States of Matter

1. Intermolecular Forces

Whether a substance is a gas, liquid or solid depends on the balance between intermolecular attraction and thermal energy ($\sim kT$).

• London (dispersion) forces: between all molecules; grow with molar mass and surface area. They explain why $I_2$ is a solid while $F_2$ is a gas.
• Dipole–dipole: between polar molecules such as HCl.
• Dipole–induced dipole: between a polar and a non-polar molecule.
• Hydrogen bonding: H attached to F, O or N. It is why $H_2O$ boils at 373 K while $H_2S$ boils at about 213 K.

2. The Gas Laws

• Boyle's law (constant $T$, $n$): $PV = $ constant.
• Charles' law (constant $P$, $n$): $V \propto T$ (T in kelvin).
• Gay-Lussac's law (constant $V$, $n$): $P \propto T$.
• Avogadro's law (constant $T$, $P$): $V \propto n$.

Combining them gives the ideal gas equation:

Useful rearrangements: density $d = \frac{PM}{RT}$, so molar mass $M = \frac{dRT}{P}$.

Dalton's Law of Partial Pressures

$P_{total} = p_1 + p_2 + \ldots$ and $p_i = x_i P_{total}$. For a gas collected over water, $P_{dry\ gas} = P_{total} - $ aqueous tension.

Graham's Law of Diffusion

$\frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}}$ at the same $T$ and $P$. Lighter gases diffuse faster; this was used to enrich $^{235}UF_6$ from $^{238}UF_6$.

3. Kinetic Theory

Average kinetic energy depends only on temperature: $KE_{avg} = \frac{3}{2}kT$ per molecule. The characteristic molecular speeds are:

The ratio $u_{mp} : u_{avg} : u_{rms} \approx 1 : 1.128 : 1.224$.

Interactive diagram: open this page in a browser to use it.

4. Real Gases

Real gases deviate from ideal behaviour at high pressure and low temperature, because molecules have finite volume and attract each other. The van der Waals equation corrects for both:

• $a$ measures intermolecular attraction; $b$ is the excluded volume.
• Compressibility factor $Z = \frac{PV}{nRT}$. $Z = 1$ for an ideal gas; $Z < 1$ when attractions dominate (gas is easier to compress); $Z > 1$ when repulsions dominate. For $H_2$ and He, $Z > 1$ at all pressures at room temperature.
• Critical temperature $T_c = \frac{8a}{27Rb}$: above $T_c$ a gas cannot be liquefied however much pressure is applied.

5. Liquids

Vapour pressure rises with temperature; a liquid boils when its vapour pressure equals the external pressure (normal boiling point at 1 atm). Surface tension and viscosity both decrease as temperature rises.

Practice questions

1. At the same temperature, the ratio of rates of diffusion of H₂ and O₂ is:

   1. 1 : 4
   2. 4 : 1
   3. 1 : 16
   4. 16 : 1
   Answer

   B. rH_₂rO_₂ = √(32/2) = 4, so 4 : 1.

2. For a real gas, the compressibility factor Z < 1 indicates that:

   1. Repulsive forces dominate
   2. Attractive forces dominate
   3. The gas is ideal
   4. The gas cannot be liquefied
   Answer

   B. Z < 1 means the gas is more compressible than an ideal gas, which happens when intermolecular attraction dominates.

3. The van der Waals constant b accounts for:

   1. Intermolecular attraction
   2. Finite volume of molecules
   3. Molecular speed
   4. Polarity
   Answer

   B. b is the excluded volume per mole, a correction for the finite size of molecules.

4. Which of the following is the correct order of molecular speeds?

   1. uᵣₘₛ < uₐᵥg < uₘₚ
   2. uₘₚ < uₐᵥg < uᵣₘₛ
   3. uₐᵥg < uₘₚ < uᵣₘₛ
   4. uₘₚ < uᵣₘₛ < uₐᵥg
   Answer

   B. The factors are √(2) < √(8/π) < √(3), so uₘₚ < uₐᵥg < uᵣₘₛ.

5. A gas has a density of 1.25 g/L at STP (273 K, 1 atm). Its molar mass is approximately:

   1. 14 g/mol
   2. 28 g/mol
   3. 32 g/mol
   4. 44 g/mol
   Answer

   B. M = dRT/P = 1.25 × 0.0821 × 273 ≈ 28 g/mol (for example N₂ or CO).

6. The boiling point of water is much higher than that of H₂S mainly because of:

   1. Higher molar mass of water
   2. London forces
   3. Hydrogen bonding in water
   4. Ionic bonding in water
   Answer

   C. Water forms extensive hydrogen bonds (O–H···O); sulphur is not electronegative enough for strong H-bonding.