Thermodynamics

🎯 Key Points

• ΔU = Q − W (W is work done BY the system); Isochoric: W=0; Isobaric: W=PΔV; Isothermal: ΔU=0; Adiabatic: Q=0, PVᵞ=const
• Carnot efficiency η = 1 − T_C/T_H (Kelvin); no real engine can exceed this between the same two temperatures
• C_p − C_v = R (Mayer's relation); γ=C_p/C_v: monoatomic 5/3, diatomic 7/5
• Second Law: heat never flows spontaneously cold→hot; no 100% efficient engine is possible; entropy of an isolated system never decreases
• On a P-V diagram: work = area under curve; in a closed cycle, net work = area enclosed by the loop, and ΔU=0 over the full cycle

Zeroth Law

• If A and B are both in thermal equilibrium with C, then A and B are in thermal equilibrium with each other (defines temperature)

First Law of Thermodynamics

• ΔU = Q - W (internal energy change = heat added - work done BY system)
• Isochoric (constant V): W = 0, ΔU = Q
• Isobaric (constant P): W = PΔV
• Isothermal (constant T): ΔU = 0, Q = W = nRT·ln(V₂/V₁)
• Adiabatic (Q = 0): PV^γ = constant; W = -ΔU

Second Law of Thermodynamics

• Heat cannot flow spontaneously from cold to hot
• No engine can be 100% efficient
• Entropy of an isolated system never decreases

Carnot Engine

• Most efficient heat engine operating between T_H and T_C
• Efficiency η = 1 - T_C/T_H (temperatures in Kelvin)
• COP of refrigerator = T_C/(T_H - T_C)

The Carnot cycle: two isothermals (Tₕ, T₃) and two adiabatics; efficiency η = 1 − T₃/Tₕ. Image: Cristian Quinzacara, CC BY-SA 4.0, via Wikimedia Commons.

Kinetic Theory

• Pressure: P = ⅓ρ·v_rms²
• v_rms = √(3RT/M); v_avg = √(8RT/πM); v_mp = √(2RT/M)
• Degrees of freedom: monoatomic=3, diatomic=5, polyatomic=6
• Mean free path: λ = 1/(√2·π·d²·n)

Work Done in Different Processes (PV Diagram)

• On a P-V diagram, work done by the gas equals the area under the curve between initial and final volumes
• Isothermal curve is less steep than adiabatic curve at any common point (adiabatic process changes temperature, so PV^gamma falls off faster than PV = constant)
• In a cyclic process, net work done = area enclosed by the closed loop on the P-V diagram; ΔU = 0 over a full cycle since the system returns to its initial state

Specific Heat Capacities of Gases

• C_p - C_v = R (Mayer's relation, per mole)
• γ = C_p/C_v; monoatomic gas γ = 5/3, diatomic gas γ = 7/5
• Molar specific heat at constant volume: C_v = (f/2)R, where f is degrees of freedom

Heat Engines and Refrigerators

• Heat engine: converts heat into work in a cyclic process; efficiency η = W/Q_H = 1 - Q_C/Q_H
• Refrigerator/heat pump: works in reverse, extracting heat from a cold body and rejecting it to a hot body using external work input
• Coefficient of performance: COP = Q_C/W = Q_C/(Q_H - Q_C)
• Carnot's theorem: no engine working between two given temperatures can be more efficient than a reversible (Carnot) engine

Heat, Internal Energy and Work

• Heat (Q): energy transferred between a system and its surroundings due only to a temperature difference; it is energy in transit, not a property stored in a body
• Internal energy (U): the total energy (kinetic + potential) of all molecules of the system; it is a state variable (depends only on the current state) and for an ideal gas depends on temperature alone
• Work (W): energy transferred when the system pushes its boundary through a displacement (W = PΔV for expansion); like heat, work is a path function, not a state function
• Both Q and W depend on the path taken between two states, but their difference Q − W = ΔU is path-independent, which is the essence of the first law

Thermodynamic State Variables and Equation of State

• State variables (P, V, T, U, entropy) describe the equilibrium state of a system and are independent of how that state was reached
• Extensive variables (V, U, mass, total entropy) depend on system size; intensive variables (P, T, density) do not
• Equation of state: a relation connecting the state variables; for an ideal gas it is PV = nRT, reducing the independent variables to two
• A system is in thermodynamic equilibrium only when mechanical, thermal, and chemical equilibrium all hold simultaneously

Quasi-static, Reversible and Irreversible Processes

• Quasi-static process: an idealised process carried out infinitely slowly so the system stays in equilibrium (uniform P and T) at every instant
• Reversible process: can be exactly retraced so both system and surroundings return to their initial states with no net change; requires it to be quasi-static and free of dissipative effects (friction, viscosity, turbulence)
• Irreversible process: all real, spontaneous processes (sudden expansion, heat flow across a finite temperature difference, friction) that cannot be reversed without leaving a change in the surroundings
• The Carnot engine uses only reversible steps, which is precisely why it sets the maximum possible efficiency between two temperatures

🚀 JEE Advanced Edge

Multi-process cycles: For cycles combining isothermal, isobaric, isochoric, and adiabatic legs (common in JEE), calculate Q, W, ΔU separately for EACH leg using the correct process formula, then sum: total ΔU over the full cycle = 0 (state function returns to start), but total Q and total W are generally non-zero and equal to each other (W=Q for a full cycle, from the first law with ΔU=0).

Polytropic process: A general process PVⁿ=constant covers all the standard cases as special values of n: n=0 is isobaric, n=1 is isothermal, n=γ is adiabatic, n→∞ is isochoric — recognising which special case a "PV^n=const" problem reduces to instantly tells you which formula set to use.

Worked problem: One mole of an ideal monoatomic gas expands adiabatically, and its temperature drops from 400 K to 300 K. Find the work done by the gas (Cv=3R/2). Approach: For adiabatic, W=−ΔU=−nCvΔT=−1×(3R/2)×(300−400)=−(3R/2)×(−100)=150R ≈ 1247 J (gas does positive work while cooling, consistent with adiabatic expansion).