States of Matter

Gases are characterized by: low density, high compressibility, complete miscibility with each other, and assuming the shape of container.

Gas Laws:

Combined Gas Law: $\dfrac{P_1V_1}{T_1} = \dfrac{P_2V_2}{T_2}$.

Ideal Gas Equation: $$PV = nRT$$

Gas constant: $R = 8.314$ J/mol·K = $0.0821$ L·atm/mol·K = $1.987$ cal/mol·K.

Standard conditions:

• STP (IUPAC modern): $0°$C, $1$ bar → $V_m = 22.7$ L
• STP (NCERT older): $0°$C, $1$ atm → $V_m = 22.4$ L
• NTP: $20°$C, $1$ atm → $V_m = 24.0$ L

Density of gas: $d = PM/(RT)$, where $M$ = molar mass.

Dalton's Law of Partial Pressures: For a mixture of non-reacting gases: $$P_{\text{total}} = P_1 + P_2 + \ldots + P_n$$

Partial pressure = mole fraction × total pressure: $P_i = x_i P_{\text{total}}$.

Graham's Law of Diffusion/Effusion: Rate of effusion $\propto 1/\sqrt{M}$: $$\frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}} = \sqrt{\frac{d_2}{d_1}}$$ (at constant $T, P$).

Postulates of Kinetic Molecular Theory:

1. Gas consists of large number of molecules in continuous random motion
2. Volume of molecules is negligible compared to container volume
3. No intermolecular forces (in ideal gas)
4. Collisions are perfectly elastic
5. Average KE $\propto T$ (in Kelvin)

Kinetic Equation: $PV = \dfrac{1}{3}mNu_{\text{rms}}^2$

Relating to KE: $PV = \dfrac{2}{3}E_K$ where $E_K = (3/2)nRT$. Average KE per molecule = $(3/2)kT$.

Molecular Speeds:

Ratio: $u_{mp} : \bar u : u_{\text{rms}} = 1 : 1.128 : 1.225$.

For nitrogen at $300$ K: $u_{\text{rms}} \approx 517$ m/s.

Maxwell-Boltzmann Distribution: Speeds of molecules follow a specific bell-shaped distribution. At higher temperature: peak shifts right, distribution broadens.

Real Gases vs Ideal: Real gases deviate from ideal at:

• High pressure (molecular volume significant)
• Low temperature (intermolecular forces significant)

Compressibility factor: $Z = PV/(nRT)$. For ideal gas $Z = 1$; for real gases, $Z \neq 1$.

• $Z < 1$: attractive forces dominate (real gas at moderate $P$)
• $Z > 1$: repulsive forces / finite molecular volume dominates (very high $P$)

van der Waals Equation: $$\left(P + \frac{an^2}{V^2}\right)(V - nb) = nRT$$

$a$: corrects for intermolecular attraction (larger for polar gases) $b$: corrects for molecular volume

For $1$ mole: $\left(P + a/V^2\right)(V - b) = RT$.

Critical Constants:

• $T_c = 8a/(27Rb)$
• $P_c = a/(27b^2)$
• $V_c = 3b$

Above $T_c$, gas cannot be liquefied however high the pressure.

Boyle Temperature ($T_B$): $T_B = a/(Rb)$. Below this, gas is more compressible than ideal; above, less.

Liquid State: Definite volume, no definite shape (takes shape of container). Properties intermediate between solid and gas. Stronger intermolecular forces than gases.

Vapour Pressure: Pressure exerted by vapour above a liquid in equilibrium at a given temperature.

• Increases with temperature
• Independent of liquid amount or container volume (at given $T$)
• Higher VP → more volatile liquid

Boiling Point: Temperature at which VP = atmospheric pressure. Liquid boils when VP equals external pressure.

• At higher altitudes, $P_{\text{atm}}$ is lower → boiling point reduces
• Pressure cookers raise BP by raising pressure

Normal Boiling Point: BP at $1$ atm. Standard BP: at $1$ bar.

Clausius-Clapeyron equation: $$\ln\left(\frac{P_2}{P_1}\right) = \frac{\Delta H_{\text{vap}}}{R}\left(\frac{1}{T_1} - \frac{1}{T_2}\right)$$

Surface Tension ($\gamma$ or $T$): Force per unit length acting tangentially on liquid surface (N/m). Arises from unbalanced cohesive forces at surface.

Properties affected:

• Spherical shape of droplets
• Capillary action ($h = 2T\cos\theta/(\rho g r)$)
• Cleaning action of soaps (reduce $\gamma$)

Surface tension decreases with temperature. Pure liquids have specific values: water $\sim 72$ mN/m at $25°$C.

Viscosity ($\eta$): Resistance to flow. Higher $\eta$ → more viscous (slower flow).

• Newton's law: $F = \eta A\,dv/dx$
• SI unit: Pa·s (or poise: $1$ P = $0.1$ Pa·s)
• Decreases with temperature (kinetic energy overcomes intermolecular forces)
• Glycerol, honey have high $\eta$; water lower; gases very low

Intermolecular Forces (IMFs): Attractive forces between molecules. Determine physical properties (BP, MP, solubility).

Types of IMFs (van der Waals):

London Dispersion Forces:

• Universal — present in all molecules
• Strength depends on polarizability (number of electrons, molecular size)
• Stronger with larger atomic/molecular surface
• Explains why I₂ is solid, Br₂ liquid, Cl₂ gas at room T

Order of strength (typically): Covalent/ionic bonds (intramolecular) > H-bonds > dipole-dipole > induced dipole > London.

Hydrogen Bond Specifics:

• Bond energy: $5-40$ kJ/mol
• Affects: BP, MP, solubility in water, structure of ice and proteins (DNA double helix)
• Intramolecular H-bonds (within molecule): o-nitrophenol — gives lower BP than p-nitrophenol
• Intermolecular H-bonds: H₂O has BP $100°$C vs H₂S $-60°$C

Effects of IMFs on Physical Properties:

• Higher IMF → higher BP, MP
• Higher IMF → lower vapour pressure
• Higher IMF → higher viscosity, surface tension
• Solubility: "like dissolves like" — polar in polar, non-polar in non-polar

Comparing Boiling Points (BP):

• Among alkanes: $C_2H_6 < C_3H_8 < C_4H_{10}$ (London ↑ with size)
• HF > HI > HBr > HCl (H-bonding dominates HF; then London for others)
• $H_2O > H_2S > H_2Se > H_2Te$ for H₂O only; then $H_2S < H_2Se < H_2Te$ for the rest (H-bonding makes H₂O anomalous; London ↑ with size for others)