Chemical engineering thermodynamics

In short

Chemical engineering thermodynamics tells you what a process *can* do — the maximum work, the minimum work, the attainable conversion and the limiting efficiency — before any equipment is sized. It supplies the property framework (U, H, S, G, fugacity, chemical potential) and the equations of state that convert measured P, V, T into the enthalpies and entropies used in every balance. In Indian university exams it is one subject of two papers (Thermodynamics I and II under most AICTE/CBCS curricula) and carries roughly 10-12% of the GATE-CH paper. On the job it governs compressor and turbine duty, refrigeration and cryogenics, reactor operating pressure and temperature, steam-system and utility design, and BEE-mandated energy audits in refineries, fertiliser and petrochemical plants.

Explained

Basic concepts and the first law

• System / surroundings / boundary: closed (no mass crossing), open or control volume (mass crosses), isolated (neither mass nor energy).
• State and path functions: U, H, S, G, A, T, P, V are state (point) functions and have exact differentials; Q and W are path functions. ∮dU = 0 but ∮dQ ≠ 0.
• Intensive properties (T, P, molar volume, density) are independent of mass; extensive (V, U, S) scale with it.
• Reversible process: infinitesimally slow, no friction, no finite driving force; it delivers maximum work and can be retraced. All real processes are irreversible.
• First law, closed system: ΔU = Q − W; for a quasi-static process W = ∫P dV.
• First law, open system (steady flow): Q − Ws = Δ[H + u²/2 + gz]; for most chemical process units kinetic and potential terms are negligible, giving Q − Ws = ΔH.
• Why H appears: flow work PV is lumped with U, so enthalpy is the natural property for flowing streams; U for rigid closed vessels.
• Heat capacities: Cp − Cv = R for an ideal gas; Cp = (∂H/∂T)P, Cv = (∂U/∂T)V. Cp is normally fitted as Cp/R = A + BT + CT² + DT⁻².
• Ideal-gas processes: isothermal (ΔU = 0, Q = W), isobaric (Q = ΔH), isochoric (W = 0, Q = ΔU), adiabatic reversible (PVγ = constant, Q = 0, W = −ΔU).
• Polytropic: PVⁿ = constant; n = 1 isothermal, n = γ isentropic, n = 0 isobaric, n = ∞ isochoric. Real compression lies between 1 and γ.

PVT behaviour of pure fluids

• On a P-V diagram the two-phase dome is bounded by saturated-liquid and saturated-vapour curves meeting at the critical point (Tc, Pc, Vc), where (∂P/∂V)T = 0 and (∂²P/∂V²)T = 0.
• Above Tc a gas cannot be liquefied by pressure alone; above both Tc and Pc the fluid is supercritical (used for CO₂ extraction of spice oleoresins and decaffeination).
• Compressibility factor Z = PV/RT: Z = 1 ideal, Z < 1 when attraction dominates (moderate pressure), Z > 1 when molecular volume dominates (high pressure).
• Theorem of corresponding states: Z is nearly a universal function of reduced properties Tr = T/Tc, Pr = P/Pc. Three-parameter form adds the acentric factor ω = −log₁₀(Pr^sat at Tr = 0.7) − 1 (ω ≈ 0 for argon, 0.344 for water, 0.011 for methane).
• Pitzer correlation: Z = Z⁰ + ωZ¹, read from Lee-Kesler tables; the virial equation Z = 1 + BP/RT suits low pressure (Pr below about 0.5 at usual Tr).

Cubic equations of state

• van der Waals: P = RT/(V − b) − a/V²; b corrects for molecular volume, a for attraction. a = 27R²Tc²/64Pc, b = RTc/8Pc. Gives Zc = 0.375 against the real 0.23-0.29, so it is quantitatively poor but conceptually central.
• Redlich-Kwong (1949): P = RT/(V − b) − a/[T^0.5 V(V + b)] — better vapour-phase volumes.
• Soave-Redlich-Kwong (1972) replaces a/T^0.5 by a·α(Tr, ω), fitting vapour pressures; Peng-Robinson (1976) changes the attraction denominator to V(V + b) + b(V − b) and predicts liquid densities better. Both are the industrial workhorses in Aspen Plus, HYSYS and DWSIM.
• A cubic EOS gives three roots in the two-phase region: largest = saturated vapour, smallest = saturated liquid, middle is physically meaningless.
• Limitations: poor near the critical point, poor for strongly polar and hydrogen-bonded fluids (water, alcohols, glycols) unless an activity-coefficient or association model is used.

Second law, entropy and exergy

• Statements: Kelvin-Planck (no cyclic device can convert all heat from a single reservoir into work); Clausius (heat cannot flow unaided from cold to hot).
• Entropy: dS = dQrev/T; Clausius inequality ∮dQ/T ≤ 0. For an isolated system ΔS ≥ 0; entropy generation Sgen ≥ 0 measures irreversibility.
• Entropy balance, steady flow: ΣṁS(out) − ΣṁS(in) − ΣQ̇j/Tj = Ṡgen.
• Third law: the entropy of a perfect crystal is zero at 0 K — the basis of absolute entropies used in reaction-equilibrium calculations.
• Carnot engine: ηCarnot = 1 − Tc/Th, the ceiling for any heat engine between two reservoirs; no engine can beat it and all reversible engines equal it.
• Exergy (availability) = maximum useful work obtainable as a stream comes to equilibrium with the dead state (T₀, P₀; usually 298-303 K, 1 atm in Indian plants): B = (H − H₀) − T₀(S − S₀).
• Lost work = T₀·Sgen (Gouy-Stodola). Second-law (exergetic) efficiency = useful exergy out / exergy in; this is what distinguishes a thermodynamically good design from one that merely closes the energy balance.
• Typical exergy destroyers: throttling valves, mixing of streams at different temperatures, large ΔT in heat exchangers, combustion itself (~30% of fuel exergy).

Property relations, residual and departure functions

• Fundamental relations (closed system, constant composition): dU = T dS − P dV; dH = T dS + V dP; dA = −S dT − P dV; dG = −S dT + V dP, with A = U − TS and G = H − TS.
• Maxwell relations follow from exactness, e.g. (∂S/∂P)T = −(∂V/∂T)P and (∂S/∂V)T = (∂P/∂T)V — they convert unmeasurable entropy derivatives into measurable PVT data.
• Residual (departure) property: M^R = M(real) − M(ideal gas) at the same T and P; computed from an EOS by integrating (∂Z/∂T) and (Z − 1)/P over pressure.
• Practical route for any real-fluid ΔH, ΔS: (1) residual at state 1 back to ideal gas, (2) ideal-gas change using Cp(T), (3) residual into state 2. Enthalpy of vaporisation is added where a phase change occurs.
• Thermodynamic consistency checks: Cp − Cv = −T(∂V/∂T)P²/(∂V/∂P)T; (∂U/∂V)T = T(∂P/∂T)V − P (zero for an ideal gas).
• Joule-Thomson coefficient μJT = (∂T/∂P)H = [T(∂V/∂T)P − V]/Cp. μJT > 0 means cooling on throttling; the locus μJT = 0 is the inversion curve. Hydrogen and helium warm on throttling at room temperature, so they must be pre-cooled before Linde liquefaction.

Chemical potential, fugacity and activity (the equilibrium framework)

• Criterion of equilibrium: at constant T and P, G is minimum, dG = 0; for every species i, μi is equal in all phases. Spontaneity: ΔG < 0.
• Chemical potential μi = (∂G/∂ni)T,P,nj — the partial molar Gibbs energy. For a pure ideal gas dG = RT d(ln P).
• Fugacity f replaces pressure for a real fluid: dG = RT d(ln f), with f → P as P → 0. Fugacity coefficient φ = f/P, obtained from ln φ = ∫₀^P (Z − 1) dP/P.
• Activity a i = f i/f i°, activity coefficient γi = a i/x i — the liquid-phase counterpart used in solution thermodynamics.
• Gibbs-Duhem equation: Σx i dμi = 0 at constant T, P — the consistency test behind all partial-molar and mixture property work.
• Gibbs phase rule: F = C − P + 2 (non-reacting); with r independent reactions, F = C − r − P + 2.

Chemical reaction equilibrium

• Standard Gibbs change: ΔG° = Σνi ΔG°f,i, with νi positive for products and negative for reactants.
• Equilibrium constant: ΔG° = −RT ln K, K = Π(â i)^νi. For gases â i = ŷ iφ̂ iP/P°, so K = Ky·Kφ·(P/P°)^Σν; at moderate pressure Kφ ≈ 1.
• Pressure effect: if Σν < 0 (ammonia, methanol synthesis) raising P raises conversion; if Σν = 0 (water-gas shift) pressure has no equilibrium effect.
• Temperature effect — van't Hoff: d(ln K)/dT = ΔH°/RT². Exothermic reactions lose K as T rises, which is why ammonia and SO₂ converters run at the lowest temperature the catalyst will tolerate, in stages with interbed cooling.
• Le Chatelier's principle is the qualitative summary; the quantitative answers come from K and the extent of reaction ε.
• Multiple reactions: solve by equating μi across r independent reactions, or by direct Gibbs energy minimisation (what RGibbs in Aspen does) — useful for reforming and combustion equilibria.

Cycles, compression and Indian practice

• Rankine cycle (steam power): pump, boiler, turbine, condenser; reheat and regenerative feed heating raise efficiency. Boilers in India are designed, inspected and certified under the Indian Boiler Regulations, 2026 (in force from October 2026; they superseded the Indian Boiler Regulations, 1950), framed under the Boilers Act, 2025, which replaced the Boilers Act, 1923 from 1 May 2025.
• Vapour-compression refrigeration: compressor, condenser, expansion valve (isenthalpic — the main irreversibility), evaporator; COP = refrigeration effect / compressor work. Refrigerant choice follows India's Montreal-Protocol obligations (HCFC phase-down continuing; HFC phase-down under the Kigali Amendment, India's schedule beginning in the 2030s — check the current ozone-cell notification for dates).
• Compressors: multistage with intercooling approaches isothermal (minimum) work; equal pressure ratio per stage minimises total work. Isentropic efficiency ηs = ideal work/actual work for compressors (reciprocal form for turbines).
• Liquefaction: Linde-Hampson (throttling), Claude (throttling plus expander) — the basis of air separation units at Indian steel plants and of LNG regasification terminals.
• Energy and efficiency regulation: the Energy Conservation Act, 2001 (amended 2022, adding carbon-credit trading) makes energy audits and BEE's PAT targets mandatory for designated consumers, including fertiliser, cement, refinery, chlor-alkali and paper plants; pinch and exergy analysis are the usual tools for meeting them. ISO 50001 is the voluntary management standard.

Key points

• State vs path - U, H, S, G are state functions; Q and W depend on the path taken.
• First law - energy is conserved: ΔU = Q − W (closed), Q − Ws = ΔH (steady flow, kinetic/potential neglected).
• Second law - entropy of an isolated system never decreases; Sgen ≥ 0 measures how bad a process is.
• Carnot limit - η = 1 − Tc/Th; quoting any engine above it signals an arithmetic error.
• Exergy - B = ΔH − T₀ΔS; lost work = T₀Sgen, so irreversibility has a rupee value.
• Compressibility factor - Z = PV/RT; deviation from 1 measures non-ideality.
• Corresponding states - Z = Z(Tr, Pr, ω); the acentric factor captures molecular shape.
• Cubic EOS - van der Waals is conceptual; SRK and Peng-Robinson are what simulators use.
• Maxwell relations - convert entropy derivatives into PVT measurements.
• Residual property - real minus ideal-gas value at the same T, P; the bridge from EOS to ΔH and ΔS.
• Joule-Thomson - throttling is isenthalpic; cooling occurs only inside the inversion curve.
• Chemical potential - equal in every phase at equilibrium; the master criterion.
• Fugacity - the "effective pressure" of a real fluid; φ = f/P, and φ → 1 as P → 0.
• ΔG° = −RT ln K - the single most used equation in reaction equilibrium; ΔG° is a function of T only.
• van't Hoff - d(ln K)/dT = ΔH°/RT²; exothermic reactions worsen with temperature.
• Phase rule - F = C − P + 2, and F = C − r − P + 2 when r reactions occur.
• Minimum work - isothermal reversible compression; multistage intercooling moves real machines towards it.
• Equilibrium ≠ kinetics - K says how far, rate says how fast; a favourable K with no catalyst gives nothing.

Formulas and examples

Example 1 — Compression work and the cost of irreversibility. Air (ideal, Cp = 3.5R) is compressed from 1 bar, 300 K to 5 bar.

• Adiabatic reversible: T₂ = 300 × 5^0.4/1.4 = 300 × 5^0.2857 = 300 × 1.584 = 475.1 K.
• Ws = Cp ΔT = 3.5 × 8.314 × 175.1 = 5,095 J/mol = 5.10 kJ/mol; per kg (M = 29 g/mol) = 5,095/0.029 = 176 kJ/kg.
• Check entropy: ΔS = 29.10 × ln(1.584) − 8.314 × ln 5 = 13.38 − 13.38 = 0 — confirms isentropic.
• Isothermal reversible: W = RT ln(P₂/P₁) = 8.314 × 300 × 1.609 = 4,014 J/mol, i.e. 21% less work, but 4,014 J/mol of heat must be removed. This is why intercoolers exist.
• If the real machine has ηs = 0.75, actual work = 5,095/0.75 = 6,793 J/mol and real discharge T₂ = 300 + 6,793/29.10 = 533 K.

Example 2 — van der Waals vs ideal gas. CO₂ at 320 K occupying 0.400 L/mol (a = 0.3640 Pa·m⁶/mol², b = 4.267 × 10⁻⁵ m³/mol).

• Ideal: P = RT/V = (8.314 × 320)/4.00 × 10⁻⁴ = 2,660.5/4.00 × 10⁻⁴ = 66.5 bar.
• vdW: RT/(V − b) = 2,660.5/(3.5733 × 10⁻⁴) = 74.46 bar; a/V² = 0.3640/(1.60 × 10⁻⁷) = 22.75 bar; P = 74.46 − 22.75 = 51.7 bar.
• Z = PV/RT = (5.17 × 10⁶ × 4.00 × 10⁻⁴)/2,660.5 = 0.777. Using the ideal-gas law here would oversize the vessel pressure rating by 29%.

Example 3 — Carnot ceiling and second-law efficiency. A boiler supplies heat at 600 K; cooling water in a Gujarat coastal plant gives a sink at 310 K. Plant output 36 MW for 100 MW heat input.

• ηCarnot = 1 − 310/600 = 0.4833 (48.3%); actual η = 36/100 = 0.360.
• Second-law efficiency = 0.360/0.4833 = 0.745, so 25.5% of the available work potential is destroyed.
• Entropy generation: Sgen = Qc/Tc − Qh/Th = 64/310 − 100/600 = 0.2065 − 0.1667 = 0.0398 MW/K; lost work = 310 × 0.0398 = 12.3 MW, matching 48.33 − 36 = 12.3 MW.

Example 4 — Reaction equilibrium and temperature. N₂ + 3H₂ ⇌ 2NH₃, with ΔG°f(NH₃, 298 K) = −16.45 kJ/mol and ΔH°f = −46.1 kJ/mol.

• ΔG°298 = 2(−16.45) = −32.9 kJ/mol; ΔH°298 = −92.2 kJ/mol.
• ln K298 = 32,900/(8.314 × 298) = 32,900/2,477.6 = 13.28 → K298 = 5.9 × 10⁵ (very favourable).
• At 700 K: ln(K₂/K₁) = (92,200/8.314)(1/700 − 1/298) = 11,090 × (−1.9272 × 10⁻³) = −21.37; ln K700 = 13.28 − 21.37 = −8.09 → K700 = 3.1 × 10⁻⁴.
• Equilibrium collapses by nine orders of magnitude, yet plants run at 700-800 K because the catalyst is inactive below it — so they compensate with 150-250 bar, since Σν = −2 and K = Ky(P/P°)⁻². Treat the constant-ΔH° assumption as an estimate only.

Example 5 — Refrigeration COP. A cold room is held at −10 °C (263 K) rejecting heat at 30 °C (303 K), duty 50 kW.

• COPCarnot = 263/(303 − 263) = 6.58; ideal power = 50/6.58 = 7.6 kW.
• A real R-134a plant at COP 3.0 draws 50/3.0 = 16.7 kW. At ₹8/kWh running 20 h/day the gap of 9.1 kW costs 9.1 × 20 × 8 ≈ ₹1,456/day, about ₹5.3 lakh a year — the standard justification for a plant upgrade in an energy audit.

Where it is used

• Refineries and petrochemicals (Reliance Jamnagar, IOCL Panipat, HMEL Bathinda): EOS-based flash and enthalpy calculations for distillation, compressor and turbine duties, hydrogen recycle loops.
• Fertilisers and heavy chemicals (IFFCO, RCF, GSFC, Tata Chemicals): ammonia, methanol and urea synthesis-loop pressure and converter temperature profiles set entirely by K and van't Hoff behaviour.
• Industrial gases and cryogenics (INOX Air Products, Linde India, air separation units at JSW and Tata Steel): Joule-Thomson and Claude-cycle design, liquid oxygen/nitrogen/argon production.
• Energy audit and utilities roles (BEE-certified energy auditors, utilities/HSE engineers in cement and paper): pinch and exergy analysis to meet PAT targets under the Energy Conservation Act, 2001 (amended 2022).
• Process simulation and licensing jobs (Aspen Plus, HYSYS, PRO/II, DWSIM; EPC firms such as L&T Hydrocarbon, Toyo India, TCE): choosing the right property package is the first and most consequential modelling decision.
• Exams and selection tests: GATE-CH, ESE, PSU recruitment (ONGC, IOCL, GAIL, BPCL, NFL), and GATE-based PSU interviews where Carnot limits, Z and ΔG° = −RT ln K are standard questions.

Interview questions

• Q: Why is enthalpy used for flowing streams and internal energy for closed vessels? A: H = U + PV absorbs the flow work needed to push fluid across the control surface, so the steady-flow energy balance reduces to Q − Ws = ΔH; a rigid closed system does no flow work, so ΔU applies.
• Q: What does a compressibility factor of 0.8 physically mean? A: The real gas occupies 80% of the volume the ideal-gas law predicts at that T and P, because attractive forces dominate; sizing on Z = 1 would oversize the vessel or undersize the pressure duty.
• Q: Why can no engine beat the Carnot efficiency? A: Any higher efficiency would require net entropy to decrease in an isolated system, violating the second law; a Carnot engine already has zero entropy generation.
• Q: Throttling is isenthalpic, so has anything been lost? A: Energy no, work potential yes — entropy is generated, so exergy equal to T₀Sgen is destroyed; that is why expanders replace valves where the duty justifies it.
• Q: Why does hydrogen heat up when throttled at room temperature? A: Its Joule-Thomson inversion temperature is about 200 K, so at ambient temperature μJT is negative; it must be pre-cooled below the inversion temperature before throttling will cool it.
• Q: Why is ammonia synthesis run at high pressure and only moderately high temperature? A: Σν = −2, so pressure pushes equilibrium towards product, while the reaction is exothermic so K falls sharply with temperature — the temperature is the minimum the catalyst needs, not a thermodynamic choice.
• Q: Is ΔG° a function of pressure? A: No — it is defined at the standard state (usually 1 bar) and depends only on temperature; the pressure effect on conversion appears through the (P/P°)^Σν term in K.
• Q: Difference between fugacity and fugacity coefficient? A: Fugacity has pressure units and is the effective pressure of a real species; the fugacity coefficient φ = f/P is dimensionless and measures the deviation, approaching 1 as pressure approaches zero.
• Q: Which property package would you pick for a natural-gas dehydration unit with glycol? A: A cubic EOS such as Peng-Robinson or SRK for the hydrocarbon/gas phase, but with an activity-coefficient or association model for the polar glycol-water liquid, since plain cubics handle hydrogen bonding badly.
• Q: What is the use of Maxwell relations in practice? A: They express entropy derivatives, which cannot be measured, in terms of P-V-T derivatives, which can — making tables and departure functions computable from experimental data or an EOS.
• Q: A reaction has K = 10⁶ but nothing happens in the reactor. Why? A: K is thermodynamic and says only how far the reaction can go; the rate may be negligible without a catalyst or at that temperature.
• Q: What does a second-law efficiency of 40% tell a plant manager that a thermal efficiency of 35% does not? A: That 60% of the genuinely available work potential is being destroyed internally and much of it is recoverable, whereas the 35% is measured against an unattainable 100%.

Common mistakes

• Using the ideal-gas law near the critical point or at high pressure instead of Z or an EOS — the single most common numerical error.
• Mixing sign conventions: W as work done *by* the system (ΔU = Q − W, used here and in GATE) versus work done *on* it.
• Writing ΔS = Cp ln(T₂/T₁) and forgetting the −R ln(P₂/P₁) term when pressure also changes.
• Forgetting to convert temperatures to kelvin in Carnot, COP and van't Hoff calculations — degrees Celsius gives nonsense.
• Treating ΔH° and Cp as constant over a 400 K range without labelling it an approximation.
• Confusing equilibrium with kinetics, or reading K as a conversion — conversion must be obtained from the extent of reaction and the mole balance.
• Picking the middle root of a cubic EOS, or quoting the liquid root for a vapour stream.
• Reporting that energy was "lost" in a throttle or a mixer; it is work potential (exergy) that is lost, not energy.

Related topics

Energy balances · Phase equilibria and VLE · Material balances · Heat transfer and heat-exchanger design · Chemical reaction engineering · Process simulation and property packages