Thermodynamics

2.1 Core concepts

• Thermodynamics is the branch of physics dealing with heat, temperature and inter-conversion of heat and other forms of energy; it is a macroscopic science using few variables like P, V, T, m, composition and U, avoiding the molecular description (NCERT §11.1, p. 227).
• Historically, heat was thought to be a fluid called "caloric" flowing from hotter to colder bodies; Count Rumford's brass-cannon boring experiment (1798) showed heat produced depended on work done and not on the sharpness of the drill, establishing heat as a form of energy (NCERT §11.1, p. 226).
• A system is in thermodynamic equilibrium when its macroscopic variables do not change with time; the nature of the wall — adiabatic (insulating) or diathermic (conducting) — decides whether two systems can reach thermal equilibrium (NCERT §11.2, p. 227–228).
• Zeroth Law of Thermodynamics: two systems in thermal equilibrium with a third system separately are in thermal equilibrium with each other; the common physical quantity having the same value is called temperature T. R.H. Fowler formulated it in 1931, after the First and Second laws were already numbered (NCERT §11.3, p. 228).
• Internal energy U is the sum of kinetic and potential energies of the molecules in the frame in which the centre of mass of the system is at rest; it excludes the bulk kinetic energy of the system and depends only on the state, not on the path (NCERT §11.4, p. 229).
• Heat and work are the two modes of energy transfer that alter U; heat arises from a temperature difference, work is transferred by other means (e.g. moving a piston). Heat and work are NOT state variables — they are "energy in transit" (NCERT §11.4, p. 229–230).
• First Law of Thermodynamics: ΔQ = ΔU + ΔW where ΔQ is heat supplied to the system, ΔW is work done by the system on the surroundings, and ΔU is the change in internal energy. It is the energy-conservation principle applied to thermodynamic systems (NCERT §11.5, p. 230, Eq. 11.1).
• For a gas in a cylinder against constant pressure P, ΔW = P ΔV; hence ΔQ = ΔU + P ΔV (NCERT §11.5, p. 231, Eq. 11.3). Worked illustration: for 1 g of water vaporising at atmospheric pressure ΔQ = 2256 J, ΔW = 169.2 J, so ΔU = 2086.8 J — most heat goes into raising U.
• Specific heat capacity s = (1/m)(ΔQ/ΔT) and molar specific heat C = (1/µ)(ΔQ/ΔT); both depend on the substance and conditions of heating (NCERT §11.6, p. 231, Eqs. 11.5–11.6).
• For an ideal gas at constant volume Cv = (ΔU/ΔT) and at constant pressure Cp = (ΔU/ΔT) + R, leading to Mayer's relation Cp − Cv = R (NCERT §11.6, p. 232, Eq. 11.8). For a mole of solid using equipartition, U = 3RT giving C = 3R, which agrees with experiment at ordinary temperatures (carbon is an exception).
• 1 calorie is defined as the heat needed to raise 1 g of water from 14.5 °C to 15.5 °C; in SI, the specific heat capacity of water is 4186 J kg⁻¹ K⁻¹, and 1 cal = 4.186 J (NCERT §11.6, p. 232).
• State variables fall into two kinds — extensive (U, V, M scale with size) and intensive (P, T, ρ remain unchanged on subdivision). The equation of state connects state variables; for an ideal gas PV = µRT (NCERT §11.7, p. 232–233).
• A quasi-static process is infinitely slow so the system remains in equilibrium with its surroundings at every stage; the pressure and temperature differ from those of the surroundings only infinitesimally (NCERT §11.8.1, p. 233–234).
• Isothermal process: T fixed, PV = constant (Boyle's law); work done W = µRT ln(V₂/V₁); since ΔU = 0 for an ideal gas, Q = W (NCERT §11.8.2, p. 234, Eq. 11.12). In isothermal expansion the gas absorbs heat and does work; in compression the reverse.
• Adiabatic process: Q = 0, PV^γ = constant where γ = Cp/Cv; work done W = (P₁V₁ − P₂V₂)/(γ − 1) = µR(T₁ − T₂)/(γ − 1); positive work by the gas lowers its temperature (NCERT §11.8.3, p. 234–235, Eqs. 11.13–11.16).
• Isochoric process: V constant ⇒ W = 0, so all heat absorbed goes to ΔU (and hence temperature change) governed by Cv (NCERT §11.8.4, p. 235).
• Isobaric process: P fixed ⇒ W = P(V₂ − V₁) = µR(T₂ − T₁); heat absorbed goes partly into ΔU and partly into work, governed by Cp (NCERT §11.8.5, p. 235, Eq. 11.17).
• Cyclic process: the system returns to its initial state; ΔU = 0, so total Q = total W (NCERT §11.8.6, p. 235).
• Second Law of Thermodynamics — Kelvin-Planck statement: no process is possible whose sole result is the absorption of heat from a reservoir and the complete conversion of that heat into work. Clausius statement: no process is possible whose sole result is the transfer of heat from a colder to a hotter body. The two statements are completely equivalent (NCERT §11.9, p. 236).
• A reversible process can be turned back so that both the system and surroundings return to their original states with no other change; it must be quasi-static AND free of dissipative effects like friction and viscosity. Spontaneous natural processes are irreversible (NCERT §11.10, p. 236–237).
• Carnot engine: a reversible heat engine operating between a hot reservoir at T₁ and a cold reservoir at T₂; its cycle has four steps — isothermal expansion at T₁, adiabatic expansion to T₂, isothermal compression at T₂, adiabatic compression back to T₁ (NCERT §11.11, p. 237–238).
• Carnot efficiency: η = 1 − Q₂/Q₁ = 1 − T₂/T₁; Carnot's theorem proves (a) no engine working between two given temperatures can be more efficient than the Carnot engine, and (b) the Carnot efficiency is independent of the working substance (NCERT §11.11, p. 238–239, Eq. 11.27). The universal relation Q₁/Q₂ = T₁/T₂ defines a thermodynamic temperature scale (Eq. 11.28).

2.2 Definitions to memorise

2.3 Diagrams / processes to remember

• Fig. 11.1, p. 227: Two gases A and B separated first by an adiabatic wall (no thermal equilibrium attained) and then by a diathermic wall (equilibrium attained with new P, V values).
• Fig. 11.2, p. 228: Schematic illustrating the Zeroth Law — A and B each in contact with C via conducting walls, then connected directly; they remain in equilibrium.
• Fig. 11.3, p. 229: Internal energy U of a gas as sum of kinetic + potential energies of molecules (translational, rotational, vibrational), and exclusion of bulk motion.
• Fig. 11.4, p. 229: Heat (from temperature difference) and work (via a piston with weight) as two distinct modes of energy transfer.
• Fig. 11.5, p. 232: Variation of specific heat capacity of water with temperature in the range 0–100 °C, motivating the precise calorie definition.
• Fig. 11.6, p. 233: Non-equilibrium states — free expansion of a gas and an explosive chemical reaction — that cannot be described by state variables.
• Fig. 11.7, p. 234: Quasi-static process — surrounding reservoir and external pressure differ from system values only infinitesimally.
• Fig. 11.8, p. 235: P–V curves comparing isothermal and adiabatic processes; the adiabatic curve is steeper because γ > 1.
• Fig. 11.9, p. 237: Carnot cycle on a P–V diagram showing the four steps — isothermal expansion 1→2, adiabatic expansion 2→3, isothermal compression 3→4, adiabatic compression 4→1.
• Fig. 11.10, p. 239: Irreversible engine I coupled to a reversible refrigerator R — used to prove Carnot's theorem by contradicting the Kelvin-Planck statement.

2.4 Common confusions / NTA trap points

• Heat vs internal energy: a body "contains" internal energy, not heat. Heat is energy in transit during a process, so statements like "a gas in a given state has a certain amount of heat" are meaningless (NCERT §11.4, p. 230).
• Sign convention in First Law: in NCERT, ΔW is the work done BY the system on the surroundings, so ΔQ = ΔU + ΔW; isothermal expansion gives W > 0 and isothermal compression gives W < 0 (NCERT §11.5/§11.8, p. 230, 234). Mixing this up with the "work done on the system" convention is a classic trap.
• Adiabatic ≠ isothermal: in adiabatic Q = 0 (the system is insulated) and PV^γ = const; in isothermal T = const and PV = const. Don't conflate the two equations or the corresponding work formulas.
• For an isothermal process of an ideal gas ΔU = 0 (because U depends only on T), so Q = W; but ΔU = 0 does NOT imply Q = 0 (NCERT §11.8.2, p. 234).
• Carnot efficiency uses absolute (Kelvin) temperatures. Plugging in Celsius values is a guaranteed wrong answer (NCERT §11.11, Eq. 11.27, p. 238).
• The Carnot efficiency is independent of the working substance — NTA likes to put "depends on the gas used" as a distractor (NCERT §11.11, p. 239).
• Zeroth Law was formulated AFTER the First and Second Laws (in 1931, by R.H. Fowler) — but logically it precedes them (NCERT §11.3, p. 228).