Study guide: States of Matter and Chemical Equilibrium

📋 Course Outline

1. Pure Substances and State Changes
2. State Diagrams and Critical Points
3. Systems and Their Properties
4. Mixtures and Composition Measures
5. Reaction Equations and Progress
6. Chemical Activities
7. Reaction Quotient and Equilibrium
8. Equilibrium Disruption and Measurement
9. Equilibrium Calculations and Applications

📖 1. Pure Substances and State Changes

🔑 Key Concepts & Definitions

• Pure substance: Contains a single chemical constituent and may be simple, such as O₂, or compound, such as H₂O.

★ Must-know

• A pure substance can occur as a solid, liquid, or gas depending on temperature and pressure.

• The state-change names are:

  • fusion (solid to liquid)
  • solidification (liquid to solid)
  • vaporization (liquid to gas)
  • liquefaction (gas to liquid)
  • sublimation (solid to gas)
  • condensation (gas to solid)

Further detail

• A solid is compact and ordered, a liquid has contacting but disordered entities, and a gas is disordered and dispersed.

• A pure substance may have several crystalline solid forms called polymorphs, a phenomenon known as polymorphism.

• At a fixed pressure, a pure substance changes state at a fixed temperature, and conversely the selected pressure fixes the change temperature.

📖 2. State Diagrams and Critical Points

🔑 Key Concepts & Definitions

• Triple point: The pressure and temperature at which the solid, liquid, and gas phases of a pure substance coexist.
• Critical point: Marks the end of the liquid–gas discontinuity; beyond it, the substance is a homogeneous supercritical fluid.

★ Must-know

• At fixed pressure, heating a pure substance takes it from the more ordered state toward the more disordered state.

📐 Formula - $T\,(\mathrm{K})=t\,(^{\circ}\mathrm{C})+273.15$ converts Celsius temperature to kelvin.

Further detail

• Water's triple point is at 0.01 °C or 273.16 K under 6 mbar and is used to define the Kelvin temperature scale.

• The vaporization and sublimation curves have positive slopes, while water's nearly vertical fusion curve has a negative slope; most pure substances have a positive fusion-curve slope.

• Water is supercritical when both its temperature and pressure exceed their critical values, and its critical pressure is 221 bar.

• Supercritical water can dissolve organic and ionic compounds and large amounts of oxygen, enabling proposed pollutant extraction and in-situ oxidation applications.

💡 Memory Hook

At the triple point, three phases coexist; beyond the critical point, liquid and gas cease to be distinct.

📖 3. Systems and Their Properties

🔑 Key Concepts & Definitions

• Physicochemical system: An assembly of constituents of defined chemical nature and physical state, bounded by a real or imaginary closed surface Σ.

★ Must-know

• An open system permits matter and energy exchange, a closed system permits no matter exchange, and an isolated system permits neither matter nor energy exchange across its boundary.

• Intensive parameters are independent of the amount of matter and non-additive, whereas extensive parameters depend on amount and are additive across subsystems.

• A homogeneous system has one uniform phase, whereas a heterogeneous system contains at least two distinct phases.

Further detail

• Changes within a system may be:

  • physical state changes
  • chemical rearrangements of matter
  • transformations of an atomic nucleus

• A closed system has constant mass even though its composition may change.

💡 Memory Hook

Closed systems exchange energy but not matter; isolated systems exchange neither.

📖 4. Mixtures and Composition Measures

🔑 Key Concepts & Definitions

• Aqueous solution: An aqueous solution has water as the solvent in large excess and contains dissolved solutes; for a dilute solution, solute concentration is $C_i=\frac{n_i}{V}$ in mol·L⁻¹.

★ Must-know

📐 Formula - For an ideal gas mixture, the total pressure, volume, amount, and temperature satisfy $PV=nRT$, with $R=8.314\ \mathrm{J\,K^{-1}\,mol^{-1}}$.

📐 Formula - The mole fraction of constituent i is $x_i=\frac{n_i}{n_{\mathrm{tot}}}$, and mole fractions sum to one within a single phase.

📐 Formula - In an ideal gas mixture, the partial pressure is $p_i=x_iP$ and the partial pressures sum to the total pressure according to Dalton's law.

Further detail

📐 Formula - The density of an ideal gas mixture relative to air is $d=\frac{\sum_i x_iM_i}{29}$ when molar masses are in g·mol⁻¹.

📐 Formula - For a homogeneous liquid mixture, the mass fraction is $w_i=\frac{m_i}{m_{\mathrm{tot}}}$ and the mass concentration is $X_i=\frac{m_i}{V}$ in g·L⁻¹.

📖 5. Reaction Equations and Progress

🔑 Key Concepts & Definitions

• Stoichiometric coefficient: A stoichiometric coefficient gives the amount ratio of a reactant or product in a balanced reaction equation; its algebraic value is negative for reactants and positive for products.
• Conversion fraction: The conversion fraction of reactant i is the amount of that reactant consumed divided by its initial amount; for a stoichiometric mixture, it is independent of which reactant is selected.

★ Must-know

📐 Formula - The amount of constituent i evolves according to $n_i(t)=n_i(t_0)+\nu_i\xi(t)$ and $dn_i=\nu_i\,d\xi$, where ξ is the reaction extent in moles.

📌 For a complete forward reaction, the limiting reactant disappears at the maximum extent ξₘₐₓ; for a reversible reaction, the final extent ξₑ is the extent at equilibrium.

Further detail

📐 Formula - A reaction written with algebraic stoichiometric numbers satisfies $\sum_i\nu_iA_i=0$.

📐 Formula - The volumetric reaction extent is $\xi_v=\frac{\xi}{V}$ and is mainly used for homogeneous equilibria in aqueous solution.

📖 6. Chemical Activities

🔑 Key Concepts & Definitions

• Chemical activity: A dimensionless intensive quantity describing the behavior of a constituent in a system and used in equilibrium laws.

★ Must-know

📐 Formula - For a pure ideal gas, activity is $a=\frac{P}{P^0}$, where $P^0=1\ \mathrm{bar}=10^5\ \mathrm{Pa}$.

• A pure condensed phase and the solvent in a dilute aqueous solution have activity equal to one.

📐 Formula - For a solute in a dilute aqueous solution, activity is $a_i=\frac{C_i}{C^0}$, with $C^0=1\ \mathrm{mol\,L^{-1}}$.

Further detail

📐 Formula - For constituent i in an ideal gas mixture, activity is $a_i=\frac{p_i}{P^0}$, where pᵢ is its partial pressure.

📖 7. Reaction Quotient and Equilibrium

🔑 Key Concepts & Definitions

• Reaction quotient: For a reaction written $\sum_i\nu_iA_i=0$, the dimensionless reaction quotient at time t is $Q_r=\prod_i a_i^{\nu_i}$ and depends on reaction notation, extent, pressure, and temperature.
• Equilibrium constant: At equilibrium, the reaction quotient equals the thermodynamic equilibrium constant $K^0(T)=\prod_i(a_{i,\mathrm{eq}})^{\nu_i}$, which depends only on temperature.
• Inversion temperature: The inversion temperature Tᵢ is the temperature at which the equilibrium constant equals one.

★ Must-know

📌 For a homogeneous system with reactants and products initially present, Qᵣ < K⁰ drives forward evolution, Qᵣ > K⁰ drives reverse evolution, and Qᵣ = K⁰ means equilibrium.

Further detail

• A pure solvent with activity one is omitted from the reaction-quotient expression.

• For an endothermic reaction, the equilibrium constant increases with temperature and heating shifts equilibrium in the forward endothermic direction.

• When only reactants are initially present, reaction can proceed only in the forward direction; a very large K⁰ indicates a nearly complete forward reaction.

💡 Memory Hook

Qᵣ below K⁰ drives the forward direction; Qᵣ above K⁰ drives the reverse direction.

📖 8. Equilibrium Disruption and Measurement

🔑 Key Concepts & Definitions

• Equilibrium disruption: Occurs when a constituent required by the reaction is no longer present in the final state.

★ Must-know

• Chemical methods determine equilibrium composition by assaying a reactant or product, while physical methods measure a physical quantity linked to concentration.

Further detail

• Dissolving 10⁻⁶ mol of AgCl(s) in 1 L of water gives Qᵣ = 10⁻¹², below K⁰ = 10⁻¹⁰, so the solid dissolves completely and the equilibrium is disrupted.

📐 Formula - For a monoprotic acid titrated by a standard monobase, equivalence satisfies $C_aV_a=C_bV_b$.

📐 Formula - For an acid–base pair AH/A⁻, acidity is given by $K_a=\frac{[A^-][H_3O^+]}{[AH]}$.

📐 Formula - In dilute aqueous solutions, conductivity follows Kohlrausch's law $\sigma=\sum_i\lambda_i^0C_i$, pH is $-\log[H_3O^+]$, and absorbance follows Beer–Lambert's law $A(\lambda)=\sum_i\varepsilon_i(\lambda)C_i$.

📖 9. Equilibrium Calculations and Applications

📝 Essential Points

📐 Formula - For butane combustion, $\mathrm{C_4H_{10}}+\frac{13}{2}\mathrm{O_2}\rightarrow4\mathrm{CO_2}+5\mathrm{H_2O}$.

• For an equimolar mixture of n mol butane and n mol oxygen, oxygen is limiting and the maximum extent is $\xi_m=\frac{2n}{13}$.

📐 Formula - For a stoichiometric butane–oxygen mixture, the conversion fraction is $\tau=\frac{\xi}{n}$ and the water mole fraction at half-reaction is 0.30.

• Air is modeled as one-fifth O₂ and four-fifths N₂, and the nitrogen must be included when calculating total gas amount.

• At 550 K, the equilibrium constant for PCl₃(g) + Cl₂(g) ⇌ PCl₅(g) is 0.30, giving an equilibrium conversion fraction of 0.12 and mole fractions of 46.8% each for PCl₃ and Cl₂ and 6.4% for PCl₅.

• For 2 SO₂(g) + O₂(g) ⇌ 2 SO₃(g) at 800 K, K⁰ = 1.21·10¹⁰ for an initial mixture of 7 mol SO₂, 10 mol O₂, and 83 mol N₂; the near-complete-reaction approximation gives residual SO₂ of 2.5·10⁻⁴ mol and partial pressure 2.5·10⁻⁶ bar.

• At the inversion temperature, ammonia synthesis from 20% N₂, 60% H₂, and 20% NH₃ at 500 bar has Qᵣ = 3.7·10⁻⁶ < K⁰ = 1 and therefore evolves forward.