Summary of Thermodynamics and Thermochemistry

Overview

• These notes are a collection of the basic principles of thermodynamics and thermochemistry.
• Main topics: system-surroundings, properties, state and path functions, major thermodynamic processes, rules for work and heat, U/H/Δ, heat capacity, entropy, Gibbs energy and bomb calorimeter practice.
• Goal: provide exam-level formulas, definitions and step-by-step solution methods.

System, Surroundings and Boundaries

• System: the part of the universe we study (e.g., a cup of hot tea).
• Surroundings: everything outside the system.
• Boundary: the surface separating system and surroundings; can be real or imaginary.
• Types of boundaries: adiabatic (no heat exchange), diathermic (heat exchange possible), rigid/non-rigid, permeable/impermeable.

System Types (open/closed/isolated)

• Open system: mass and heat can change (Δm ≠ 0, q ≠ 0).
• Closed system: mass does not change, heat can change (Δm = 0, q ≠ 0).
• Isolated system: neither mass nor heat is exchanged (Δm = 0, q = 0).

Extensive and Intensive Properties

• Extensive: depend on amount (mass, volume, U, H, S).
• Intensive: do not depend on amount (T, p, mole fraction, pH).
• Identification: if a property changes with scaling → extensive; per mole or specific → intensive; ratio of two extensive properties → intensive.

State Functions and Path Functions

• State functions: P, V, T, U, H, S, G — depend only on initial/final states; ΔU, ΔH etc.
• Path functions: W, Q — depend on the path; expressed with δQ, δW.
• Infinitesimal changes: dU, dH (state); δQ, δW (path).

Thermodynamic Processes (definitions and formulas)

• Isothermal (T constant): PV = constant; reversible W = - nRT ln(V2/V1); ΔU = 0 ⇒ Q = -W.
• Isochoric / isovolumetric (V constant): ΔV = 0 ⇒ W = 0; Q = ΔU = n CV ΔT.
• Isobaric (P constant): W = - P (V2 - V1); Q = ΔH = n CP ΔT.
• Adiabatic (Q = 0): ΔU = W; reversible: PV^γ = constant, TV^(γ-1) = constant; γ = CP/CV, R = CP - CV; W = (P2V2 - P1V1)/(γ - 1) (with appropriate sign).
• Cyclic process: start = end ⇒ all state function Δ = 0 but Q_net and W_net are generally not 0.

Work (details, sign convention, formulas)

• Infinitesimal work: δW = - P_ext dV (P_ext = external pressure).
• Sign convention:
  • Q positive when the system absorbs heat; Q negative when it releases heat.
  • W negative when the system does work; W positive when work is done on the system.
• Specifics:
  • Reversible isothermal: W = - nRT ln(V2/V1).
  • Irreversible (constant external P): W = - P_ext (V2 - V1).
  • On a PV graph W = area under the curve.
  • Free expansion (vacuum): P_ext = 0 ⇒ W = 0.

Internal Energy (U) and Enthalpy (H)

• Internal energy U: the system's internal energy (translation, rotation, vibration, electronic). For an ideal gas U = function(T).
• Enthalpy H = U + PV; both are state functions.
• General relation: ΔH = ΔU + Δ(PV); if P is constant then ΔH = ΔU + P ΔV.
• For an ideal gas Δ(PV) = nR ΔT ⇒ ΔH = ΔU + nR ΔT.
• For changes in gaseous moles ΔH = ΔU + R T Δn_gas (assuming constant T).

Heat Capacity

• C = dq/dT; specific heat = per gram; molar heat capacity = per mole.
• Constant-volume: ΔU = n CV ΔT.
• Constant-pressure: ΔH = n CP ΔT.
• Relation: CP - CV = R ; γ = CP / CV.
• Typical values:
  • Monoatomic: CV = 3/2 R, CP = 5/2 R, γ = 5/3.
  • Diatomic: CV = 5/2 R, CP = 7/2 R, γ = 7/5.
  • Polyatomic (approx): γ ≈ 4/3.

Mixtures (mixed heat capacity)

• Molar average for mixtures:
  • CP_mix = (Σ ni CP,i) / (Σ ni) ; CV_mix = (Σ ni CV,i) / (Σ ni).
• If γ_i are given: CV_i = R / (γ_i - 1) then form the mixture; then γ_mix = CP_mix / CV_mix.

Thermochemistry — Enthalpy and ways to get ΔH

• ΔH_reaction = Σ ΔHf(products) − Σ ΔHf(reactants).
• Alternative: ΔH = Eact(fwd) − Eact(bwd).
• Combustion dataset: ΔH_reaction = Σ(combustion of reactants) − Σ(combustion of products).
• Bond energies: ΔH_reaction = Σ(BDE broken) − Σ(BDE formed).
• Conditions for heats of formation: 1 mol product, elements in most stable standard state, usually 298 K and 1 bar.

Entropy (ΔS) — properties and formulas

• Qualitative: entropy = disorder; gas > liquid > solid.
• Rules: increase in moles of gas → ΔS > 0; mixing increases ΔS; greater atomicity → larger S.
• Spontaneity: ΔS_universe = ΔS_system + ΔS_surroundings; if >0 → spontaneous.
• Mathematical:
  • dS = δQ_rev / T.
  • Ideal gas: ΔS = n Cv ln(T2/T1) + n R ln(V2/V1).
  • Special paths: isothermal: ΔS = n R ln(V2/V1); isochoric: ΔS = n Cv ln(T2/T1); isobaric: ΔS = n Cp ln(T2/T1).
  • Phase change: ΔS = ΔH_transition / T_transition.
  • Adiabatic reversible: ΔS = 0 (isentropic).

Gibbs Free Energy (ΔG) and spontaneity

• ΔG = ΔH − T ΔS.
• Rules: ΔG < 0 → spontaneous (at constant T,P); ΔG = 0 → equilibrium; ΔG > 0 → non-spontaneous.
• Temperature dependence: if ΔH and ΔS have the same sign → spontaneity depends on T; cutoff T = ΔH/ΔS.
• Relation: ΔG° = − R T ln K.

Bomb Calorimeter — principle and calculations (step-by-step)

• Purpose: measure ΔE (internal energy change) from combustion/reaction at constant volume.
• Basic steps:
  1. Calculate n from the given mass/moles.
  2. Total heat Q = C_total × ΔT (if C is given).
  3. If molar heat capacities are given use Q = Σ (n_i × C_molar,i × ΔT).
  4. ΔU per mole = - Q / n_reactant (negative because the system released heat).
  5. If ΔH is needed and gaseous moles change then ΔH = ΔU + Δn_gas · R · T (T in K).
• In bomb calorimetry Q_released = C_cal × ΔT ; ΔE per mol ≈ − Q_released / n.
• Use Δn_gas and RT to adjust ΔU → ΔH.

Example: Benzoic acid bomb calorimeter (steps)

• Given: benzoic acid 12.2 g (molar mass 122 g/mol → n = 0.100 mol ≈ 1 mol if the problem states 1 mol).
• C_total = 50 (units), ΔT = 77 − 27 = 50°C.
• Q = C × ΔT = 50 × 50 = 2500 (units); instructor example also showed 25000 J — check units carefully.
• ΔU per mol = - Q / n (negative sign).
• Determine Δn_gas (moles of gaseous products − moles of gaseous reactants). Example had Δn_gas = -1/2.
• ΔH = ΔU + Δn_gas · R · T (T = 300 K if 27°C).

Useful formulas (collected summary)

Thermochemistry: Hess's Law, Bond Energies, Formation, Combustion

• Hess's Law: the ΔH of a reaction is the sum of ΔH of the steps that lead to it.
• Formation enthalpy rule: ΔH_reaction = Σ ΔHf(products) − Σ ΔHf(reactants).
• Combustion enthalpy: can use ΔH_reaction = Σ (combustion reactants) − Σ (combustion products).
• Bond method: broken bonds (positive) − formed bonds (negative) = ΔH_reaction.

Action Items / Next Steps (practice suggestions)

• Practice areas on PV graphs: areas and interconversion for isothermal, isobaric, isochoric, adiabatic curves.
• Solve several numerical problems with reversible isothermal and irreversible constant-P work.
• Memorize CP, CV, γ values (mono-, di-, polyatomic) and use them in ΔU/ΔH problems.
• Thermochemistry: solve 3–5 problems using ΔH formation/combustion and bond energy methods.
• Bomb calorimeter problems: work through full steps C, ΔT → Q → ΔE per mol → ΔH adjustment.
• Pay special attention to units (e.g., atm·L → J; temperature in K when using R).