Thermodynamics— Notes, Formulas & Revision

Isothermal process: temperature stays CONSTANT throughout (ΔT = 0).

For an ideal gas: ΔU = 0 (U depends only on T).

First law: Q = W. All heat in becomes work out (or vice versa).

On a PV diagram: HYPERBOLA (PV = constant, Boyle's law).

Work done by gas: W = nRT·ln(V_f/V_i) = nRT·ln(P_i/P_f).

Slow expansion in contact with thermal reservoir at constant T ⇒ approximately isothermal.

Slope on PV diagram: less steep than adiabatic (because P drops less for same V increase).

Used in: refrigeration cycles, theoretical engine analysis (Carnot cycle).

Adiabatic process: NO heat exchange with surroundings (Q = 0).

Achieved by: thermal insulation OR fast process (no time for heat to flow).

First law: ΔU = −W. If gas expands (W > 0), T DROPS. Compression heats it up.

For ideal gas: PV^γ = constant, where γ = C_p/C_v.

Equivalently: TV^(γ−1) = const, or T·P^((1−γ)/γ) = const.

Adiabatic curve on PV diagram is STEEPER than isotherm (P drops faster for same V increase).

Examples: rapid compression in a diesel engine (no time for heat loss), atmospheric air parcels rising.

Adiabatic expansion is how gas in clouds cools and condenses (rising air, lower P, lower T).

Isochoric (or isovolumetric) process: volume stays CONSTANT (ΔV = 0).

No work done: W = P·ΔV = 0.

First law: Q = ΔU. All heat goes into internal energy.

Heat absorbed: Q = nC_v·ΔT.

On PV diagram: VERTICAL line.

Pressure-temperature link: P/T = constant (Gay-Lussac's law).

Examples: heating gas in a sealed rigid container, pressure cooker partially.

Used in Otto cycle (gasoline engine) — constant-volume heat addition (idealized).

Isobaric process: pressure stays CONSTANT (ΔP = 0).

On PV diagram: HORIZONTAL line.

Work done by gas: W = P·ΔV — simple area calculation.

Heat absorbed: Q = nC_p·ΔT, where C_p = (f/2 + 1)R = molar specific heat at constant P.

Change in internal energy: ΔU = nC_v·ΔT (always, for ideal gas).

Volume and temperature linked: V/T = constant (Charles's law).

Examples: water boiling in an open pot (P_atm constant), gas expansion against atmospheric pressure.

C_p > C_v because some heat goes into expansion work (Mayer's relation: C_p − C_v = R).

First law of thermodynamics: ΔU = Q − W. Energy conservation applied to a thermodynamic system.

Q = heat ADDED to the system (positive when heat enters).

W = work done BY the system on surroundings (positive when gas expands).

ΔU = change in internal energy.

Alternative sign convention: ΔU = Q + W where W = work done ON the system. Use one convention consistently.

For an ideal gas: ΔU = nC_v·ΔT, regardless of process.

Different processes redistribute Q and W differently — but ΔU = nC_v·ΔT always (for ideal gas).

Cannot create or destroy energy — only transform between heat, work, and internal energy.

Work done by gas in a process: W = ∫P·dV. Always equals the AREA UNDER the curve on a PV diagram.

Sign convention: V increases (expansion) ⇒ W > 0 (gas does work). V decreases (compression) ⇒ W < 0 (work done on gas).

Different processes give different work for same initial-final V — work is a PATH FUNCTION.

Isobaric: W = PΔV (rectangle area).

Isothermal: W = nRT·ln(V_f/V_i) (under hyperbola).

Adiabatic: W = (P_iV_i − P_fV_f)/(γ−1).

Isochoric: W = 0 (ΔV = 0).

Cycle: W_net = enclosed area.

Heat engine: device that converts HEAT into WORK over a cycle.

Takes heat Q_h from a hot reservoir, dumps heat Q_c to a cold reservoir, produces work W = Q_h − Q_c.

Efficiency: η = W/Q_h = 1 − Q_c/Q_h.

Real engines: gasoline (Otto cycle), diesel (Diesel cycle), gas turbine (Brayton), steam (Rankine).

Maximum possible efficiency: Carnot's η = 1 − T_c/T_h.

Heat engines are CYCLIC: return to initial state, ΔU = 0 over cycle, Q_net = W_net.

On PV diagram: clockwise closed loop. Enclosed area = net work output.

Kelvin-Planck statement (2nd law): No engine can convert 100% of heat to work in a cycle.

Refrigerator: device that EXTRACTS heat from a cold reservoir and DUMPS it to a hot reservoir, using work INPUT.

Reverse of heat engine — runs the cycle counter-clockwise.

Work IN, heat OUT to atmosphere. Q_h = W + Q_c (energy conservation).

Coefficient of Performance: COP = Q_c/W = heat extracted per unit work input.

Carnot refrigerator COP: COP_max = T_c/(T_h − T_c). Can exceed 1.

Heat pump: same machine but operated for HEATING. COP_heat = Q_h/W = 1 + COP_ref.

Real refrigerators: COP ~ 2-4 for household, 3-5 for AC. Heat pumps: 3-5 for heating.

Clausius statement (2nd law): Cannot transfer heat from cold to hot without work input.

Carnot engine: theoretical ideal engine — most efficient possible between two reservoirs at temperatures T_hot and T_cold.

Cycle: isothermal expansion at T_hot → adiabatic expansion → isothermal compression at T_cold → adiabatic compression.

Carnot efficiency: η_Carnot = 1 − T_cold/T_hot (temperatures in KELVIN!).

All reversible engines between same two reservoirs have SAME efficiency = η_Carnot.

No real engine can exceed Carnot efficiency (2nd law of thermodynamics).

Engine output = work; engine sucks heat Q_h from hot, dumps Q_c to cold, produces W = Q_h − Q_c.

Carnot engine is REVERSIBLE — can run backward as a refrigerator with max COP.

Higher T_hot or lower T_cold ⇒ higher efficiency.

Ideal (Carnot) engine: reversible, infinitely slow, max efficiency.

Real engines: irreversible, finite speed, lower efficiency.

Sources of inefficiency: friction, turbulence, heat leakage, fast non-quasi-static processes, incomplete combustion.

Real engine efficiency = η_real < η_Carnot = 1 − T_c/T_h.

Carnot's theorem: NO engine can exceed Carnot efficiency; only reversible engines reach it.

Practical efficiencies: gasoline ~25-35%, diesel ~35-45%, gas turbine ~30-40%, combined cycle ~50-60%.

Higher T_hot (more efficient) requires advanced materials — engineering trade-off.

Engine output power = efficiency × heat input rate.

PV diagram: pressure (P) on y-axis vs volume (V) on x-axis. Represents thermodynamic processes graphically.

Each point: a specific state with P, V, T (and n) defined.

Each curve: a process. Slope and shape tell what kind of process.

Isothermal (PV = const): hyperbola.

Adiabatic (PV^γ = const): steeper than isothermal.

Isobaric: horizontal line.

Isochoric: vertical line.

Area under curve = work done by gas: W = ∫P·dV.

Cyclic process: closed loop on PV. Enclosed area = net work over cycle.

TS diagram: temperature (T) vs entropy (S). Useful complement to PV diagram.

Each point: a thermodynamic state. Each curve: a process.

Isothermal: HORIZONTAL line (T = const).

Adiabatic (reversible): VERTICAL line (S = const, isentropic).

Area under T-S curve = heat absorbed: Q = ∫T·dS.

Cyclic process: closed loop. Enclosed area = net heat absorbed = net work output (for an engine).

Carnot cycle on TS: PERFECT RECTANGLE — two isotherms (horizontal) + two adiabats (vertical).

TS diagram makes 2nd law of thermodynamics visually direct: entropy can only stay constant (reversible) or increase.

Cyclic process: gas returns to its initial state after a series of processes.

Net change in any state function over a cycle is ZERO: ΔU_cycle = 0, ΔT_cycle = 0, ΔP_cycle = 0, ΔV_cycle = 0.

First law for a cycle: Q_net = W_net. Net heat input = net work output.

On PV diagram: closed curve. Area enclosed = net work done by gas.

Clockwise cycle on PV: net work > 0 (heat engine).

Counter-clockwise: net work < 0 (refrigerator/heat pump).

Examples: Carnot, Otto (gasoline), Diesel, Brayton, Stirling cycles — all engine/refrigerator cycles.

Efficiency of a heat engine cycle: η = W_net/Q_hot — fraction of input heat converted to useful work.

Internal energy U: total energy of all microscopic motions and interactions inside a system.

For an ideal gas, U depends ONLY on temperature: U = (f/2)·n·R·T, where f = degrees of freedom.

Monatomic gas (He, Ar): f = 3, U = (3/2)nRT. Diatomic gas (N₂, O₂): f = 5, U = (5/2)nRT (translation + rotation).

Polyatomic: more degrees of freedom (translation + rotation + vibration).

ΔU depends only on initial and final STATES — not on path (state function).

Real gases also have potential energy of intermolecular forces; U depends on V slightly.

First law of thermodynamics: ΔU = Q − W. U changes when heat enters or work is done by gas.

Equipartition theorem: each degree of freedom contributes ½k_BT per molecule (½RT per mole).