Chemical thermodynamics

In short

Chemical thermodynamics studies the energy changes, heat flow and direction of chemical and physical processes using state functions: internal energy (U), enthalpy (H), entropy (S), Helmholtz energy (A) and Gibbs energy (G). It answers two questions a chemist always faces - how much heat does a reaction give or take, and will it happen on its own - without needing to know the mechanism or the rate. It is the highest-weight unit in B.Sc physical chemistry and in CSIR-UGC NET, GATE Chemistry (CY), IIT-JAM and TIFR papers, and carries into process design in fertiliser, refinery, pharmaceutical and metallurgical plants. Interviewers at Reliance, IOCL, Tata Steel, Hindalco, Dr Reddy's, Sun Pharma and NTPC use it to test whether a candidate can reason about feasibility, heat duty and energy balance. All data here is in SI units; Indian calorimetry and fuel-testing laboratories follow the relevant IS and ASTM bomb-calorimeter methods and report calorific values in MJ/kg.

Explained

Systems, states and basic terms

• System and surroundings: the part under study and everything else. Open (exchanges matter and energy - a distillation column), closed (energy only - a sealed reaction flask), isolated (neither - an ideal thermos).
• State function: depends only on the present state, not the path - U, H, S, G, A, T, P, V. Path function: depends on the route - q (heat) and w (work). A cyclic integral of a state function is zero.
• Extensive properties scale with amount (V, U, H, S); intensive ones do not (T, P, density, molar quantities, chemical potential).
• Processes: isothermal (T constant), isobaric (P constant), isochoric (V constant), adiabatic (q = 0), cyclic, reversible (infinitesimally slow, always at equilibrium, gives maximum work) and irreversible (real, finite rate).
• Sign convention (IUPAC/SI): heat absorbed by the system and work done on the system are positive. So ΔU = q + w, with w = −P_ext dV for expansion work.
• Equilibrium: thermal, mechanical and chemical equality across the system; the zeroth law defines temperature through thermal equilibrium and justifies the thermometer.

First law, enthalpy and thermochemistry

• First law (conservation of energy): ΔU = q + w. Energy can be converted but not created or destroyed; a perpetual-motion machine of the first kind is impossible.
• For a cyclic process ΔU = 0, so q = −w. At constant volume w = 0, hence q_V = ΔU; at constant pressure q_P = ΔH, where H = U + PV.
• Heat capacities: C_V = (∂U/∂T)_V and C_P = (∂H/∂T)_P. For an ideal gas C_P − C_V = R (per mole). Monatomic: C_V = 3R/2; diatomic (translation + rotation): C_V = 5R/2.
• Relation of ΔH and ΔU: ΔH = ΔU + Δn_g RT, where Δn_g is the change in moles of gaseous species. They are equal when Δn_g = 0.
• Hess's law of constant heat summation: ΔH of a reaction is the same by any route, because H is a state function. Basis of all thermochemical cycles - ΔH°_rxn = Σ ΔH°_f (products) − Σ ΔH°_f (reactants).
• Standard state: pure substance at 1 bar (the current IUPAC standard pressure; older books use 1 atm) and the stated temperature, usually 298.15 K. ΔH°_f of an element in its reference form is zero by definition.
• Named enthalpies: formation, combustion, neutralisation (strong acid + strong base ≈ −57.3 kJ/mol), solution, hydration, atomisation, bond dissociation, sublimation, fusion, vaporisation, lattice enthalpy (from a Born-Haber cycle).
• Bond enthalpy is an averaged value over many molecules; estimates from bond enthalpies are approximate (±10-20 kJ/mol typical).
• Kirchhoff's equation: temperature dependence of reaction enthalpy, ΔH°(T₂) = ΔH°(T₁) + ∫ΔC_P dT.
• Calorimetry: a bomb calorimeter (constant volume) gives ΔU of combustion for solid and liquid fuels; a coffee-cup or water calorimeter at constant pressure gives ΔH. Bomb calorimetry is the standard route to gross calorific value of Indian coal and petroleum products.

Second law, entropy and spontaneity

• Need for the second law: the first law permits a cold body to heat a hot one; experience forbids it. The second law fixes direction.
• Statements: *Clausius* - heat cannot flow of itself from a colder to a hotter body. *Kelvin-Planck* - no cyclic engine can convert heat completely into work with a single reservoir. Both rule out a perpetual-motion machine of the second kind.
• Entropy: dS = dq_rev/T (J K⁻¹ mol⁻¹), a state function. Clausius inequality: dS ≥ dq/T, equality only for a reversible change.
• Entropy criterion: for an isolated system (or universe) ΔS_total > 0 for a spontaneous change, = 0 at equilibrium, < 0 impossible. ΔS_universe = ΔS_system + ΔS_surroundings.
• Boltzmann (statistical) view: S = k_B ln W, with W the number of microstates; entropy measures molecular disorder or the spread of energy over available levels.
• Carnot cycle: two isothermals and two adiabatics; efficiency η = 1 − T_C/T_H depends only on temperatures, not on the working substance (Carnot's theorem). No engine between the same reservoirs can beat it.
• Entropy rises on fusion, vaporisation, mixing, expansion, dissolution of a solid and any increase in moles of gas; it falls on condensation, crystallisation, polymerisation and hydration of ions.
• Third law: the entropy of a perfect crystalline substance is zero at 0 K. This makes absolute entropies S° measurable from heat-capacity data; residual entropy survives in substances with disorder frozen in (CO, ice).
• Entropy of mixing of ideal gases: ΔS_mix = −nR Σ x_i ln x_i, always positive, ΔH_mix = 0.

Free energy, spontaneity and chemical potential

• Helmholtz energy A = U − TS: −ΔA is the maximum total work obtainable in an isothermal (constant T) process; ΔA < 0 is the spontaneity criterion at constant T and V.
• Gibbs energy G = H − TS: −ΔG is the maximum non-expansion (useful) work at constant T and P - directly the electrical work in a cell, −ΔG = nFE.
• Criteria at constant T, P: ΔG < 0 spontaneous, ΔG = 0 equilibrium, ΔG > 0 non-spontaneous (reverse is spontaneous).
• ΔG = ΔH − TΔS splits spontaneity into an enthalpy and an entropy part. Four cases: ΔH < 0, ΔS > 0 → always spontaneous; ΔH > 0, ΔS < 0 → never; ΔH < 0, ΔS < 0 → spontaneous at low T; ΔH > 0, ΔS > 0 → spontaneous at high T (the basis of carbon reduction of oxides at high temperature in a blast furnace).
• Fundamental equations: dU = TdS − PdV; dH = TdS + VdP; dA = −SdT − PdV; dG = −SdT + VdP. From the last, (∂G/∂T)_P = −S and (∂G/∂P)_T = V.
• Maxwell relations follow from exactness, e.g. (∂S/∂V)_T = (∂P/∂T)_V and (∂S/∂P)_T = −(∂V/∂T)_P - used to convert unmeasurable entropy derivatives into measurable P-V-T data.
• Gibbs-Helmholtz equation: [∂(ΔG/T)/∂T]_P = −ΔH/T², which gives ΔH from the temperature variation of ΔG or of an equilibrium constant.
• Chemical potential μ_i = (∂G/∂n_i)_{T,P,n_j}: the partial molar Gibbs energy, the driving force for mass transfer and the quantity equal in all phases at equilibrium.
• Gibbs-Duhem equation: Σ n_i dμ_i = 0 at constant T and P - the chemical potentials of a mixture are not independent.
• ΔG and equilibrium: ΔG = ΔG° + RT ln Q and at equilibrium ΔG° = −RT ln K. ΔG° fixes K; ΔG tells the direction at the actual composition.
• Van't Hoff equation: d(ln K)/dT = ΔH°/RT², so ln K falls with T for exothermic reactions.
• Fugacity f replaces pressure for real gases (μ = μ° + RT ln f/f°); fugacity coefficient φ = f/P → 1 as P → 0. Activity a plays the same role for solutions.

Real gases, phase changes and partial molar quantities

• Joule-Thomson effect: throttling at constant enthalpy; μ_JT = (∂T/∂P)_H. Cooling occurs below the inversion temperature, heating above it - hydrogen and helium warm on throttling at room temperature. Basis of the Linde liquefaction process used in Indian industrial gas plants.
• For an ideal gas μ_JT = 0; the effect arises from intermolecular attraction (cooling) and finite molecular size (heating).
• Clapeyron equation: dP/dT = ΔH_trans/(TΔV_trans) for any two-phase equilibrium. The negative slope of the ice-water line (ΔV < 0) explains why ice melts under pressure.
• Clausius-Clapeyron equation (liquid-vapour, vapour treated as ideal): ln(P₂/P₁) = −(ΔH_vap/R)(1/T₂ − 1/T₁) - used to get ΔH_vap from vapour-pressure data.
• Trouton's rule: ΔS_vap ≈ 85-88 J K⁻¹ mol⁻¹ at the normal boiling point for non-associated liquids; water and alcohols deviate because of hydrogen bonding.
• Gibbs phase rule: F = C − P + 2 (non-reacting system, only T and P as variables) - the backbone of one- and two-component phase diagrams.
• Thermodynamic equation of state: (∂U/∂V)_T = T(∂P/∂T)_V − P (internal pressure), zero for an ideal gas, = a/V² for a van der Waals gas.
• Partial molar volume V̄_i = (∂V/∂n_i): why mixing ethanol and water contracts the volume.
• Third-law / statistical link: absolute entropies tabulated in data books come from ∫(C_P/T)dT plus transition entropies, cross-checked against spectroscopic (statistical) entropies.

Limitations and what thermodynamics cannot tell you

• It says nothing about rate - diamond → graphite has ΔG° ≈ −2.9 kJ/mol yet takes geological time; kinetics is a separate subject.
• It deals with macroscopic bulk systems at (or near) equilibrium, not with single molecules or far-from-equilibrium states.
• It gives no mechanism and no molecular picture by itself; statistical thermodynamics supplies that bridge.
• A negative ΔG means feasibility, not that a usable amount of product will form in a usable time, nor that a catalyst is unnecessary.

Key points

• State function - value fixed by the state (U, H, S, G, A); path functions q and w are not.
• First law - ΔU = q + w; energy is conserved, ΔU = 0 over a cycle.
• q_V = ΔU, q_P = ΔH - the reason bomb calorimetry gives ΔU and open-vessel calorimetry gives ΔH.
• C_P − C_V = R - per mole of ideal gas; C_P > C_V because constant-pressure heating also does expansion work.
• ΔH = ΔU + Δn_g RT - converts between the two reaction heats.
• Hess's law - reaction enthalpy is route-independent; the tool for Born-Haber and formation-enthalpy problems.
• Kirchhoff's equation - gives ΔH at another temperature from ΔC_P.
• Entropy - dS = dq_rev/T; ΔS_universe > 0 for every spontaneous change.
• Carnot efficiency - η = 1 − T_C/T_H with absolute temperatures; the ceiling for any heat engine.
• Third law - S → 0 for a perfect crystal at 0 K, which makes absolute entropy measurable.
• Gibbs energy - G = H − TS; ΔG < 0 is the spontaneity test at constant T and P.
• −ΔG = w_max,non-expansion; −ΔA = w_max,total.
• ΔG° = −RT ln K - thermodynamics sets the position of equilibrium.
• Van't Hoff / Gibbs-Helmholtz - temperature dependence of K and of ΔG, source of ΔH from slopes.
• Maxwell relations - replace entropy derivatives by P-V-T measurements.
• Chemical potential - equal in all phases at equilibrium; drives diffusion and phase change.
• Clapeyron and Clausius-Clapeyron - slope of phase boundaries and ΔH_vap from vapour pressures.
• Joule-Thomson cooling - below the inversion temperature only; zero for an ideal gas; basis of gas liquefaction.

Formulas and examples

Example 1 - ΔH and ΔU for a combustion. C₂H₅OH(l) + 3O₂(g) → 2CO₂(g) + 3H₂O(l), ΔU = −1,364 kJ/mol at 298 K. Find ΔH.

• Δn_g = 2 − 3 = −1.
• Δn_g RT = (−1)(8.314)(298) = −2,478 J = −2.48 kJ.
• ΔH = ΔU + Δn_g RT = −1,364 − 2.48 = −1,366.5 kJ/mol.

Example 2 - Hess's law. Given ΔH°_f: CH₄(g) = −74.8, CO₂(g) = −393.5, H₂O(l) = −285.8 kJ/mol. Find ΔH°_c of methane.

• CH₄ + 2O₂ → CO₂ + 2H₂O(l).
• ΔH° = [−393.5 + 2(−285.8)] − [−74.8 + 0] = (−965.1) + 74.8 = −890.3 kJ/mol (the standard value; 1 m³ of methane, ≈ 44.6 mol at STP, releases about 39.7 MJ).

Example 3 - Spontaneity and the switch-over temperature. CaCO₃(s) → CaO(s) + CO₂(g); ΔH° = +178.3 kJ/mol, ΔS° = +160.5 J K⁻¹ mol⁻¹.

• At 298 K: ΔG° = 178.3 − 298(0.1605) = 178.3 − 47.8 = +130.5 kJ/mol → not spontaneous.
• Switch-over: T = ΔH°/ΔS° = 178,300/160.5 = 1,111 K (≈ 838 °C); above this ΔG° < 0.
• Hence lime kilns in Indian cement and sugar plants run at roughly 900-1,000 °C.
• K at 298 K: ln K = −ΔG°/RT = −130,500/(8.314 × 298) = −52.7 → K ≈ 1.3 × 10⁻²³.

Example 4 - Isothermal reversible expansion: w, q, ΔU, ΔS. 2 mol of an ideal gas expands reversibly and isothermally at 300 K from 5 L to 20 L.

• w = −nRT ln(V₂/V₁) = −2(8.314)(300) ln 4 = −4,988.4 × 1.3863 = −6,915 J (work done by the gas = 6.92 kJ).
• Isothermal ideal gas: ΔU = 0, so q = −w = +6,915 J absorbed.
• ΔS_system = nR ln(V₂/V₁) = 2(8.314)(1.3863) = +23.05 J K⁻¹; ΔS_surr = −6,915/300 = −23.05 J K⁻¹ → ΔS_universe = 0 (reversible).
• If instead it expanded freely into vacuum, w = 0, q = 0, ΔU = 0, but ΔS_system is still +23.05 J K⁻¹ (state function) and ΔS_universe = +23.05 J K⁻¹ - irreversible.

Example 5 - Carnot engine and ΔG° from a cell. (a) An engine works between 500 °C (773 K) and 30 °C (303 K). η = 1 − 303/773 = 1 − 0.392 = 60.8%. Heat supplied 1,000 kJ → maximum work 608 kJ, heat rejected 392 kJ. A real NTPC-class steam plant achieves about 38-42% because of irreversibilities. (b) Daniell cell, E° = 1.10 V, n = 2: ΔG° = −nFE° = −2(96,485)(1.10) = −212.3 kJ/mol - spontaneous.

Where it is used

• Fertiliser and heavy chemicals: ammonia synthesis at IFFCO, RCF and Chambal plants is run at the pressure and temperature chosen by ΔG°-K and Le Chatelier reasoning; heat of reaction fixes converter cooling duty.
• Refineries and petrochemicals: IOCL, BPCL, Reliance and HPCL use enthalpy and entropy balances for distillation reboiler/condenser duty, reformer heat loads and Joule-Thomson gas chilling in LPG recovery.
• Metallurgy: Tata Steel, SAIL and Hindalco rely on Ellingham diagrams (ΔG° vs T) to pick carbon, hydrogen or electrolytic reduction, and to set blast-furnace and smelter temperatures.
• Power and energy: Carnot and Rankine limits set plant efficiency at NTPC stations; gross and net calorific values of coal from bomb calorimetry decide tariffs and boiler design, and ΔG drives fuel cells and battery voltages.
• Pharmaceuticals and formulation: Sun Pharma, Cipla and Dr Reddy's use solubility (ΔG_solution), polymorph stability, DSC/TGA melting and crystallisation enthalpies for drug and excipient selection.
• Roles and labs: process engineer, R&D chemist, QC analyst, thermal-analysis specialist, energy auditor (BEE-certified), and research at CSIR laboratories (NCL Pune, IICT Hyderabad) and BARC.

Interview questions

• Q: Why is ΔH used more than ΔU in chemistry? A: Most reactions are run in open vessels at constant atmospheric pressure, where the heat exchanged equals ΔH.
• Q: A reaction has ΔH negative but does not occur. Does this break the first law? A: No. The first law only accounts for energy; direction is decided by ΔG (or ΔS_universe), and a large kinetic barrier can stall even a feasible reaction.
• Q: Can entropy of a system ever decrease in a spontaneous process? A: Yes - freezing of water below 0 °C lowers the system's entropy, but the surroundings gain more, so ΔS_universe is positive.
• Q: Why is work a path function? A: Its value depends on how the volume changes - reversible expansion gives maximum work, free expansion gives zero, for the same initial and final states.
• Q: What is the physical meaning of −ΔG? A: The maximum non-expansion work obtainable at constant T and P, realised for example as electrical work in a galvanic cell, −ΔG = nFE.
• Q: Difference between ΔG and ΔG°? A: ΔG° refers to all species in their standard states and fixes K; ΔG applies to the actual composition, ΔG = ΔG° + RT ln Q, and is zero at equilibrium.
• Q: Why does the melting line of ice slope backwards on a P-T diagram? A: Water is denser than ice, so ΔV of fusion is negative, and dP/dT = ΔH/(TΔV) becomes negative.
• Q: Why do hydrogen and helium warm up on throttling at room temperature? A: Their Joule-Thomson inversion temperatures are far below room temperature, so the repulsive (molecular-size) contribution dominates and μ_JT is negative.
• Q: State the third law and its practical use. A: The entropy of a perfect crystal is zero at 0 K; it provides the reference that lets absolute entropies be obtained from heat-capacity integration.
• Q: Why does Carnot efficiency not depend on the working substance? A: Carnot's theorem shows any reversible engine between the same two reservoirs must have the same efficiency, else the second law could be violated by coupling two engines.
• Q: How is ΔH of a reaction obtained from equilibrium data? A: From a van't Hoff plot of ln K against 1/T; the slope is −ΔH°/R (Gibbs-Helmholtz gives the same result from ΔG data).
• Q: Why is reduction of a metal oxide by carbon favoured only at high temperature? A: The reaction has a large positive ΔS (CO gas formed), so the −TΔS term eventually overcomes the positive ΔH - the crossing of lines on an Ellingham diagram.

Common mistakes

• Mixing sign conventions - using w = +PΔV (old convention) with ΔU = q + w, which flips the answer.
• Writing ΔS in kJ while ΔH is in kJ and ΔS data are tabulated in J K⁻¹ mol⁻¹; in ΔG = ΔH − TΔS, ΔS must be converted to kJ K⁻¹ mol⁻¹.
• Treating ΔG < 0 as proof that a reaction is fast, or as proof it goes to completion.
• Applying ΔS_system > 0 as the spontaneity test instead of ΔS_universe, or ΔG < 0 outside constant T and P.
• Forgetting that ΔH = ΔU when Δn_g = 0, and counting liquids or solids in Δn_g.
• Using Celsius in η = 1 − T_C/T_H, in ln(K₂/K₁) or in ΔG = ΔH − TΔS - all need kelvin.
• Assuming ΔH and ΔS are temperature-independent in all problems - acceptable over a small range only; Kirchhoff's equation is the correction.

Related topics

Chemical and ionic equilibrium · Chemical kinetics and catalysis · Electrochemistry · Solutions and colligative properties · Phase rule and phase equilibria · Statistical thermodynamics