States of Matter and Gas Laws

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States of Matter and Gas Laws — Key Facts for WAEC WASSCE

• Boyle’s Law: At constant temperature, the pressure of a fixed mass of gas is inversely proportional to its volume: $P_1V_1 = P_2V_2$ or $PV = \text{constant}$.
• Charles’ Law: At constant pressure, the volume of a fixed mass of gas is directly proportional to its absolute temperature: $\frac{V_1}{T_1} = \frac{V_2}{T_2}$. Note: temperature must be in Kelvin (K), where $T(\text{K}) = t(°\text{C}) + 273$.
• Pressure Law (Gay-Lussac’s Law): At constant volume, pressure is directly proportional to absolute temperature: $\frac{P_1}{T_1} = \frac{P_2}{T_2}$.
• Combined Gas Law: $\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$ for a fixed mass of gas.
• Ideal Gas Equation: $PV = nRT$, where $n$ = number of moles, $R$ = universal gas constant = 8.314 J mol⁻¹ K⁻¹ (or 0.0821 dm³ atm mol⁻¹ K⁻¹).
• Molar Volume: At standard temperature and pressure (STP: 273 K, 1 atm), 1 mole of any ideal gas occupies 22.4 dm³.

⚡ WAEC Exam Tip: Always convert Celsius to Kelvin by adding 273. A common WAEC error is using Celsius directly in gas law calculations — this will give wrong answers every time! Also note that 0°C = 273 K, not 0 K.

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States of Matter and Gas Laws — WAEC WASSCE Chemistry Study Guide

Kinetic Theory of Gases — Assumptions

The kinetic theory treats a gas as a large number of particles (molecules) in continuous random motion. The key assumptions are:

1. Gas particles are point masses (negligible volume compared to the container)
2. Collisions between particles and with container walls are perfectly elastic
3. There are no intermolecular forces except during collisions
4. The average kinetic energy of particles depends only on temperature
5. The time spent in collisions is negligible compared to time between collisions

Real gases deviate from ideal behaviour at high pressure (particles are close together) and low temperature (intermolecular forces become significant).

Boyle’s Law — Graphical Representation

• $P$ vs $V$ at constant $T$: hyperbolic curve (rectangular hyperbola)
• $P$ vs $1/V$ at constant $T$: straight line through the origin (direct proportion)

Charles’ Law — Graphical Representation

• $V$ vs $T$ at constant $P$: straight line through the origin when extrapolated. The intercept on the temperature axis is at $-273°C$ (absolute zero), which is the basis for the Kelvin scale.

Avogadro’s Law

At constant temperature and pressure, equal volumes of gases contain equal numbers of molecules. This leads to $V \propto n$ and explains why the molar volume is the same for all gases at the same $T$ and $P$.

Dalton’s Law of Partial Pressures

In a mixture of non-reacting gases, the total pressure is the sum of the individual partial pressures: $$P_{\text{total}} = P_1 + P_2 + P_3 + \ldots$$

The partial pressure of gas $A$ is $P_A = x_A \times P_{\text{total}}$, where $x_A = \frac{n_A}{n_{\text{total}}}$ is the mole fraction.

Graham’s Law of Diffusion

The rate of diffusion of a gas is inversely proportional to the square root of its density (or molar mass): $$\frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}} = \sqrt{\frac{d_2}{d_1}}$$

This explains why ammonia ($\mathrm{NH_3}$, $M = 17$) diffuses faster than hydrogen chloride ($\mathrm{HCl}$, $M = 36.5$) — the classic ‘two cotton wool plugs’ experiment demonstrating that gases meet and form a white ring of ammonium chloride closer to the HCl end.

Intermolecular Forces and States of Matter

State	Intermolecular Forces	Particle Spacing	Particle Motion	Shape/Volume
Solid	Strong (close-packed)	Very close	Vibrational only	Fixed shape, fixed volume
Liquid	Moderate	Close	Vibrational + translational	Takes container shape, fixed volume
Gas	Very weak/negligible	Far apart	Rapid random motion	Fills entire container

Types of Intermolecular Forces (in order of increasing strength)

1. Van der Waals forces (London dispersion forces): Temporary induced dipole-induced dipole attractions; present in all molecules. Strength increases with number of electrons.
2. Dipole-dipole forces: Attractions between permanent dipoles in polar molecules; stronger than London forces for molecules of similar size.
3. Hydrogen bonding: Special strong dipole-dipole attraction when H is bonded to F, O, or N (high electronegativity). Responsible for water’s high boiling point, ice floating, and many biological structures.

Changes of State

• Melting (fusion): Solid → Liquid. The temperature at which this occurs at 1 atm is the melting point.
• Boiling: Liquid → Gas. The temperature at which the saturated vapour pressure equals atmospheric pressure is the boiling point.
• Evaporation: Surface phenomenon occurring at any temperature; molecules with sufficient kinetic energy escape from the surface.
• Sublimation: Direct conversion from solid to gas without passing through the liquid phase. Examples: iodine, dry ice ($\mathrm{CO_2}$), mothballs (naphthalene).

The Triple Point

The triple point is the unique combination of temperature and pressure at which all three phases (solid, liquid, gas) coexist in equilibrium. For water, this occurs at 0.01°C (273.16 K) and 0.006 atm (611 Pa).

Comparison Table — Gas Laws

Law	Constant Conditions	Relationship	Equation
Boyle’s	Temperature, $n$	$P \propto 1/V$	$P_1V_1 = P_2V_2$
Charles’	Pressure, $n$	$V \propto T$	$V_1/T_1 = V_2/T_2$
Pressure (Gay-Lussac)	Volume, $n$	$P \propto T$	$P_1/T_1 = P_2/T_2$
Avogadro’s	$T$, $P$	$V \propto n$	$V_1/n_1 = V_2/n_2$
Combined	$n$	—	$P_1V_1/T_1 = P_2V_2/T_2$
Ideal Gas	—	$PV = nRT$	—

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States of Matter and Gas Laws — Comprehensive WAEC Chemistry Notes

Derivation of the Ideal Gas Equation

From Boyle’s Law: $V \propto 1/P$ (at constant $T, n$) From Charles’ Law: $V \propto T$ (at constant $P, n$) From Avogadro’s Law: $V \propto n$ (at constant $T, P$)

Combining: $V \propto \frac{nT}{P}$ → $PV = nRT$

Where $R$ (universal gas constant) = 8.314 J mol⁻¹ K⁻¹ = 0.0821 dm³ atm mol⁻¹ K⁻¹.

Derivation of Boyle’s Law from Kinetic Theory

For a gas molecule colliding with a wall, the change in momentum per collision $\Delta p = 2mv_x$ (for molecules moving with velocity component $v_x$ in the $x$-direction). The pressure force on the wall comes from the rate of change of momentum of all molecules colliding with it.

For $N$ molecules in a container of volume $V$: Pressure $P = \frac{1}{3}\frac{Nm\bar{v}^2}{V} = \frac{2}{3}\frac{N}{V}\times \text{kinetic energy per molecule}$

Since kinetic energy $\propto T$ for an ideal gas, and $\bar{v}^2$ is proportional to $T$, we get:

$$PV = \frac{1}{3}Nm\bar{v}^2 = \text{constant} \times T$$

At constant $T$, $\bar{v}^2$ is constant, so $PV = \text{constant}$ — Boyle’s Law.

Root Mean Square Speed

The root mean square speed of gas molecules is: $$c_{\text{rms}} = \sqrt{\frac{3RT}{M}}$$

where $M$ = molar mass in kg mol⁻¹, $R$ = 8.314 J mol⁻¹ K⁻¹, $T$ = temperature in K.

This shows that lighter gases (smaller $M$) have higher average speeds at the same temperature. This explains why hydrogen and helium escape from Earth’s atmosphere more readily than heavier gases — relevant to discussions about the absence of hydrogen in Earth’s atmosphere.

Graham’s Law — Detailed Derivation

From kinetic theory, the rate of diffusion $r$ is proportional to the average speed $\bar{v}$ and inversely proportional to $\sqrt{M}$: $$r \propto \frac{1}{\sqrt{M}}$$

$$\frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}}$$

For the Effusion of gases through a pinhole: Graham’s law of effusion is the same expression. This principle is used in the separation of uranium isotopes ($\mathrm{^{235}UF_6}$ vs $\mathrm{^{238}UF_6}$) by gas diffusion — a very slow process requiring thousands of stages.

Deviation of Real Gases from Ideal Behaviour

Real gases deviate from ideality because:

1. Finite molecular volume: At high pressure, molecules are close together and the available volume for movement is less than $V$. The effective volume is $(V - nb)$ where $b$ is the excluded volume per mole.
2. Intermolecular attractions: At high pressure, attractions between molecules reduce the force of collisions with the wall, reducing pressure below the ideal value.

The van der Waals equation corrects for both: $$\left(P + \frac{an^2}{V^2}\right)(V - nb) = nRT$$

Where $a$ accounts for intermolecular attractions and $b$ accounts for molecular volume.

At high pressure (small volume): $P_{\text{ideal}} \approx P + \frac{an^2}{V^2}$ (pressure correction) At low pressure (large volume): behaviour approaches ideal gas.

Critical Temperature and Liquefaction of Gases

Every gas has a critical temperature $T_c$ — the temperature above which the gas cannot be liquefied by pressure alone, regardless of how high the pressure. This is because intermolecular attractions are too weak (overcome by thermal kinetic energy) to allow condensation.

• $T < T_c$: gas can be liquefied by pressure alone — called a vapour
• $T > T_c$: true gas that cannot be liquefied by pressure alone

Examples: $\mathrm{CO_2}$ has $T_c = 31°C$ (can be liquefied at room temperature); $\mathrm{O_2}$ has $T_c = -118°C$ (requires cooling before liquefaction by pressure).

Liquid State — Surface Tension and Viscosity

• Surface tension: The force per unit length acting parallel to the surface of a liquid that opposes expansion of the surface area. It is caused by intermolecular cohesive forces pulling surface molecules inward. This explains why water droplets are spherical (minimum surface area for a given volume).
• Viscosity: The resistance to flow of a liquid. It arises from intermolecular forces impeding the sliding of one layer over another. Viscosity decreases with increasing temperature (molecules have more kinetic energy to overcome intermolecular attractions).

Solid State — Crystal Lattices

Solids may be crystalline (ordered repeating pattern) or amorphous (disordered). Crystalline solids include:

• Ionic crystals: e.g., NaCl — high melting points, conduct electricity when molten or dissolved
• Covalent (molecular) crystals: e.g., iodine, diamond — discrete molecules held by intermolecular forces; diamond is very hard with an extremely high melting point due to covalent network
• Metallic crystals: e.g., copper — electrons delocalised in a ‘sea’; good conductors of heat and electricity
• Giant covalent structures: e.g., diamond, silicon dioxide — each atom is bonded to neighbours by strong covalent bonds in a 3D network

The hexagonal close-packed (hcp) and face-centred cubic (fcc) structures achieve the maximum packing efficiency of 74% — atoms occupy 74% of the available space, which is the most efficient packing arrangement.

WAEC Past Question Patterns

Typical WAEC questions include:

1. Applying the ideal gas equation $PV = nRT$ to calculate any variable given the others
2. Using gas collection by downward displacement of water and correcting for water vapour pressure
3. Calculating the volume of a gas at STP or given conditions
4. Applying Graham’s law to diffusion/effusion problems
5. Explaining deviations from ideal gas behaviour using kinetic theory
6. Describing and explaining the three states of matter in terms of kinetic theory
7. Interpreting heating curves (temperature vs time for a substance undergoing state changes)
8. Calculating partial pressures using Dalton’s law

⚡ WAEC Exam Tip: When a gas is collected over water, the pressure of the wet gas equals the pressure of dry gas plus the water vapour pressure: $P_{\text{dry gas}} = P_{\text{total}} - P_{\text{water vapour}}$. The water vapour pressure depends on temperature and must be looked up from a table — this correction is a common WAEC question. Also remember that 1 atm = 760 mmHg = 101,325 Pa = 101.325 kPa.

Numerical Worked Example (WAEC-style)

2.0 g of a gas at 27°C and 750 mmHg occupies a volume of 1.5 dm³. Calculate the molar mass of the gas. (R = 0.0821 dm³ atm mol⁻¹ K⁻¹)

Solution: $T = 27 + 273 = 300$ K $P = 750/760 = 0.987$ atm $V = 1.5$ dm³ $n = PV/RT = (0.987 \times 1.5)/(0.0821 \times 300) = 1.4805/24.63 = 0.0601$ mol

Molar mass $= \frac{\text{mass}}{n} = \frac{2.0}{0.0601} = 33.3$ g mol⁻¹

Answer: Molar mass ≈ 33 g mol⁻¹

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