1. First Law of Thermodynamics & Sign Conventions

First Law (Conservation of Energy): ΔQ = ΔU + ΔW (or dQ = dU + dW). State function: Internal energy U depends solely on temperature: ΔU = n Cv ΔT. Sign Conventions: Heat absorbed by system: ΔQ > 0 (positive); Heat released: ΔQ < 0 (negative). Work done BY system (expansion): ΔW > 0 (positive); Work done ON system (compression): ΔW < 0 (negative). In cyclic process: Initial and final states are identical, so ΔU = 0 and net heat absorbed equals net work done: Σ Q = Σ W.

2. Work Done in Thermodynamic Processes

General work formula: W = ∫ P dV. (1) Isobaric Process (constant P): W = P(V2 - V1) = n R(T2 - T1). (2) Isochoric Process (constant V): dV = 0 -> W = 0, so ΔQ = ΔU = n Cv ΔT. (3) Isothermal Process (constant T, ideal gas PV = constant): ΔU = 0, so Q = W = n R T ln(V2 / V1) = n R T ln(P1 / P2). (4) Adiabatic Process (no heat exchange, ΔQ = 0, PV^γ = constant): ΔU = -W; Work done: W = (P1 V1 - P2 V2) / (γ - 1) = [n R (T1 - T2)] / (γ - 1). Slope of adiabatic curve on P-V diagram is γ times steeper than isothermal curve: (dP/dV)_adiabatic = γ (dP/dV)_isothermal.

3. Molar Heat Capacities & Mayer's Relation

Mayer's Relation for ideal gas: Cp - Cv = R. Specific heats ratio (adiabatic index): γ = Cp / Cv = 1 + (2 / f), where f is degrees of freedom. (1) Monoatomic Gas (He, Ne, Ar): f = 3 (translational), Cv = (3/2)R, Cp = (5/2)R, γ = 5/3 ≈ 1.67. (2) Diatomic Gas at room temperature (O2, N2, H2): f = 5 (3 translational + 2 rotational), Cv = (5/2)R, Cp = (7/2)R, γ = 7/5 = 1.40. (3) Non-linear Polyatomic Gas (H2O, NH3): f = 6 (3 translational + 3 rotational), Cv = 3R, Cp = 4R, γ = 4/3 ≈ 1.33.

4. Second Law & Carnot Heat Engine Efficiency

Carnot cycle is an ideal reversible cycle operating between heat source at absolute temperature T1 and sink at T2. Efficiency: η = W / Q1 = (Q1 - Q2) / Q1 = 1 - (Q2 / Q1) = 1 - (T2 / T1). Maximum theoretical efficiency depends solely on reservoir temperatures in Kelvin. For an ideal refrigerator (reverse Carnot cycle): Coefficient of Performance (COP, β): β = Q2 / W = Q2 / (Q1 - Q2) = T2 / (T1 - T2). Relation between engine efficiency and refrigerator COP: β = (1 - η) / η.

5. Heat Transfer: Conduction, Radiation & Cooling

Thermal Conduction through slab of area A, thickness l, thermal conductivity k: Heat current H = dQ/dt = [k A (T1 - T2)] / l. Thermal resistance: R_th = l / (k A). Stefan-Boltzmann Law for blackbody radiation: Total radiant power E = σ A T^4, where σ ≈ 5.67 × 10^-8 W/(m^2·K^4). For non-blackbody with emissivity e: E = e σ A T^4. Net loss in environment of temperature T0: E_net = e σ A (T^4 - T0^4). Wien's Displacement Law: Peak wavelength λ_max · T = b = 2.898 × 10^-3 m·K. Newton's Law of Cooling (for small ΔT = T - T0): dT/dt = -K(T - T0).

6. Teacher's Exam Hall Hack: Temperature Scale Unit Traps

In all thermodynamic formulas involving temperature ratios or products—including Carnot efficiency η = 1 - T2/T1, ideal gas law PV = nRT, Stefan's law E ∝ T^4, and Wien's law λ_max T = b—temperature MUST be converted to absolute Kelvin: T(K) = T(°C) + 273.15. The ONLY time Celsius can be used directly is when dealing with temperature differences (ΔT = T2 - T1), because 1 °C change equals exactly 1 K change.