General Chemistry II: Comprehensive Study Guide (CHEM 6B Learning Goals)

Gases

Properties and Behavior of Gases

Gases are one of the three primary states of matter, characterized by their low density, high compressibility, and significant thermal expansivity compared to liquids and solids.

• Density: Gases have much lower densities than liquids and solids due to the large distances between particles.

• Compressibility: Gases can be compressed easily because their particles are far apart.

• Thermal Expansivity: Gases expand significantly when heated.

Temperature and Pressure

• Temperature (T): A measure of the average kinetic energy of gas particles, typically in Kelvin (K).

• Pressure (P): The force exerted by gas particles per unit area, measured in atmospheres (atm), Pascals (Pa), or torr.

• Barometer: Device to measure atmospheric pressure.

• Manometer: Device to measure the pressure of a gas in a container.

Gas Laws

Empirical relationships describe how two physical properties of gases relate when others are held constant.

• Boyle’s Law: At constant temperature and amount, pressure and volume are inversely related.

  • $P \propto \frac{1}{V}$ or $PV = \text{constant}$

• Charles’ Law: At constant pressure and amount, volume is directly proportional to temperature.

  • $V \propto T$ or $\frac{V}{T} = \text{constant}$

• Avogadro’s Law: At constant temperature and pressure, volume is directly proportional to the number of moles.

  • $V \propto n$ or $\frac{V}{n} = \text{constant}$

Ideal Gas Law

The ideal gas law combines the simple gas laws into a single equation of state:

• $PV = nRT$

• Where P = pressure, V = volume, n = moles, R = gas constant, T = temperature (K).

• Applications:

  • Solve for any variable when others are known.

  • Calculate molar mass and density of gases.

  • Determine partial pressures and mole fractions in mixtures (Dalton’s Law).

  • Relate to reaction stoichiometry for gaseous reactants/products.

Kinetic-Molecular Theory (KMT) of Gases

• Gases consist of particles in constant, random motion.

• Collisions are elastic; no energy is lost.

• Volume of particles is negligible compared to container volume.

• No intermolecular forces between particles.

• Average kinetic energy is proportional to temperature (K).

Maxwell-Boltzmann Distribution

• Describes the distribution of molecular speeds in a gas.

• Root-mean-square speed: $u_{rms} = \sqrt{\frac{3RT}{M}}$

• Average kinetic energy: $\langle KE \rangle = \frac{3}{2}RT$ per mole

• Higher temperature or lower molar mass increases average speed.

Effusion and Diffusion

• Effusion: Escape of gas through a small hole.

• Diffusion: Mixing of gases due to random motion.

• Graham’s Law: Rate of effusion is inversely proportional to the square root of molar mass:

  • $\text{Rate}_1/\text{Rate}_2 = \sqrt{M_2/M_1}$

Real Gases and van der Waals Equation

• Real gases deviate from ideal behavior at high pressures and low temperatures.

• van der Waals equation: $\left(P + a\frac{n^2}{V^2}\right)(V - nb) = nRT$

• Parameters a (attractive forces) and b (finite volume of particles) correct for non-ideal behavior.

Intermolecular Forces and Condensed Phases of Matter

Types of Intermolecular Forces (IMFs)

• Intramolecular forces: Bonds within molecules (covalent, ionic, metallic).

• Intermolecular forces: Forces between molecules or ions.

  • Ion-dipole

  • Dipole-dipole

  • Dipole-induced dipole

  • London dispersion (induced dipole-induced dipole)

• Relative strength: Ion-dipole > Hydrogen bonding > Dipole-dipole > London forces

Physical Properties and IMFs

• Stronger IMFs lead to higher surface tension, viscosity, and capillary action.

• Vapor pressure and boiling point increase as IMFs decrease.

Types of Crystalline Solids

• Ionic solids (e.g., NaCl): High melting points, hard, brittle.

• Molecular solids (e.g., ice): Lower melting points, soft.

• Covalent network solids (e.g., diamond): Very high melting points, hard.

• Metallic solids (e.g., Fe): Variable melting points, malleable, conductive.

Thermodynamics – First Law

Systems, Surroundings, and State Functions

• System: Part of the universe under study.

• Surroundings: Everything else.

• Types: Open (exchange matter/energy), closed (energy only), isolated (no exchange).

• State function: Property dependent only on current state (e.g., U, H, S, G).

• Path function: Depends on process path (e.g., work, heat).

Work and Heat

• Work ($w$): Energy transfer by force over distance. For gases, $w = -P_{ext}\Delta V$.

• Heat ($q$): Energy transfer due to temperature difference.

• Sign conventions: $q > 0$ (system gains heat), $w > 0$ (work done on system).

• Adiabatic process: No heat exchange ($q = 0$).

Exothermic vs. Endothermic

• Exothermic: Releases heat ($q < 0$).

• Endothermic: Absorbs heat ($q > 0$).

• Heat vs. temperature: Heat is energy transfer; temperature is average kinetic energy.

Heat Capacity

• Heat capacity ($C$): Amount of heat to raise temperature by 1 K.

• Molar heat capacity ($C_m$): Per mole.

• Specific heat ($c$): Per gram.

• Calculation: $q = m c \Delta T$ or $q = n C_m \Delta T$

Internal Energy and Enthalpy

• Internal energy ($U$): Total energy of a system.

• First Law: $\Delta U = q + w$

• Enthalpy ($H$): $H = U + PV$

• At constant pressure, $\Delta H = q_p$

Calorimetry

• Coffee-cup calorimeter: Measures $\Delta H$ (constant pressure).

• Bomb calorimeter: Measures $\Delta U$ (constant volume).

Thermochemistry

Thermochemical Equations and Hess’s Law

• Thermochemical equation: Balanced equation with enthalpy change ($\Delta H$).

• $\Delta H$ is extensive; depends on amount of substance.

• Hess’s Law: Enthalpy change is independent of path; sum enthalpy changes for steps.

Energy/Enthalpy Diagrams

• Visualize energy changes during reactions or phase changes.

Standard States and Enthalpy of Formation

• Standard state: Most stable form at 1 bar and 25°C.

• Standard enthalpy of formation ($\Delta H_f^\circ$): Enthalpy change for forming 1 mol from elements in standard states.

Phase Changes and Heating Curves

• Six phase transitions: Melting, freezing, vaporization, condensation, sublimation, deposition.

• Enthalpy changes: $\Delta H_{fus}$, $\Delta H_{vap}$, etc.

• Heating/cooling curve: Shows temperature vs. heat added; plateaus at phase changes.

Calculating Reaction Enthalpies

• By Hess’s Law, standard enthalpies of formation, or average bond enthalpies.

• Born-Haber cycle: Used to derive lattice energies for ionic compounds.

Thermodynamics – Second and Third Laws

Spontaneity and Entropy

• Spontaneous: Occurs without external input (may be slow or fast).

• First Law cannot predict spontaneity; Second Law is required.

• Second Law: $\Delta S_{universe} > 0$ for spontaneous processes.

Entropy (S)

• Statistical definition: $S = k \ln W$

• Change in entropy: $\Delta S = k \ln \left(\frac{W_f}{W_i}\right)$

• Thermodynamic definition: $\Delta S = \frac{q_{rev}}{T}$

• For temperature change: $\Delta S = nC_m \ln \left(\frac{T_f}{T_i}\right)$

Third Law of Thermodynamics

• At 0 K, entropy of a perfect crystal is zero ($S^\circ = 0$).

• Standard entropies ($S^\circ$) are always positive.

Predicting and Calculating Entropy Changes

• Factors: Temperature, physical state, dissolution, atomic size, molecular complexity.

• Phase changes: $\Delta S_{vap} > \Delta S_{fus}$ (Trouton’s Rule).

• For reactions: $\Delta S^\circ_{rxn} = \sum S^\circ_{products} - \sum S^\circ_{reactants}$

Gibbs Free Energy (G)

• $G = H - TS$

• Change in free energy: $\Delta G = \Delta H - T\Delta S$

• Standard free energy: $\Delta G^\circ = \Delta H^\circ - T\Delta S^\circ$

• Spontaneity: $\Delta G < 0$ (spontaneous), $\Delta G > 0$ (non-spontaneous)

• Relationship to maximum work: $\Delta G$ is the maximum non-expansion work obtainable.

Temperature Dependence and Coupled Reactions

• When $\Delta H$ and $\Delta S$ have the same sign, temperature determines spontaneity.

• At $\Delta G = 0$, the reaction is at equilibrium.

• Non-spontaneous processes can be driven by coupling to spontaneous ones.

Physical Equilibria

Phase Equilibrium and Vapor Pressure

• Phase equilibrium: Dynamic balance between phases (e.g., liquid and vapor).

• Vapor pressure: Pressure exerted by vapor in equilibrium with liquid or solid.

• Clausius-Clapeyron equation: Describes temperature dependence of vapor pressure:

  • $\ln P = -\frac{\Delta H_{vap}}{R} \cdot \frac{1}{T} + C$

Phase Diagrams

• Show regions of stability for solid, liquid, and gas phases as a function of P and T.

• Critical point: End of liquid-gas boundary.

• Triple point: All three phases coexist.

Solutions and Colligative Properties

Solution Formation and Concentration Units

• Solutions can be formed from gases, liquids, or solids.

• "Like dissolves like": Polar solvents dissolve polar solutes; nonpolar dissolve nonpolar.

• Enthalpy of solution: $\Delta H_{solution} = \Delta H_{lattice} + \Delta H_{hydration}$

• Entropy change ($\Delta S_{solution}$) also affects solubility.

• Concentration units: Molarity (M), molality (m), mole fraction ($X$), parts by mass/volume.

Solubility and Saturation

• Saturated solution: Equilibrium between dissolved and undissolved solute.

• Temperature increases solubility of solids, decreases for gases.

• Henry’s Law: $C = k_P P_{gas}$ (solubility of gas proportional to its pressure).

Colligative Properties

• Depend on number of solute particles, not identity.

• Vapor pressure lowering (Raoult’s Law): $P_{solution} = X_{solvent} P^\circ_{solvent}$

• Boiling point elevation: $\Delta T_b = i K_b m$

• Freezing point depression: $\Delta T_f = i K_f m$

• Osmotic pressure: $\Pi = iMRT$

• van’t Hoff factor ($i$): Accounts for dissociation of electrolytes.

Phase Diagrams of Solutions

• Show changes in boiling/freezing points compared to pure solvent.

• Vapor over solution is richer in more volatile component (Raoult’s and Dalton’s Laws).

Chemical Equilibria

Law of Mass Action and Equilibrium Constants

• At equilibrium, the ratio of product and reactant concentrations is constant ($K$).

• Homogeneous equilibria: All species in same phase.

• Heterogeneous equilibria: Species in different phases; pure solids/liquids omitted from $K$.

• Equilibrium constant expressions:

  • $K_c$ (concentration), $K_p$ (pressure)

• Relationship: $K_p = K_c(RT)^{\Delta n_{gas}}$

Manipulating Equilibrium Constants

• Reversing reaction: $K_{reverse} = 1/K$

• Multiplying equation: $K_{new} = K^{n}$

• Adding reactions: $K_{overall} = K_1 \times K_2$

Reaction Quotient (Q) and Direction of Change

• Reaction quotient ($Q$): Calculated like $K$ but with current concentrations.

• If $Q < K$, reaction proceeds forward; if $Q > K$, proceeds in reverse.

Gibbs Free Energy and Equilibrium

• $\Delta G = \Delta G^\circ + RT \ln Q$

• At equilibrium: $\Delta G = 0$, $Q = K$, so $\Delta G^\circ = -RT \ln K$

Temperature Dependence (van’t Hoff Equation)

• Describes how $K$ changes with temperature:

  • $\ln \left(\frac{K_2}{K_1}\right) = -\frac{\Delta H^\circ}{R} \left(\frac{1}{T_2} - \frac{1}{T_1}\right)$

Equilibrium Calculations

• Use ICE tables (Initial, Change, Equilibrium) to solve for unknowns.

• For small $K$ and large initial reactant, $x$ can be neglected if error < 5%.

• Quadratic equations may be required for exact solutions.

Le Chatelier’s Principle

• System at equilibrium responds to disturbances (concentration, pressure, temperature) to counteract the change.

• Temperature changes affect $K$; catalysts do not affect equilibrium composition.

Solubility Equilibria

Slightly Soluble Ionic Compounds

• Equilibrium between solid and dissolved ions in water.

• Solubility product constant ($K_{sp}$):

  • For $A_mX_n$: $K_{sp} = [A^{n+}]^m [X^{m-}]^n$

• Molar solubility: Moles of solute dissolved per liter.

• Mass solubility: Grams of solute dissolved per liter.

Reaction Quotient ($Q_{sp}$) and Saturation

• If $Q_{sp} < K_{sp}$: Unsaturated (more can dissolve).

• If $Q_{sp} = K_{sp}$: Saturated (at equilibrium).

• If $Q_{sp} > K_{sp}$: Supersaturated (precipitation occurs).

Factors Affecting Solubility

• Common ion effect: Presence of a common ion decreases solubility.

• Other factors (e.g., pH, complex ion formation) covered in later courses.

Gas Law	Equation	Variables Held Constant
Boyle's Law	$P_1V_1 = P_2V_2$	n, T
Charles' Law	$\frac{V_1}{T_1} = \frac{V_2}{T_2}$	n, P
Avogadro's Law	$\frac{V_1}{n_1} = \frac{V_2}{n_2}$	P, T
Ideal Gas Law	$PV = nRT$	None

Example: Calculate the pressure exerted by 2.0 mol of an ideal gas in a 5.0 L container at 300 K. $P = \frac{nRT}{V} = \frac{2.0 \times 0.0821 \times 300}{5.0} = 9.85$ atm

Additional info: This guide expands on the learning goals by providing definitions, equations, and examples for each topic. For more detailed derivations, refer to your course textbook or lecture notes.