States of matter: arrangement, motion, energies of particles, changes of state, gas laws

Physical Chemistry – States of Matter (Cambridge AS & A Level 9701)

1. Particle Arrangement in the Three States

State	Particle arrangement	Typical motion	Key properties
Solid	Particles are closely packed. Either a regular, repeating lattice (crystalline) or a fixed random network (amorphous).	Predominantly vibrational about fixed equilibrium positions.	Definite shape & volume; high density; low compressibility.
Liquid	Particles are close together but not ordered in a lattice.	Vibrational + translational (sliding past one another).	Definite volume, no fixed shape; surface tension; moderate compressibility.
Gas	Particles are far apart; no permanent arrangement.	Free, rapid translational motion; collisions with each other and container walls.	Indefinite shape & volume; low density; high compressibility.

2. Kinetic Theory of Gases

2.1 Translational kinetic energy

The average translational kinetic energy of one mole of particles is

\[ \langle E_{\text{kin}}\rangle = \frac{3}{2}\,RT \]

where R = 8.314 J mol⁻¹ K⁻¹ and T is the absolute temperature (K).

2.2 Origin of pressure

For an ideal gas the pressure can be derived from the momentum change on the container walls:

\[ p = \frac{2}{3}\,\frac{N}{V}\,\langle E_{\text{kin}}\rangle \]

Substituting \(\langle E_{\text{kin}}\rangle = \tfrac{3}{2}k_{\mathrm B}T\) (or the molar form above) gives the familiar ideal‑gas equation

\[ pV = nRT \]

2.3 Energy distribution

• Kinetic energy: translational (dominant for gases), rotational and vibrational (important for liquids and solids).
• Potential energy: arises from intermolecular forces.
  • Solids – strong attractive forces, high potential energy.
  • Liquids – moderate attractive forces.
  • Ideal gases – potential energy is taken as zero (no intermolecular forces).

3. Phase Changes (Phase Transitions)

During a phase change the temperature remains constant while energy is absorbed or released – the latent heat.

Process	Direction	Heat flow	Enthalpy symbol	Typical ΔH (kJ mol⁻¹)
Melting (fusion)	Solid → Liquid	Endothermic	ΔHfus	≈ 6 (ice)
Freezing	Liquid → Solid	Exothermic	–ΔHfus	≈ –6 (ice)
Vapourisation (boiling)	Liquid → Gas	Endothermic	ΔHvap	≈ 41 (water)
Condensation	Gas → Liquid	Exothermic	–ΔHvap	≈ –41 (water)
Sublimation	Solid → Gas	Endothermic	ΔHsub	≈ 30 (dry ice)
Deposition	Gas → Solid	Exothermic	–ΔHsub	≈ –30 (dry ice)

Useful relationships:

• ΔH_sub = ΔH_fus + ΔH_vap (approximate for many substances).
• Latent heat per gram = ΔH (kJ mol⁻¹) / M (g mol⁻¹).

4. Gases – Ideal and Real Behaviour

4.1 Ideal‑Gas Assumptions (syllabus requirement 4.1)

1. Particle volume is negligible compared with the container volume.
2. No intermolecular attractions or repulsions; collisions are perfectly elastic.
3. All kinetic energy is translational and depends only on temperature.

4.2 Individual Gas Laws (derived from the ideal‑gas equation)

• Boyle’s Law (T, n constant): \(pV = \text{constant}\)
• Charles’s Law (p, n constant): \(\displaystyle\frac{V}{T} = \text{constant}\)
• Gay‑Lussac’s Law (V, n constant): \(\displaystyle\frac{p}{T} = \text{constant}\)
• Avogadro’s Law (p, T constant): \(\displaystyle\frac{V}{n} = \text{constant}\)

4.3 Combined Gas Law

\[ \frac{p_{1}V_{1}}{T_{1}} \;=\; \frac{p_{2}V_{2}}{T_{2}} \]

4.4 Ideal‑Gas Equation

\[ pV = nRT \]

• p – pressure (Pa or atm)
• V – volume (L or m³)
• n – amount of substance (mol)
• R – 8.314 J mol⁻¹ K⁻¹ (or 0.0821 L atm mol⁻¹ K⁻¹)
• T – temperature (K)

4.5 When the Ideal Equation Fails (high P / low T)

Two main reasons:

1. Finite molecular volume – reduces the space available for motion.
2. Intermolecular attractions – lower the pressure exerted on the walls.

4.6 Real‑Gas Equation – van der Waals

\[ \left(p + \frac{a}{V_{m}^{2}}\right)(V_{m} - b) = RT \]

• V_m – molar volume (L mol⁻¹).
• a (L² atm mol⁻²) corrects for attractive forces.
• b (L mol⁻¹) corrects for the finite size of molecules.

4.7 Other Useful Real‑Gas Concepts (syllabus 4.1 b)

• Compressibility factor \(Z = \dfrac{pV_{m}}{RT}\). \(Z = 1\) for an ideal gas; deviations indicate non‑ideality.
• Critical temperature (T_c) – above which a gas cannot be liquefied by pressure alone.
• Boyle temperature (T_B) – temperature at which a real gas most closely follows Boyle’s law (Z ≈ 1 over a wide pressure range).
• Reduced variables \(\displaystyle p_{r}= \frac{p}{p_{c}},\; T_{r}= \frac{T}{T_{c}},\; V_{r}= \frac{V_{m}}{V_{c}}\) – useful for comparing different gases.

4.8 Sample Calculation – Ideal vs. van der Waals

Calculate the volume of 1 mol CO₂ at 300 K and 50 atm.

1. Ideal gas: \(V = \dfrac{nRT}{p} = \dfrac{(1)(0.0821)(300)}{50} = 0.492\;\text{L}\).
2. van der Waals (a = 3.59 L² atm mol⁻², b = 0.0427 L mol⁻¹):
   Iterate \(V\) in \(\left(p + \dfrac{a}{V^{2}}\right)(V-b)=RT\):
   Solution ≈ 0.523 L.
3. Interpretation: The ideal equation under‑estimates the volume because it neglects the finite size (b) and attractions (a). The corrected value is much closer to the experimental value (≈ 0.525 L).

4.9 Diagrammatic Summary

5. Lattice Structures and Their Physical Properties (syllabus requirement 4.2)

Lattice type	Typical example	Key structural features	Properties explained by structure
Giant ionic	NaCl, MgO	3‑D array of alternating cations & anions; strong electrostatic attraction.	Very high melting points, brittle, soluble in polar solvents, conduct electricity only when molten or aqueous.
Simple molecular	I₂ (solid), CO₂ (dry ice)	Discrete molecules held together by weak van der Waals forces.	Low melting/boiling points, soft, poor conductivity, often volatile.
Giant covalent (network)	SiO₂ (quartz), diamond	Each atom covalently bonded to several neighbours in a 3‑D network.	Extremely high melting points, very hard, generally non‑conductive (unless doped), insoluble.
Giant metallic	Cu, Al	Metal cations in a lattice surrounded by a delocalised “sea of electrons”.	High electrical & thermal conductivity, malleable, ductile, moderate‑high melting points.

6. Exam‑Style Checklist (What you must be able to do)

1. Describe particle arrangement and typical motion in solids, liquids and gases.
2. State and use the kinetic‑theory relationship \(\langle E_{\text{kin}}\rangle = \tfrac{3}{2}RT\) and derive \(p = \tfrac{2}{3}\dfrac{N}{V}\langle E_{\text{kin}}\rangle\).
3. List the six phase changes, indicate endo‑ or exothermic nature, and write the appropriate ΔH symbol.
4. Recall and manipulate the four individual gas laws, the combined gas law, and the ideal‑gas equation.
5. State the three ideal‑gas assumptions; explain qualitatively why they fail at high pressure/low temperature.
6. Perform a short calculation showing the failure of the ideal equation and apply the van der Waals correction (including use of the compressibility factor Z).
7. Identify the four lattice types, give one example of each, and link structure to melting point, conductivity and solubility.
8. Use the pressure‑origin expression from kinetic theory to justify \(pV = nRT\) and to discuss real‑gas deviations (Z, critical temperature, Boyle temperature).