Thermal physics and thermodynamics

In short

Thermal physics is the study of heat, temperature, work and the bulk properties of matter; thermodynamics is its macroscopic framework — four laws plus a set of state functions (U, H, S, F, G) that predict what a system can and cannot do without any knowledge of atoms. It fixes the ceiling on every engine, refrigerator, power plant and heat pump, and explains phase changes, gas liquefaction and thermal radiation. In B.Sc Physics it is a full paper; it is heavily examined in GATE (PH), IIT-JAM, CSIR-UGC NET, JEST and TIFR, and in state PSU and PSC technical papers. Employers who test it directly include BARC, ISRO, DRDO, CSIR-NPL, IMD, NTPC, BHEL, cryogenic and HVAC companies, and semiconductor/thermal-design teams.

Explained

Systems, state, equilibrium and the zeroth law

• System / surroundings / boundary. *Open* exchanges matter and energy, *closed* only energy, *isolated* neither. *Adiabatic* boundary blocks heat; *diathermic* allows it.
• State variables are *extensive* (V, U, S, m — scale with size) or *intensive* (P, T, ρ, μ). An equation of state links them, e.g. PV = nRT.
• Thermodynamic equilibrium = mechanical + thermal + chemical equilibrium simultaneously; only then are P, T uniquely defined.
• Zeroth law: if A is in thermal equilibrium with C and B is in thermal equilibrium with C, then A is in equilibrium with B. This is what makes *temperature* a valid property and thermometry possible.
• Thermometry. Thermometric properties: gas pressure/volume (constant-volume gas thermometer — the primary standard), resistance (Pt-100, thermistor), thermo-emf (thermocouple), radiance (pyrometer). The absolute (Kelvin) scale is defined by a reversible engine and is independent of working substance.
• SI since 20 May 2019: the kelvin is fixed by the Boltzmann constant k = 1.380649 × 10⁻²³ J K⁻¹ exactly; the old triple-point-of-water definition (273.16 K) is no longer the definition, though the triple point is still 273.16 K to within measurement uncertainty. Practical realisation follows ITS-90 (and PLTS-2000 below 1 K); CSIR-NPL, New Delhi maintains India's national temperature standards and traceable calibration.
• Process types: isothermal, isobaric, isochoric, adiabatic, isenthalpic (throttling), polytropic (PVⁿ = const), cyclic. Quasi-static = infinitely slow, always near equilibrium; reversible = quasi-static *and* free of dissipation (no friction, no finite ΔT heat flow, no mixing). All real processes are irreversible.

First law: energy, work, heat and specific heats

• Internal energy U is a state function; Q and W are path functions. First law (physics sign convention): dU = δQ − δW, with δW = P dV the expansion work done *by* the system (negative when the gas is compressed).
• Perpetual motion of the first kind (work with no energy input) is forbidden by the first law.
• Specific heats: C = δQ/dT per mole. C_V = (∂U/∂T)_V, C_P = (∂H/∂T)_P, with H = U + PV (enthalpy). For an ideal gas C_P − C_V = R; γ = C_P/C_V = 1.67 (monatomic), 1.40 (diatomic, rigid), ≈1.33 (polyatomic/vibrating).
• Why C_P > C_V: at constant pressure part of the heat does expansion work. C_P is the one measured for solids and liquids (hard to hold V fixed against huge pressures).
• Joule free expansion (into vacuum, rigid insulated vessel): Q = 0, W = 0, so ΔU = 0. For an ideal gas T is unchanged; for a real gas T falls because work is done against intermolecular attraction.
• Ideal-gas processes: isothermal W = nRT ln(V₂/V₁); adiabatic reversible PVᵍ = const, TVᵍ⁻¹ = const, W = (P₁V₁ − P₂V₂)/(γ−1).
• Specific heat of solids: Dulong–Petit C ≈ 3R ≈ 24.9 J mol⁻¹K⁻¹ at high T; it fails at low T, where Einstein's model gives an exponential fall and Debye's T³ law matches experiment. Electronic contribution in metals is linear in T.
• Calorimetry: method of mixtures, latent heat (L_f of ice = 334 kJ/kg, L_v of water at 100 °C = 2257 kJ/kg), water equivalent of the calorimeter, Newton's law of cooling correction.
• Thermal expansion: α (linear), β (area ≈ 2α), γ_V (volume ≈ 3α) for isotropic solids; water's anomalous expansion below 4 °C. Bimetallic strips, expansion joints in rails and bridges, invar for instruments.

Second law, entropy and the Carnot cycle

• Kelvin–Planck statement: no cyclic device can convert heat *entirely* into work from a single reservoir. Clausius statement: heat cannot flow from a colder to a hotter body with no other effect. The two are equivalent (violating one lets you violate the other).
• Carnot cycle: isothermal expansion at T_H → adiabatic expansion → isothermal compression at T_C → adiabatic compression. Fully reversible.
• Carnot's theorem: no engine between two reservoirs beats a reversible one, and all reversible engines have the same efficiency, independent of working substance — this defines the thermodynamic temperature scale through Q_H/Q_C = T_H/T_C.
• Clausius inequality: ∮ δQ/T ≤ 0, equality for reversible cycles. Hence dS = δQ_rev/T defines entropy as a state function, and ΔS_universe ≥ 0 (principle of increase of entropy) — the arrow of time.
• Entropy generation: heat flow across finite ΔT, friction, free expansion, mixing of different gases and throttling all create entropy. Available work lost = T₀ × ΔS_gen.
• Standard cycles examined: Otto (η = 1 − r^{1−γ}), Diesel, Brayton (η = 1 − r_p^{(1−γ)/γ}), Stirling and Ericsson (externally reversible, Carnot efficiency with regeneration), Rankine for steam plants.
• Refrigerator / heat pump: COP_ref = Q_C/W, COP_hp = Q_H/W = COP_ref + 1; Carnot limits are T_C/(T_H−T_C) and T_H/(T_H−T_C). Real split ACs sold in India carry a BEE star label, rated by ISEER (Indian Seasonal Energy Efficiency Ratio) under BEE's labelling programme — the thresholds are revised periodically, so quote the current schedule rather than a fixed number.
• Third law (Nernst): the entropy of a perfect crystal → 0 as T → 0, so absolute zero is unreachable in a finite number of steps; also C_P, C_V, α → 0 as T → 0.

Thermodynamic potentials, Maxwell relations and TdS equations

• Four potentials, each natural for a different constraint: U(S,V), H(S,P) = U+PV, Helmholtz F(T,V) = U−TS, Gibbs G(T,P) = H−TS.
• Spontaneity criteria: at constant T,V the system minimises F; at constant T,P it minimises G (this is the one chemists and materials engineers use). Maximum work from an isothermal process = −ΔF; maximum *non-expansion* work = −ΔG.
• The four Maxwell relations follow from exactness of these differentials and let you replace an unmeasurable entropy derivative by a measurable P–V–T derivative — the single most useful trick in the exam.
• TdS equations give C_P − C_V, the adiabatic/isothermal compressibility ratio, and the energy equation (∂U/∂V)_T = T(∂P/∂T)_V − P (= 0 for an ideal gas, = a/V² for van der Waals).
• Gibbs–Helmholtz, chemical potential μ = (∂G/∂n)_{T,P}, and the Gibbs–Duhem relation extend the scheme to open and multicomponent systems.

Real gases, liquefaction and the Joule–Thomson effect

• Deviations from ideality: finite molecular size and attraction. van der Waals: (P + a n²/V²)(V − nb) = nRT; *a* for attraction, *b* for excluded volume.
• Critical point: (∂P/∂V)_T = (∂²P/∂V²)_T = 0 gives V_c = 3nb, P_c = a/27b², T_c = 8a/27Rb, and the universal ratio RT_c/P_cV_c = 8/3 ≈ 2.67 (real gases ≈ 3.4). Law of corresponding states in reduced variables.
• Andrews' experiments on CO₂ showed a gas can be liquefied only below T_c; the Maxwell equal-area construction removes the unphysical van der Waals loop.
• Joule–Thomson (throttling) is isenthalpic; μ_JT = (∂T/∂P)_H. Cooling if μ_JT > 0 (below the inversion temperature), heating if negative. van der Waals gives T_i = 2a/Rb = 6.75 T_c. H₂ (T_i ≈ 200 K) and He (T_i ≈ 45 K) must be pre-cooled before throttling — the reason Linde's air process works for air but not directly for helium.
• Liquefaction techniques: Linde–Hampson (throttling + regenerative counter-flow heat exchanger), Claude (adds an expansion engine, doing external work), cascade cooling, and adiabatic demagnetisation of paramagnetic salts for the millikelvin range. Indian use: cryogenic upper stages at ISRO (LH₂/LOX), liquid-helium cryostats at TIFR/IUAC, LNG regasification terminals.

Phase transitions, equilibrium and thermal radiation

• Gibbs phase rule: F = C − P + 2. For pure water, the triple point (P = 3) has F = 0 — hence it is a fixed point.
• First-order transition: ΔG = 0 but S and V jump → latent heat (melting, boiling, sublimation). Second-order: no latent heat, discontinuity in C_P, compressibility or α (superconducting transition, He-I → He-II λ-point at 2.17 K, ferromagnetic Curie point); described by the Ehrenfest equations.
• Clausius–Clapeyron equation gives the slope of a phase boundary; it explains why the ice line has a *negative* slope (ice contracts on melting, so pressure lowers the melting point) and why water boils near 94 °C in Shimla.
• Blackbody radiation (thermodynamic treatment): Kirchhoff's law (good absorber = good emitter), Stefan–Boltzmann E = σT⁴, Wien's displacement law λ_max T = 2.898 × 10⁻³ m·K, radiation pressure P = u/3, and the adiabatic relation T³V = const for a photon gas. Planck's law resolves the ultraviolet catastrophe of the Rayleigh–Jeans form.
• Heat transfer modes: conduction (Fourier Q = −kA dT/dx), convection (Newton's law of cooling), radiation; thermal resistance in series/parallel for walls and lagging. Building thermal performance in India is governed by ECBC (commercial) and Eco-Niwas Samhita (residential) under BEE, which set U-values and envelope performance.

Key points

• Zeroth law — makes temperature meaningful and thermometers transitive.
• First law — dU = δQ − δW; U is a state function, Q and W are not.
• Second law — ΔS_universe ≥ 0; forbids 100% heat-to-work conversion in a cycle.
• Third law — S → 0 as T → 0; absolute zero unattainable in finite steps.
• Reversible vs quasi-static — reversible needs quasi-static *and* zero dissipation.
• Carnot efficiency — η = 1 − T_C/T_H, the upper bound for any two-reservoir engine, independent of substance.
• Entropy — dS = δQ_rev/T; for an irreversible path, compute ΔS along *any* reversible path with the same end states.
• Enthalpy H = U + PV — heat at constant pressure; conserved in throttling.
• F and G — minimised at constant (T,V) and constant (T,P) respectively; give maximum extractable work.
• Maxwell relations — convert entropy derivatives into measurable P–V–T derivatives.
• C_P − C_V = R for an ideal gas (per mole); generally C_P − C_V = TVβ²/κ_T.
• van der Waals constants — *a* attraction, *b* volume; T_c = 8a/27Rb.
• Joule–Thomson — isenthalpic; cools only below the inversion temperature T_i = 6.75 T_c.
• Clausius–Clapeyron — dP/dT = L/(T Δv); negative slope for the ice–water line.
• Phase rule — F = C − P + 2; triple point is invariant.
• Stefan–Boltzmann — E = σT⁴, σ = 5.67 × 10⁻⁸ W m⁻²K⁻⁴; net exchange ∝ (T⁴ − T_s⁴).
• Wien's law — λ_max T = 2.898 × 10⁻³ m·K; hotter bodies peak at shorter wavelengths.
• Kelvin since 2019 — defined by fixing k = 1.380649 × 10⁻²³ J/K; realised via ITS-90, traceable in India to CSIR-NPL.

Formulas and examples

1. Carnot engine and the real-plant gap. Steam at 540 °C (813 K) rejecting to a condenser at 40 °C (313 K).

• η_Carnot = 1 − 313/813 = 1 − 0.3850 = 61.5 %.
• A real Indian supercritical unit achieves about 40–42 % because of finite-ΔT heat transfer, friction, throttling and flue-gas loss — roughly two-thirds of the ideal.
• If it takes 1,000 MW of heat at 42 % efficiency, output = 420 MW and 580 MW is rejected to the condenser (hence the cooling tower).

2. Reversible adiabatic expansion. 2 mol of a diatomic ideal gas (γ = 1.4) at 300 K expands reversibly and adiabatically to twice its volume.

• T₂ = T₁(V₁/V₂)^{γ−1} = 300 × (0.5)^{0.4}. Since 2^{0.4} = 1.3195, T₂ = 300/1.3195 = 227.4 K.
• C_V = R/(γ−1) = 8.314/0.4 = 20.79 J mol⁻¹K⁻¹.
• W = −ΔU = nC_V(T₁ − T₂) = 2 × 20.79 × 72.6 = 3,019 J ≈ 3.02 kJ; Q = 0, ΔS = 0.

3. Entropy generated by mixing hot and cold water. 200 g at 80 °C mixed with 300 g at 20 °C, c = 4,186 J kg⁻¹K⁻¹, insulated.

• Final T = (200×80 + 300×20)/500 = 22,000/500 = 44 °C = 317.15 K.
• ΔS_hot = 0.2 × 4186 × ln(317.15/353.15) = 837.2 × (−0.1075) = −90.0 J/K.
• ΔS_cold = 0.3 × 4186 × ln(317.15/293.15) = 1255.8 × (0.0787) = +98.8 J/K.
• ΔS_total = +8.8 J/K > 0 — irreversible, as expected for heat flow across a finite ΔT.

4. Clausius–Clapeyron: boiling point at altitude. Water at 100 °C: L_v = 2,257 kJ/kg, v_g = 1.673 m³/kg, v_l = 0.001 m³/kg.

• dP/dT = 2.257 × 10⁶ / [373.15 × 1.672] = 2.257 × 10⁶ / 623.9 = 3,618 Pa/K ≈ 3.62 kPa/K.
• At Shimla, P ≈ 80 kPa, so ΔP = −21.3 kPa → ΔT ≈ −21,300/3,618 = −5.9 K, boiling point ≈ 94 °C (a linear estimate; the true value is close because the range is small). This is why pressure cookers are needed in the hills.

5. Radiative loss from a hot surface. A furnace wall patch of 0.05 m², emissivity 0.8, at 500 K in a 300 K room.

• P = 0.8 × 5.67×10⁻⁸ × 0.05 × (500⁴ − 300⁴) = 2.268×10⁻⁹ × (6.25×10¹⁰ − 8.1×10⁹) = 2.268×10⁻⁹ × 5.44×10¹⁰ = 123 W.
• λ_max at 500 K = 2.898×10⁻³/500 = 5.8 µm (infrared) — invisible, which is why a 500 K body looks dark but burns.

Where it is used

• Thermal power and renewables: NTPC's supercritical coal units, combined-cycle gas plants, and solar-thermal/CSP plants are designed on Rankine and Brayton cycle analysis with entropy-based loss accounting.
• Cryogenics and space: ISRO's LH₂/LOX cryogenic stages, LNG terminals at Dahej and Kochi, and liquid-helium systems at BARC/TIFR/IUAC use Linde–Claude liquefaction and Joule–Thomson cooling.
• HVAC and cold chain: refrigeration COP, psychrometry and ISEER-based BEE star ratings drive split-AC, chiller and vaccine cold-chain design (Voltas, Blue Star, Daikin India).
• Metrology and calibration: CSIR-NPL and NABL-accredited laboratories realise ITS-90 fixed points and calibrate Pt resistance thermometers, thermocouples and pyrometers for industry.
• Materials, metallurgy and semiconductors: Gibbs free energy and phase diagrams govern heat treatment at Tata Steel/SAIL, and thermal-budget and heat-sink design in electronics and ISM-funded fab projects.
• Research and teaching roles: JRF/SRF positions at BARC, DAE, DRDO, IMD (atmospheric thermodynamics — lapse rates, potential temperature) and university faculty posts test exactly this syllabus.

Interview questions

• Q: Why is the zeroth law a *law* and not a definition? A: It is an experimental fact that thermal equilibrium is transitive; only because of it can a single number — temperature — be assigned to a body and compared with any other.
• Q: Is entropy a state function even for an irreversible process? A: Yes. S depends only on the state, so ΔS is found by integrating δQ_rev/T along *any* reversible path joining the same two states, even if the actual path was irreversible.
• Q: Can entropy of a system decrease? A: Yes — a refrigerated body loses entropy — but the surroundings gain more, so the universe's entropy never decreases.
• Q: Why does free expansion of an ideal gas not change temperature, while a real gas cools? A: Q = W = 0 so ΔU = 0; for an ideal gas U depends only on T, but a real gas spends internal energy doing work against intermolecular attraction, so T falls.
• Q: Difference between free expansion and throttling? A: Free expansion is at constant U (isolated, rigid); throttling is at constant H (steady flow). Both are irreversible and generate entropy.
• Q: Why must hydrogen and helium be pre-cooled before throttling? A: Their inversion temperatures (≈200 K and ≈45 K) are below room temperature, so at 300 K throttling *heats* them; cooling needs μ_JT > 0.
• Q: Why does the ice line on the water phase diagram slope backwards? A: Ice is less dense than water, so Δv on melting is negative; by Clausius–Clapeyron dP/dT is negative, and pressure lowers the melting point.
• Q: Can a 100 % efficient engine exist? A: Only if the sink were at 0 K, which is unattainable; a cyclic device drawing from one reservoir and producing only work violates the Kelvin–Planck statement.
• Q: Why is C_P greater than C_V? A: At constant pressure the substance also does expansion work, so more heat is needed for the same temperature rise; generally C_P − C_V = TVβ²/κ_T, which is non-negative.
• Q: When would you use G rather than F? A: When the process runs at constant temperature and pressure — laboratory and plant conditions — where equilibrium corresponds to minimum G; F is for constant T and V.
• Q: What does the third law imply for specific heats? A: C_P and C_V, and the expansion coefficient, must all vanish as T → 0, which is why Dulong–Petit fails and the Debye T³ law takes over.
• Q: Two Carnot engines in series between T_H and T_C, with intermediate T_m — does the total efficiency depend on T_m? A: No. The combined efficiency is 1 − T_C/T_H regardless of T_m, since the intermediate heat is passed on fully.

Common mistakes

• Treating Q and W as state functions and writing ΔQ or ΔW instead of δQ, δW.
• Using η = 1 − T_C/T_H with temperatures in °C instead of kelvin.
• Applying PVᵍ = const to an *irreversible* adiabatic process (only ΔU = −W holds there).
• Assuming ΔS = 0 for every adiabatic process — true only if it is also reversible (isentropic).
• Using C_P − C_V = R per kg; it is per mole (per kg it is the specific gas constant R/M).
• Forgetting the latent-heat term (mL/T) when the mixture involves melting or boiling.
• Saying throttling is isothermal (it is isenthalpic) or that free expansion does work on the surroundings.
• Mixing units in Clausius–Clapeyron — L in J/kg demands v in m³/kg, giving Pa/K.

Related topics

Statistical mechanics basics · Kinetic theory of gases · Heat transfer: conduction, convection and radiation · Power cycles: Otto, Diesel, Rankine and Brayton · Blackbody radiation and quantum origins · Low-temperature physics and cryogenics