J mol-1 K-1.

Real gases and deviations
At high pressures and low temperatures, real gases deviate from ideality because particle volumes and intermolecular attractions become significant. van der Waals equation corrects for these: (P + a(n/V)^2)(V - nb) = nRT, where a corrects attractive forces and b corrects finite molecular volume.

When to use the ideal gas equation
PV = nRT is reliable at low to moderate pressure and temperatures well above condensation points. For high precision near liquefaction or at high pressures, use real-gas models (van der Waals or empirical compressibility factors).

Important notes for CBSE Class 11
Be able to: apply PV = nRT to solve mole, mass, volume and density problems; convert temperature to Kelvin; apply combined gas law for transformations; use Dalton's law for gas mixtures; understand kinetic basis of temperature and derive or use u_rms formula; recognize limits of ideal behavior and the purpose of van der Waals corrections.

Fig 3 — Educational Diagram: Dalton's Law of Partial Pressures

Statement: Dalton's law of partial pressures states that for a mixture of non-reacting gases occupying the same volume at the same temperature, the total pressure exerted by the mixture equals the sum of the partial pressures of the individual gases. The partial pressure of a gas is the pressure it would exert if it alone occupied the entire volume at the same temperature.

Derivation (using ideal gas law): For gas i, ideal gas law gives Pi = niRT/V. For a mixture of gases 1, 2, ..., n in the same V and T, the total pressure is

PTotal = P1 + P2 + ... + Pn = (n1 + n2 + ... + nn) RT / V = ntotal RT / V

Define mole fraction Xi = ni / ntotal. Then the partial pressure of component i can be written as

Pi = ni RT / V = Xi (ntotal RT / V) = Xi PTotal

Key assumptions and limitations:

• Gases behave ideally (valid at low pressure and moderate temperature).
• No chemical reactions or strong interactions between components.
• Deviations occur at high pressures, low temperatures, or with polar/associating gases.

Physical meaning: Each gas in a mixture acts independently and contributes to the total pressure in proportion to its amount (moles or mole fraction). Measuring total pressure and one component's partial pressure allows determination of composition.

Practical use: Dalton's law is used to correct measured gas pressures when vapour is present (e.g., collecting gas over water), to calculate partial pressures from composition (e.g., air at sea level), and in breathing-gas management (scuba diving, medical gas mixes).

Fig 4 — Educational Diagram: Kinetic Energy and Temperature

Core idea: Temperature is a measure of the average translational kinetic energy of the particles (atoms or molecules) in a substance. For gases, especially ideal gases, the connection between kinetic energy and temperature is direct and quantitative.

Microscopic picture: A gas consists of a large number of particles moving in random directions with a distribution of speeds. Each particle of mass m and speed v has instantaneous translational kinetic energy (1/2)mv². Because speeds vary, we use averages: the mean kinetic energy per particle is directly proportional to the absolute temperature (Kelvin).

Key relationships (conceptual):

• Temperature (T, in K) measures the average translational kinetic energy per particle: higher T → higher average molecular speeds.
• At the same temperature, lighter molecules move faster on average than heavier ones (because 1/2 mv² must be similar for different m at same T).
• Kelvin scale is required because zero kinetic energy (classically) corresponds to 0 K.

Equipartition of energy (brief): For each degree of freedom that appears quadratically in the energy, the average energy contribution is (1/2)kT per particle (k = Boltzmann constant). For a monatomic gas, there are three translational degrees → average translational energy = (3/2)kT per particle.

Consequences for macroscopic properties: Because pressure in a gas arises from particle collisions with container walls, higher average kinetic energy (higher T) increases the momentum transfer per collision and/or collision frequency, giving higher pressure (at fixed volume) or larger volume (at fixed pressure).

Limitations: The simple proportionality between translational kinetic energy and temperature holds exactly for ideal gases and for translational motion. Internal (rotational/vibrational/electronic) energy of molecules also contributes to total internal energy, and equipartition gives how that contribution depends on temperature and molecular structure.

Fig 5 — Educational Diagram: Maxwell–Boltzmann Distribution of Molecular Speeds

What it describes
The Maxwell–Boltzmann distribution gives the probability distribution of speeds of molecules in an ideal (classical) gas at a given temperature. It shows how many molecules have a particular speed v. The distribution is continuous and depends on temperature (T) and the mass of a molecule (m).

Distribution function and meaning
If f(v) is the distribution function, then f(v)dv is the fraction (or probability) of molecules with speeds between v and v + dv. The functional form (for a single species of molecules) is:

f(v) = 4π (m / 2πkT)^(3/2) v^2 exp(−mv^2 / 2kT)

Here k is Boltzmann's constant and m is the mass of one molecule. The factor v^2 makes the distribution start at 0 for v = 0, rise to a peak, then fall off exponentially for large v.

Key characteristic speeds

• Most probable speed (v_mp) — the speed at the peak of the distribution: v_mp = sqrt(2kT/m) = sqrt(2RT/M).
• Average (mean) speed (v_avg) — arithmetic mean of speeds: v_avg = sqrt(8kT/πm) = sqrt(8RT/πM).
• Root-mean-square speed (v_rms) — related to kinetic energy: v_rms = sqrt(3kT/m) = sqrt(3RT/M).

These satisfy v_rms > v_avg > v_mp.

Physical implications
At a fixed temperature, lighter molecules (smaller m) move faster on average than heavier ones. Increasing temperature broadens the distribution and shifts the peak to higher speeds: more molecules have higher speeds. The area under the whole curve is 1 (normalization, representing 100% of molecules).

Assumptions and limitations
This classical distribution assumes non-interacting point-like particles (ideal gas) and is valid when quantum effects are negligible (typical for gases at ordinary temperatures and pressures). It does not apply to condensed phases or quantum-degenerate gases at very low temperatures.

Numerical illustration (air — approximated by N₂ at 300 K)
For N₂ (M ≈ 28.0 g mol⁻¹ = 0.028 kg mol⁻¹) at T = 300 K:
v_mp ≈ 422 m s⁻¹, v_avg ≈ 476 m s⁻¹, v_rms ≈ 517 m s⁻¹. These show typical molecular speeds in air are several hundred metres per second.

Fig 6 — Educational Diagram: Collision Frequency, Mean Free Path and Transport Phenomena

Overview: In a gas, molecules move randomly and collide with one another. Two related microscopic quantities are collision frequency (how often a molecule collides) and mean free path (the average distance a molecule travels between collisions). These microscopic properties govern macroscopic transport phenomena such as diffusion, viscosity and thermal conductivity.

Collision frequency: Consider identical hard-sphere molecules of diameter d and number density n (molecules per unit volume). The effective collision cross-section for two spheres is σ = πd². The average relative speed between two molecules is approximately v_rel,avg = √2 · v_avg, where v_avg is the average molecular speed. The collision frequency (collisions per second experienced by one molecule) is

z = n · σ · v_rel,avg ≈ √2 · n · π d² · v_avg

Thus collision frequency increases with number density (pressure) and with average speed (temperature).

Mean free path: The mean free path λ is the average distance a molecule travels between successive collisions. For identical hard spheres

λ = 1 / (√2 · π · d² · n)

Using ideal-gas relations (n = p / (k_B T)), this becomes

λ = k_B T / (√2 · π · d² · p)

Because z and λ are related through molecular speed, we have

z = v_avg / λ

Transport phenomena (overview):

• Diffusion (D): random molecular motion causes net mass transfer from high to low concentration. A simple kinetic estimate is D ≈ (1/3) · v_avg · λ. Since v_avg ∝ √T and λ ∝ T/p, D ∝ T^(3/2) / p.
• Viscosity (η): momentum transport by molecules causes internal resistance to flow. For a gas, η ≈ (1/3) · ρ · v_avg · λ (ρ = mass density). Using ideal-gas relations, η ∝ √T and is approximately independent of pressure at low densities.
• Thermal conductivity (κ): energy transport by molecules. κ has a similar dependence on v_avg and λ and thus increases with √T.

Physical meaning and trends:

• At higher pressure (larger n) molecules are closer → λ decreases and collision frequency z increases.
• At higher temperature molecules move faster → v_avg and z increase; λ also increases because thermal expansion reduces n (if pressure held constant), so transport coefficients change accordingly.
• Because λ is typically very small (nanometre scale at STP), continuum descriptions (fluid mechanics) work well for most macroscopic situations. When λ becomes comparable to system size (high vacuum), free-molecular behaviour appears (effusion, molecular beams).

Connection to macroscopic observables: Pressure arises from momentum transfer in collisions with container walls. Transport properties (D, η, κ) measured in experiments can be explained quantitatively using λ and v_avg.

Fig 7 — Educational Diagram: Graham's Law of Diffusion and Effusion

Definition: Diffusion is the spontaneous mixing of gases (or liquids) due to the random motion of molecules. Effusion is the process by which gas molecules escape through a very small hole (smaller than the mean free path) into a vacuum or much lower pressure region without collisions in the hole.

Qualitative idea: Lighter gas molecules move faster (at the same temperature) than heavier ones. Because rate of spreading (diffusion) or escaping (effusion) depends on molecular speed, lighter gases diffuse and effuse faster than heavier gases.

Key result (Graham's law): At the same temperature, the rate of diffusion or effusion of a gas is inversely proportional to the square root of its molar mass. If r1 and r2 are rates of two gases of molar masses M1 and M2,

r1 / r2 = sqrt(M2 / M1)

Derivation (simple kinetic-theory argument):

• Root-mean-square speed of molecules: vrms = sqrt(3RT / M) (R = gas constant, T = absolute temperature, M = molar mass in kg mol⁻¹). Thus vrms ∝ 1 / sqrt(M).
• For effusion (and approximately for diffusion in dilute gases), rate r ∝ average molecular speed (vrms). Therefore r ∝ 1 / sqrt(M).
• Combining for two gases gives r1 / r2 = sqrt(M2 / M1). Equivalently, the times taken to effuse are inversely proportional to rates: t1 / t2 = sqrt(M1 / M2).

Assumptions and limitations:

• Applies best at low pressure and when the hole is small enough that molecules pass without collisions (effusion) or when gases are ideal and dilute (for diffusion approximation).
• Does not hold well when intermolecular forces are important, gases are non-ideal, pores are large (bulk flow), or when chemical interactions occur.

Applications: Explains why helium-filled balloons lose lift faster than air-filled ones, why the smell of perfume is detected across a room (diffusion), and was historically used in gaseous diffusion methods for isotope separation (e.g., enrichment of uranium hexafluoride).

Connection with Maxwell-Boltzmann distribution: At a given temperature, the speed distribution is shifted to higher values for lighter molecules. This distribution picture explains why lighter gases have both larger mean speeds and higher likelihood to escape or spread faster.

Fig 8 — Educational Diagram: Real Gases and Deviations from Ideal Behaviour

Overview: An ideal gas is a theoretical gas that follows the ideal gas equation PV = nRT under all conditions. Real gases depart from ideal behaviour at high pressures and low temperatures because (1) gas molecules occupy a finite volume and (2) intermolecular forces (mostly attractive) become significant. These causes lead to measurable deviations captured by corrective equations and parameters.

Causes of Deviation:

• Finite molecular volume: Molecules have finite size, so the free volume available for motion is less than the container volume. This effect becomes important at high pressures (small volumes).
• Intermolecular forces: Attractive forces reduce the momentum transfer to container walls (lower pressure than ideal), especially at moderate pressures and low temperatures; at very high pressures repulsive forces dominate.

Quantitative Measures:

• Compressibility factor Z = PV/(nRT) (or Z = PV_m/RT per mole). For an ideal gas Z = 1; for real gases Z differs from 1. If Z < 1 attractions dominate; if Z > 1 repulsions/finite volume effects dominate.
• Virial equation (expansion to account for interactions): Z = 1 + B(T)/V_m + C(T)/V_m^2 + ... or equivalently Z = 1 + B'(T)P + C'(T)P^2 + ... where B, C are virial coefficients dependent on temperature. B(T) (second virial coefficient) indicates pairwise interactions: positive for net repulsion, negative for net attraction. The temperature where B(T) = 0 is the Boyle temperature, at which gas behaves ideally over a wide range of pressures.
• Van der Waals equation: A simple correction for real gas behavior: (P + a(n/V)^2)(V - nb) = nRT. Per mole form: (P + a/V_m^2)(V_m - b) = RT. Here a accounts for attractive forces and b for finite molecular volume. When a = b = 0 this reduces to the ideal gas law.

Physical consequences & phenomena:

• At low temperatures near the boiling point, gases can liquefy; isotherms deviate strongly from ideal behaviour and show a flat portion (phase coexistence) in experimentally measured P–V isotherms. The van der Waals isotherms predict such behaviour and a critical point where liquid and gas become indistinguishable.
• At very high pressures, the finite volume of molecules (b) forces Z > 1.
• Near moderate pressures and temperatures below a certain range, attractive forces cause Z < 1.

Critical behavior: Van der Waals theory gives critical constants (where the isotherm has an inflection point): T_c = 8a/(27bR), P_c = a/(27b^2), V_c (per mole) = 3b. Above T_c there is no liquid–gas transition (supercritical fluid).

Summary: Real gases behave approximately ideally at low pressure and high temperature. Deviations are described by compressibility factor Z, virial expansion, and empirical/theoretical equations like van der Waals. Understanding a and b (or virial coefficients) helps predict when gases will liquefy, how pressure changes with volume, and properties of supercritical fluids.

Fig 9 — Educational Diagram: van der Waals Equation of State

Overview
The van der Waals equation is a modified form of the ideal gas law that accounts for finite molecular size and intermolecular attractions. It provides a better description of real gases, especially at high pressures and low temperatures where deviations from ideal behaviour are significant.

Equation (per mole)
(P + a/V_m²)(V_m - b) = RT
where P = pressure, V_m = molar volume (V/n), T = temperature, R = gas constant, and a, b are gas-specific van der Waals constants.

Physical meaning of constants
- a corrects for intermolecular attractive forces: the internal pressure is effectively reduced by a/V_m², so the measured pressure is increased by this term to give the idealized pressure.
- b corrects for finite molecular volume (excluded volume): the available free volume is reduced from V_m to V_m − b.

Alternate forms
For n moles: (P + a n²/V²)(V − n b) = nRT. Equivalent pressure form: P = RT/(V_m − b) − a/V_m².

Key predictions and features
- At low pressures and high temperatures (large V_m), the a and b corrections are small and the equation reduces to the ideal gas law PV = RT.
- The equation predicts non-ideal isotherms with an inflection point at the critical point. It can describe liquefaction and a critical temperature above which no liquid can form.
- Critical constants (in terms of a and b):
   P_c = a/(27 b²) , V_c = 3 b , T_c = 8 a/(27 b R).

Compressibility factor and law of corresponding states
The compressibility factor Z = P V_m/(R T) deviates from 1 for real gases. Using reduced variables (P_r=P/P_c, V_r=V_m/V_c, T_r=T/T_c), the van der Waals equation takes a universal reduced form, which leads to the law of corresponding states: gases at the same reduced conditions have approximately similar behaviour.

Limitations
van der Waals is a simple improvement over ideal gas law but remains approximate: it does not account for molecular shape, polar interactions in detail, or accurate quantitative predictions near criticality for many gases. More complex equations of state (Redlich–Kwong, Peng–Robinson) are used where higher accuracy is needed.

Use in practice
It explains qualitative behaviour of real gases (deviation from ideality, liquefaction, critical phenomena) and provides a basis for estimating conditions for liquefaction, storage, and design calculations where moderate accuracy is acceptable.

Fig 10 — Educational Diagram: Critical Phenomena and Liquefaction of Gases

Overview: Critical phenomena describe the behaviour of a substance near the critical point — the unique temperature and pressure at which the distinction between liquid and gas disappears. Liquefaction of gases is the process of converting a gas into liquid by cooling and/or compressing it below its critical temperature and above its critical pressure.

Critical point and critical constants
The critical point is defined by three critical constants: critical temperature (T_c), critical pressure (P_c) and critical molar volume (V_c). For T > T_c no amount of pressure alone can liquefy the gas. At T = T_c the P–V isotherm has an inflection point where gas and liquid become identical (density of liquid = density of vapour).

Van der Waals description and mathematical conditions
A simple real‑gas model is the van der Waals equation: (P + a/v2)(v - b) = RT (v = molar volume). The critical point is found where the P–V isotherm has a horizontal inflection, so:

• (∂P/∂V)T = 0
• (∂2P/∂V2)T = 0

From these conditions (for van der Waals) one obtains:

• Tc = 8a / (27Rb)
• Pc = a / (27b2)
• Vc = 3b

Reduced variables and law of corresponding states
Define reduced variables: P_r=P/P_c, T_r=T/T_c, V_r=V/V_c. The law of corresponding states states that all fluids at the same reduced conditions have similar behaviour.

Physical features near the critical point
- The latent heat of vaporisation becomes zero at the critical point.
- Liquid and vapour densities merge continuously (no phase boundary).
- Above T_c the substance is a supercritical fluid — it has liquid‑like density and gas‑like viscosity/diffusivity.

Liquefaction of gases (practical methods)
To liquefy a gas you must bring it below its T_c and apply sufficient pressure (or remove heat while compressing). Common methods include:

• Compression and cooling: Compress the gas and remove heat (isobaric cooling or successive compression with intercooling) until T < T_c, then further compression liquefies it.
• Joule–Thomson (throttling) cooling: A high‑pressure gas is expanded through a throttling valve (porous plug) to a lower pressure at constant enthalpy; if the Joule–Thomson coefficient μ_JT = (∂T/∂P)_H > 0 at the starting temperature, the gas cools. Repeated cycles in a counter‑current heat exchanger (Linde process) achieve liquefaction.
• Adiabatic expansion / rapid expansion: Gas expands in a turbine (doing work) and cools (Claude process uses expansion work plus Joule‑Thomson stage for efficiency).

Joule–Thomson details
The Joule–Thomson coefficient is defined as: μJT = (∂T/∂P)H. Most gases cool on throttling at ambient temperatures if they are below their inversion temperature; hydrogen and helium at room temperature warm on throttling unless precooled.

Practical significance & examples
Industrial liquefaction (air separation) produces liquid O₂, N₂ and Ar by cycles using compression, counter‑current heat exchange and throttling. Supercritical CO₂ is widely used as a solvent in decaffeination and extraction because it has tunable density and diffusion properties.

Summary: The critical point marks the end of the liquid–vapour coexistence curve; below T_c gases can be liquefied by cooling and/or compression, while above T_c only a supercritical fluid exists. Practical liquefaction combines compression, heat removal and Joule–Thomson or expansion cooling cycles.

Fig 11 — Educational Diagram: Properties of Liquids and Intermolecular Forces

Overview
Liquids have a definite volume but no fixed shape. Their macroscopic behaviour is governed by intermolecular forces (IMFs) — interactions between molecules — which determine properties such as vapour pressure, boiling point, surface tension, viscosity, capillarity and diffusion.

Intermolecular Forces (IMFs)

• London dispersion forces (instantaneous dipole–induced dipole): present in all molecules; strength increases with polarizability (molar mass, surface area).
• Dipole–dipole interactions: between polar molecules; orientation-dependent and stronger than dispersion for comparable sizes.
• Hydrogen bonding: a special, strong dipole interaction when H is bonded to F, O, or N and interacts with a lone pair on another F/O/N. Responsible for many anomalous properties of water (high b.p., surface tension).
• Ion–dipole: between ions and polar molecules; important in solutions (e.g., salt dissolving in water).

Stronger IMFs → lower vapour pressure, higher boiling point, higher surface tension and viscosity, slower diffusion and lower compressibility.

Key Properties of Liquids

• Vapour pressure: equilibrium pressure of vapour above a liquid. More volatile liquids have higher vapour pressure. Vapour pressure increases with temperature. The Clausius–Clapeyron relation (approximate) links vapour pressure and temperature: ln P = -ΔH_vap/(R T) + C.
• Boiling point: temperature where vapour pressure equals external pressure. At high altitude (lower external pressure), boiling point decreases.
• Surface tension (γ): energy required to increase surface area or force per unit length at surface. Caused by net inward cohesive forces on surface molecules. Surface tension decreases with temperature and is increased by strong IMFs (e.g., H-bonding).
• Viscosity (η): resistance to flow due to internal friction. Higher for stronger IMFs and for larger/chain-like molecules. Viscosity decreases with increasing temperature for liquids.
• Capillary action: rise or fall of a liquid in a narrow tube due to balance of surface tension and weight; depends on adhesive vs cohesive forces and contact angle θ.
• Diffusion: slower in liquids than in gases because of closer packing and stronger IMFs.
• Compressibility: liquids are nearly incompressible due to close packing of molecules.

Microscopic picture
At molecular level, molecules in a liquid are closely packed but not fixed; they have short-range order. IMFs create potential-energy minima at an equilibrium intermolecular distance; thermal energy allows molecules to move around these positions, enabling flow but resisting separation into gas (requires overcoming IMFs).

Temperature dependence
Increasing temperature increases kinetic energy relative to IMF strength: vapour pressure and diffusion increase; surface tension and viscosity decrease; hydrogen bonds may break at higher temperatures.

Measurement and practical notes
Surface tension measured by capillary rise, drop-weight or ring (Du Noüy) methods. Viscosity measured by flow times in viscometers or by measuring terminal velocity (Stokes law for small spheres).

Summary
Properties of liquids result from a balance between thermal motion and intermolecular attractions. Identifying the dominant IMF (dispersion, dipole, H-bonding, ion–dipole) explains trends across substances and practical phenomena (e.g., why water wets glass but mercury doesn't, why oils are viscous, why surfactants change surface tension).

Fig 12 — Educational Diagram: Vapor Pressure, Evaporation and Boiling

Overview
Vaporization is the process by which molecules leave the liquid phase and enter the gas phase. It occurs in two ways: evaporation (surface phenomenon at any temperature) and boiling (bulk phenomenon at a definite temperature when vapor pressure equals external pressure).

Vapor pressure
Vapor pressure of a liquid is the pressure exerted by its vapor when the liquid and vapor are in dynamic equilibrium in a closed system. At equilibrium the rate of evaporation equals the rate of condensation. Vapor pressure depends strongly on temperature and on the strength of intermolecular forces: weaker intermolecular forces → higher vapor pressure.

Microscopic idea
In a liquid molecules have a distribution of kinetic energies. Molecules with enough energy to overcome intermolecular attractions escape from the surface (evaporation). In a closed container some vapor molecules return to the liquid (condensation). When the rates become equal, a saturated vapor is established with a characteristic vapor pressure at that temperature.

Evaporation — key points

• Occurs at the surface and at temperatures below the boiling point.
• Causes cooling of the remaining liquid because higher-energy molecules leave first (evaporative cooling).
• Factors affecting rate: temperature (↑ → ↑), surface area (↑ → ↑), humidity of surrounding air (↑ humidity → ↓ evaporation), wind/blowing air (↑ → ↑), and vapour pressure of the liquid (lower vapour pressure → faster evaporation under given conditions).

Boiling — key points

• Boiling occurs when the vapour pressure of the liquid equals the external (ambient) pressure; bubbles form throughout the liquid and rise to the surface.
• The boiling point depends on external pressure. The normal boiling point is the temperature at which vapour pressure = 1 atm (760 mmHg).
• At high altitude (lower atmospheric pressure) boiling occurs at lower temperature; in a pressure cooker (higher pressure) boiling temperature is higher, cooking faster.

Thermodynamic relations
Clausius–Clapeyron relationship describes how vapour pressure changes with temperature. Differential form:
d ln P / dT = ΔH_vap / (R T^2)
Integrated form (approximate if ΔH_vap ~ constant):
ln P = −ΔH_vap / (R T) + C
where P is vapour pressure, T absolute temperature, ΔH_vap molar enthalpy (heat) of vaporization, R gas constant and C is integration constant. This leads to the useful plot ln P vs 1/T which is approximately a straight line.

Vapour pressure of solutions
For ideal solutions, Raoult's law: partial vapour pressure of component A above the solution = x_A × P_A°, where x_A is mole fraction in liquid and P_A° is vapour pressure of pure A at that temperature. Vapor pressure lowering for solvent A: ΔP = P_A° − P_A = x_B × P_A° (x_B = mole fraction of solute). This is the basis of colligative effects (boiling point elevation, freezing point depression).

Practical consequences
Evaporative cooling (sweating), drying of wet clothes, perfume spread, distillation, boiling-point changes with altitude, operation of pressure cookers and vacuum distillation are all consequences of vapour pressure, evaporation and boiling concepts.

Fig 13 — Educational Diagram: Surface Tension and Capillary Action

Surface tension is the property of a liquid surface that makes it behave like a stretched elastic membrane. It arises because molecules at the surface experience a net inward cohesive force (from neighbouring liquid molecules) that molecules in the bulk do not. Surface tension is the force per unit length acting along a line on the surface, or equivalently the excess free energy per unit area of the surface.

Definitions and expressions

• Mechanical definition: γ = F / L, where F is the force acting along a line of length L on the surface (units: N m⁻¹).
• Energy definition: γ = dE / dA, the increase in surface free energy E when the surface area A is increased.
• Dimensions: [γ] = M T⁻².

Cause: Cohesive forces between liquid molecules (e.g., hydrogen bonding in water) pull surface molecules inward. Adhesive forces between liquid and solid control wetting behaviour.

Contact angle and wetting

• Contact angle θ is the angle between the liquid surface and the solid at the contact line. If θ < 90° the liquid wets (concave meniscus), if θ > 90° it does not wet (convex meniscus).
• Young's relation (qualitative for contact angle): γ_sv = γ_sl + γ cosθ, where γ_sv, γ_sl, γ are surface tensions of solid–vapor, solid–liquid and liquid–vapor interfaces respectively.

Surface curvature and pressure difference

• For a spherical droplet of radius R, the excess pressure inside is given by the Young–Laplace relation: Δp = 2γ / R.
• For a soap bubble (two surfaces) Δp = 4γ / R.

Capillary action (capillarity)

When a thin tube (capillary) is placed in a liquid, the liquid will rise or fall in the tube relative to the external liquid level because of the balance between surface tension (acting along the contact circumference) and the weight of the column of liquid. The meniscus shape (concave or convex) depends on the contact angle θ.

Derivation (outline): Upward surface-tension force along the circumference = 2πr γ cosθ. Weight of liquid column = πr² h ρ g. Equate to get:

h = (2 γ cosθ) / (ρ g r)

This is known as Jurin's law: capillary rise h ∝ 1/r.

Factors affecting surface tension

• Temperature: γ decreases with temperature and goes to zero at the critical temperature.
• Impurities and surfactants: substances like soaps lower γ and change wetting and capillary rise.
• Nature of liquid and solid (polarity, hydrogen bonding) affects contact angle and meniscus shape.

Typical values: Water at 20°C: about 72.8 mN·m⁻¹. Mercury is high (>400 mN·m⁻¹) and shows capillary depression on glass (θ > 90°).

Applications and significance: Surface tension controls droplet formation and breakup, stabilises bubbles and foams, enables insects to walk on water, drives capillary transport in porous media and plants, and is crucial in many industrial processes (painting, printing, wetting, detergency).

Fig 14 — Educational Diagram: Viscosity and Flow of Liquids

Definition: Viscosity is a measure of internal friction in a fluid that resists relative motion between adjacent layers. It quantifies the force required to slide one layer of liquid over another at a given velocity gradient.

Molecular origin: In liquids, molecules are close together and interact strongly. When layers move past one another, momentum transfer between molecules (via intermolecular forces) opposes relative motion, producing viscous resistance. Stronger intermolecular attractions or larger molecular size generally increase viscosity.

Mathematical statement (Newtonian fluids): For many simple liquids (called Newtonian), the shear stress τ (force per unit area) is proportional to the velocity gradient (rate of shear) du/dy:

τ = η (du/dy)

Here η is the (dynamic) viscosity. The velocity gradient du/dy is the change of flow velocity u in the direction y perpendicular to the flow. This linear relation defines Newtonian behaviour; non-Newtonian fluids deviate from it (shear-thinning, shear-thickening, Bingham plastics, etc.).

Types of viscosity:

• Dynamic (absolute) viscosity η: units SI = Pa·s (1 Pa·s = 1 N·s/m²). CGS unit: poise (P), where 1 P = 0.1 Pa·s; centipoise (cP) = 0.001 Pa·s.
• Kinematic viscosity ν = η/ρ: units m²/s; ρ is density.

Flow regimes: Flow can be laminar (smooth layers, predictable) or turbulent (chaotic eddies). The Reynolds number Re predicts the regime:

Re = (ρ v d)/η

where ρ is fluid density, v a characteristic velocity, and d a characteristic length (e.g., pipe diameter). For flow in pipes, laminar flow generally occurs for Re < ~2000; turbulent flow occurs for larger Re.

Important results for flow of liquids:

• Stokes' law (terminal velocity of a small sphere in a viscous liquid, valid at low Re):

v_t = (2/9) (r^2 (ρ_s - ρ_f) g) / η

• Hagen–Poiseuille equation (laminar flow of an incompressible Newtonian fluid through a circular pipe or capillary):

Q = (π r^4 ΔP) / (8 η L)

Here Q is volumetric flow rate, r is pipe radius, ΔP is pressure difference, L is pipe length. Note the strong dependence on radius (r^4).

Temperature dependence: For liquids, viscosity typically decreases rapidly with increasing temperature because thermal motion reduces the relative effect of intermolecular attractions. Empirical/Arrhenius-like expressions are often used, e.g. η = A exp(E/(RT)), where E is an activation energy for flow.

Non-Newtonian behaviour (brief): Many complex fluids (paints, ketchup, blood, polymer solutions) do not have a constant η. Shear-thinning (viscosity decreases with shear rate) and shear-thickening (viscosity increases with shear rate) are common.

Measurement of viscosity: Common methods include capillary viscometers (measure flow time under gravity), falling-sphere viscometers (use Stokes' law), and rotational viscometers (measure torque required to rotate an object in the fluid).

Importance: Viscosity controls how liquids flow in pipes, lubrication, blood circulation, industrial processing (pumping, spraying, coating), and affects diffusion and heat transport in fluids.

Fig 15 — Educational Diagram: Comparative Summary and Applications

Overview: This topic compares the properties of gases and liquids, contrasts ideal and real gases, explains why real gases deviate from ideal behaviour, and outlines important applications arising from their properties (viscosity, surface tension, vapour pressure, compressibility, diffusion, etc.).

Comparative summary (gases vs liquids):

• Shape and volume: Gases have neither definite shape nor definite volume (they expand to fill the container). Liquids have definite volume but take the shape of the container.
• Intermolecular forces: Much weaker in gases (molecules far apart); stronger in liquids (molecules closer, appreciable cohesion).
• Density: Gases have low density (typically 1/1000 that of liquids); liquids are much denser.
• Compressibility: Gases are highly compressible; liquids are nearly incompressible.
• Kinetic energy and motion: Gas molecules have higher average translational kinetic energy at the same temperature; liquids show translational + restricted rotational motion.
• Diffusion: Diffusion is rapid in gases, slower in liquids.
• Surface phenomena: Liquids show surface tension and capillarity; gases do not.
• Vapour pressure and volatility: Liquids with higher vapour pressure evaporate faster; gases already in vapour state depend on pressure and temperature.

Ideal vs real gases:

• Ideal gas assumptions: Particles have negligible volume, no intermolecular forces, elastic collisions. Ideal gas equation: PV = nRT.
• Real gas behaviour: At high pressure and low temperature, molecular volume and intermolecular attractions become important; gases deviate from ideality. Deviations measured by compressibility factor Z = PV/RT (Z = 1 for ideal gas).
• van der Waals correction: a corrects for attractions, b for finite molecular volume; (P + a(n/V)^2)(V - nb) = nRT models real gas behaviour qualitatively and predicts liquefaction and critical phenomena.
• Critical temperature and pressure: Above critical temperature (Tc) a gas cannot be liquefied by pressure alone; below Tc liquefaction is possible. Critical constants are important in gas liquefaction processes.

Key physical properties and their significance:

• Viscosity (η): Resistance to flow. Liquids: viscosity decreases with temperature; gases: viscosity increases with temperature. Important for lubrication, flow in pipes, blood flow.
• Surface tension (γ): Energy per unit area at liquid surface. Determines drop shape, capillary rise, wetting. Decreases with temperature; surfactants reduce γ (soaps, detergents).
• Vapour pressure and evaporation: Vapour pressure increases with temperature; determines boiling point and volatility. Clausius–Clapeyron relation links vapour pressure to enthalpy of vaporization.
• Diffusion and effusion: Rates depend on molecular mass and temperature (Graham's law). Explain smell spreading, gas leakage, effusion through small pores.

Practical relevance and applications: Understanding differences and real gas behaviour allows design of refrigeration, liquefaction plants, aerosol systems, industrial gas handling, and many everyday technologies (detailed examples given below).