Chapter 12 — Thermodynamics

Fig 1 — Educational Diagram: Basic concepts and definitions

Introduction: Thermodynamics is the study of energy, its transformations, and the macroscopic properties of matter that describe those transformations. The subject begins with basic concepts that define what is being studied and how we measure changes.

System, Surroundings and Boundary: A system is the specific portion of the universe chosen for study (e.g. gas in a cylinder). Everything else is the surroundings. The boundary separates system and surroundings and may be fixed or movable, real or imaginary.

Types of Systems:

• Isolated system: No exchange of mass or energy (heat/work) with surroundings (ideal thermos bottle).
• Closed system: No mass exchange but energy (heat/work) can cross boundary (piston-cylinder with fixed mass).
• Open system: Both mass and energy can cross boundary (flowing fluid in a pipe).

Macroscopic State and State Variables: The macroscopic state of a system is described by measurable state variables (pressure P, volume V, temperature T, internal energy U, number of moles n, etc.). A set of state variables that uniquely specifies the state is called the state. An equation of state relates state variables (e.g. ideal gas law PV = nRT).

Extensive and Intensive Variables:

• Extensive: depend on system size (mass, volume, internal energy, entropy).
• Intensive: independent of size (temperature, pressure, density).

Thermodynamic Equilibrium: A system is in thermodynamic equilibrium if it has mechanical, thermal and chemical equilibrium simultaneously (no net macroscopic change). In equilibrium, state variables are uniform and unchanging in time. The Zeroth Law of Thermodynamics formalizes thermal equilibrium: if A is in thermal equilibrium with B, and B with C, then A is with C; this allows the definition of temperature.

Process and Path: A process is any change of state. A process may be described by the path traced in state-space (e.g. P-V plane). If the process proceeds infinitely slowly through a succession of equilibrium states it is called quasi-static. Reversible processes are ideal quasi-static processes that can be reversed without leaving net changes in system and surroundings; most real processes are irreversible.

State Functions and Path Functions:

• State functions (U, P, V, T, entropy S) depend only on the state, not on how the state was reached.
• Path functions (heat Q and work W) depend on the process path; their values between two states depend on the route taken.

Heat and Work: Heat is energy transfer due to temperature difference. Work is energy transfer when a force acts through a distance (mechanical work, electrical work, etc.). For a simple compressible system, mechanical work done by the system in a quasi-static process is W = ∫ P dV (area under P–V curve).

Internal Energy: Internal energy U is the total microscopic energy (kinetic + potential) of the particles in the system. For an ideal gas, U depends only on temperature (U = nCvT for ideal gas, where Cv is molar specific heat at constant volume).

Heat Capacity and Specific Heat: Heat capacity C is the amount of heat required to change the temperature by 1 K. Specific heat c (per unit mass) and molar heat capacities (Cv, Cp) are commonly used. For small temperature change dQ = C dT (for systems held at constant parameter corresponding to that C).

Common Elementary Processes (defined by which variable is held fixed): isothermal (T constant), isochoric/isochoric (V constant), isobaric (P constant), adiabatic (no heat exchange Q = 0). Each has characteristic relations (e.g., isothermal ideal gas: PV = constant; reversible adiabatic ideal gas: PV^γ = constant, where γ = Cp/Cv).

Why these definitions matter: These basic definitions let us classify problems, write appropriate relations (equations of state, energy balances), draw process paths on P–V or T–S diagrams, and apply the laws of thermodynamics (e.g. first and second laws) correctly.

Fig 2.1 — Educational Diagram: Newton's Laws of Motion & Free Body Diagrams

Zeroth law (statement): If two systems A and B are each in thermal equilibrium with a third system C, then A and B are in thermal equilibrium with each other. In symbols: if A ~ C and B ~ C (thermal equilibrium), then A ~ B.

Thermal equilibrium means no net exchange of heat between systems when they are brought into thermal contact. Thermal equilibrium is a transitive relation — the content of the Zeroth law — and this transitivity allows the concept of temperature to be a well-defined physical property of a system.

Temperature as a physical property: The Zeroth law implies there exists a scalar quantity (temperature) that is the same for all systems in mutual thermal equilibrium. Temperature characterizes the hotness/coldness of a body and predicts the direction of spontaneous heat flow (from higher temperature to lower temperature).

Thermometers and temperature scales: A thermometer is a device that reaches thermal equilibrium with the system whose temperature is to be measured; the thermometer’s reading (after calibration) gives the temperature of the system. Empirical scales (Celsius, Fahrenheit) are based on fixed reference points; the thermodynamic (absolute) scale is the Kelvin scale, where 0 K is absolute zero. Relation: T(K) = t(°C) + 273.15.

Ideal gas thermometer and absolute temperature: For an ideal gas at fixed volume, pressure is proportional to absolute temperature (p ∝ T). Using a reference point (e.g., triple point of water), one defines the Kelvin scale so that the gas law gives an absolute measure of temperature independent of substance used in the thermometer.

Microscopic meaning: In kinetic theory for an ideal monoatomic gas, the average translational kinetic energy per particle is proportional to absolute temperature: <KE> = (3/2) k_B T, where k_B is Boltzmann’s constant. This provides a microscopic interpretation of temperature as a measure of average molecular kinetic energy.

Importance: The Zeroth law underpins all temperature measurement and thermometry. Without it, a single consistent scale of temperature that applies universally would not be possible.

Fig 3 — Educational Diagram: Heat and internal energy

Definitions

Heat (Q) is energy in transfer between a system and its surroundings due to a temperature difference. Heat is a process quantity (path function) — it is not a property of the system.

Internal energy (U) is the total microscopic energy of a system: the sum of kinetic and potential energies of all its molecules (translational, rotational, vibrational kinetic energies and intermolecular potential energies). Internal energy is a state function: its value depends only on the state (e.g., temperature, pressure, volume, phase) of the system.

Microscopic picture

For a collection of N particles, internal energy U = sum of molecular kinetic energies + sum of potential energies due to interactions. For an ideal monatomic gas (no intermolecular potential), U is entirely kinetic and depends only on temperature: U = (3/2) nRT (or (3/2) NkT).

Key conceptual differences

• Heat is energy in transit caused by ΔT; internal energy is energy contained in the system.
• Heat (Q) and work (W) are process quantities; internal energy (U) is a state quantity.
• Adding heat to a system can change its internal energy and/or do work on the surroundings.

First law of thermodynamics (energy conservation)

ΔU = Q − W

Here ΔU is the change in internal energy, Q is heat added to the system, and W is work done by the system on the surroundings (physics sign convention). This summarizes that heat supplied may increase internal energy and/or be converted to work.

Temperature vs internal energy

For ideal gases: internal energy depends only on temperature. For real substances (esp. during phase changes or when intermolecular forces are important) U depends on temperature and other variables (e.g., volume, phase).

Heat capacity and latent heat

• Specific heat capacity (c): heat required per unit mass to raise temperature by 1 K. Q = m c ΔT
• Molar heat capacity (Cv, Cp): heat per mole per K at constant volume or pressure. For ideal gas at constant volume: ΔU = n Cv ΔT.
• Latent heat (L): heat required per unit mass for a phase change at constant temperature. Q = m L (during phase change temperature remains constant).

Important remarks and sign conventions

• Heat supplied to the system: Q > 0. Heat removed: Q < 0.
• Work done by the system: W > 0 (so ΔU = Q − W). If using the engineers’/chemists’ convention (W = work done on the system) sign changes accordingly — be consistent.
• Because U is a state function, ΔU between two states is independent of path; Q and W individually depend on the path.

Examples of processes

• Isothermal (ΔT = 0 for ideal gas): ΔU = 0, so Q = W (heat supplied equals work done by gas).
• Adiabatic: Q = 0, so ΔU = −W (work done by gas reduces its internal energy and temperature).
• At constant volume: W = 0, so ΔU = Q (all heat changes internal energy).

Units

SI unit of heat and internal energy: joule (J). Specific heat: J kg−1 K−1. Molar heat capacity: J mol−1 K−1. Latent heat: J kg−1.

Fig 4 — Educational Diagram: Work in thermodynamics

What is work in thermodynamics?
In thermodynamics, work refers to the energy transferred when a macroscopic force acts through a distance due to a change in the system's macroscopic variables (most commonly volume). For a gas in a piston, the mechanical (PV) work done by the gas during a small quasi‑static change of volume dV against the pressure P is dW = P dV. The total work in a process from initial volume Vi to final volume Vf is W = ∫(Vi->Vf) P dV.

Sign convention
Common physics sign convention: W = ∫ P dV is the work done BY the system on the surroundings. If the system expands (dV > 0) W > 0. Work done ON the system is negative of this. Always state the sign convention before using formulas.

Reversible versus irreversible processes
For a reversible (quasi‑static) process the internal gas pressure equals the external pressure so P in the integral is the gas pressure. For an irreversible expansion against a constant external pressure Pext, the work is W = ∫ Pext dV = Pext (Vf - Vi). Reversible work is the maximum work obtainable for given end states.

Path dependence
Work is a path function (not a state function): the value of W depends on the process path connecting the same end states. This contrasts with internal energy and state functions that depend only on state variables.

Special cases for ideal gases

• Isobaric (constant pressure): W = P (Vf - Vi).
• Isothermal (T constant) reversible for ideal gas: W = nRT ln(Vf/Vi). Here internal energy change ΔU = 0, so heat Q = W.
• Adiabatic (Q = 0), reversible: for ideal gas with γ = CP/CV and PV^γ = constant, W = (Pi Vi - Pf Vf)/(γ - 1). This follows from integrating P = constant × V^{-γ}.
• Free (Joule) expansion into vacuum: Pext ≈ 0 so W ≈ 0 (no work done).

Graphical meaning
On a P–V diagram the work done by the gas during a process is the area under the process curve between Vi and Vf (with pressure on the vertical axis and volume on the horizontal axis). For a cyclic process the net work equals the area enclosed by the cycle (clockwise area positive for expansion-first cycles).

Units
SI unit of work: joule (J). Note 1 Pa·m^3 = 1 J.

Connections to first law
First law: ΔU = Q - W (with W the work done by the system). Thus W appears as energy leaving the system by mechanical work; sign conventions must be consistent.

Fig 5 — Educational Diagram: Ideal gas and kinetic interpretation

What is an ideal gas?
An ideal gas is a model in which a large number of identical point-like particles move randomly and obey Newton's laws. Intermolecular forces are neglected except during perfectly elastic collisions. The ideal gas law relates macroscopic variables: pressure (P), volume (V) and temperature (T) by PV = nRT.

Basic assumptions of kinetic theory (ideal gas):

• The gas consists of a very large number of identical molecules moving in random directions.
• Molecules are point particles (their size is negligible compared with the distances between them).
• There are no intermolecular attractive or repulsive forces except during collisions.
• Collisions between molecules and with the walls are perfectly elastic.
• The time spent in collisions is negligible compared with the time between collisions.
• The average kinetic energy of molecules is proportional to the absolute temperature.

Derivation linking microscopic motion to pressure (sketch):
Consider N molecules of mass m in a cubical container of side L (volume V = L^3). For one molecule with x-component of velocity v_x, the momentum change on hitting a wall is 2m v_x and the time between successive hits on the same wall is 2L / v_x. So the average force contributed by that molecule on the wall is m v_x^2 / L. Summing over all molecules and dividing by wall area L^2 gives

p = (m N / V) average(v_x^2). By isotropy average(v_x^2) = (1/3) average(v^2), hence

p = (1/3) (N m / V) v_rms^2, where v_rms = sqrt(average(v^2)).

Using the ideal gas law in microscopic form pV = N k T (k = Boltzmann constant) we get

(1/2) m average(v^2) = (3/2) k T. This shows that the average translational kinetic energy per molecule is directly proportional to absolute temperature.

Consequences and physical meaning

• Average translational kinetic energy per molecule: (1/2) m v_rms^2 = (3/2) k T.
• Internal energy of a monatomic ideal gas (only translational KE) U = N (3/2 k T) = (3/2) n R T.
• Temperature is a measure of the average kinetic energy of molecules, not of total energy of the gas (for ideal monatomic gas all internal energy is kinetic).

Limits of the model: Real gases deviate from ideal behavior at high pressure and low temperature because molecular volume and intermolecular forces become important. Ideal-gas kinetic interpretation is most accurate for dilute, high-temperature gases.

Fig 6 — Educational Diagram: Thermodynamic processes — types and equations

Overview: A thermodynamic process is a change of state of a system (e.g. a gas) that involves heat (Q) transfer, work (W) done, and change in internal energy (ΔU). The First Law of Thermodynamics relates them: ΔU = Q - W (using the convention W = work done by the system).

Key concepts:

• Internal energy for an ideal monoatomic gas: U = (3/2)nRT; in general ΔU = n C_v ΔT.
• Specific heats: C_p - C_v = R and γ = C_p / C_v (ratio of specific heats).

Main types of processes (for an ideal gas):

• Isothermal (T = constant): Temperature stays constant so ΔU = 0. Heat added equals work done by the gas: Q = W. Equation: PV = constant (hyperbola). Work: W = nRT ln(V_f / V_i).
• Adiabatic (Q = 0): No heat exchange. ΔU = -W (work comes from internal energy). For a reversible adiabatic (ideal gas): PV^γ = constant, TV^(γ-1) = constant, or T^{γ}P^{1-γ}=const. Work between states 1 and 2: W = (P_1V_1 - P_2V_2)/(γ - 1).
• Isobaric (P = constant): Pressure fixed. Work: W = P ΔV. Heat: Q = n C_p ΔT. Change in internal energy: ΔU = n C_v ΔT.
• Isochoric / isovolumetric (V = constant): Volume fixed so W = 0. Heat goes into internal energy: Q = ΔU = n C_v ΔT.
• Polytropic (PV^n = constant): A general family that includes isothermal (n=1) and adiabatic (n=γ). Work: W = (P_2V_2 - P_1V_1)/(1 - n) for n ≠ 1.
• Cyclic process: System returns to initial state, so ΔU = 0. Net work done in a cycle equals net heat absorbed: W_net = Q_net. Heat engines operate in cyclic processes (Carnot, Otto, Diesel).
• Free (Joule) expansion: Irreversible expansion into vacuum. For an ideal gas in an insulated container, Q = 0 and W = 0 so ΔU = 0 (temperature unchanged).

Reversible vs Irreversible / Quasi-static: A quasi-static (infinitesimally slow) process is nearly reversible and is represented by a smooth path on PV diagrams. Real processes often are irreversible.

Energy bookkeeping (First Law) — summary:

• ΔU = Q - W
• For ideal gas: ΔU = n C_v ΔT
• Relate Q and W depending on the constraint (T const, P const, V const, Q=0, etc.).

Fig 7.1 — Educational Diagram: Newton's Laws of Motion & Free Body Diagrams

Statement (Kelvin–Planck): It is impossible to build a heat engine which, operating in a cycle, extracts heat from a single heat reservoir and converts all of it into work. Some heat must be rejected to a colder reservoir.

Statement (Clausius): Heat cannot spontaneously flow from a colder body to a hotter body without external work being done on the system.

Meaning and consequence: The first law (energy conservation) allows many processes that are not observed; the second law introduces directionality (time-asymmetry) in natural processes. It defines limits on the efficiency of heat engines and explains why some processes are irreversible.

Heat engines and refrigerators: A heat engine absorbs heat Q1 from a hot reservoir at temperature T1, does work W, and rejects heat Q2 to a cold reservoir at temperature T2. A refrigerator or heat pump uses work to transfer heat from cold to hot.

Reversible and irreversible processes: A reversible process is an idealization that can be reversed without net change to the system and surroundings. All real (natural) processes are irreversible and increase the total entropy of the universe.

Entropy (S): Entropy is a state function that measures energy dispersal or the unavailability of a system's energy to do work. For a reversible process, the infinitesimal change of entropy is defined as dS = dQ_rev / T. For a finite reversible change between states A and B, ΔS = ∫(A→B) dQ_rev / T.

Clausius inequality: For any cyclic process, ∮ dQ / T ≤ 0, with equality only for a reversible cycle. This leads to the conclusion that for an isolated system, entropy never decreases: ΔS_total ≥ 0.

Carnot theorem and limit on efficiency: Among all heat engines operating between the same two temperatures T1 (hot) and T2 (cold), a reversible (Carnot) engine has the maximum possible efficiency. No engine can be more efficient than a Carnot engine operating between the same reservoirs.

Intuitive examples of irreversibility: Heat flowing from hot to cold, free expansion of a gas, mixing of two different gases, friction converting organized kinetic energy to thermal energy — all increase entropy.

Applications: Establishes the theoretical maximum efficiency of engines, explains why perpetual motion machines of the second kind are impossible, and provides the basis for refrigerators, heat pumps, and understanding spontaneous processes.

Notes on temperature in Carnot relations: Temperatures T1 and T2 must be in an absolute scale (Kelvin) when used in efficiency or COP formulas.

Fig 8 — Educational Diagram: Heat engines, refrigerators and heat pumps

Overview

Heat engines, refrigerators and heat pumps are devices that transfer heat and perform work governed by the laws of thermodynamics. A heat engine converts part of heat from a hot reservoir into work while rejecting remaining heat to a cold reservoir. A refrigerator absorbs heat from a cold space and dumps it to a hotter space by doing work. A heat pump is like a refrigerator but is used to deliver heat to the hot space.

Heat engines

Working substance (gas or vapour) undergoes a cyclic process. From the first law for a full cycle, internal energy change ΔU = 0, so net work done by the system W = Q_h − Q_c, where Q_h is heat absorbed from the hot reservoir and Q_c is heat rejected to the cold reservoir.

Second law (Kelvin–Planck statement): It is impossible to construct an engine that, operating in a cycle, produces no other effect than the absorption of heat from a reservoir and the performance of an equivalent amount of work. Hence some heat must be rejected.

Maximum efficiency — Carnot engine

A reversible engine operating between two reservoirs at absolute temperatures T_h and T_c (Kelvin) is the most efficient. Its efficiency is

η_Carnot = 1 − T_c/T_h

For any engine operating between same T_h and T_c, η ≤ η_Carnot. For a general engine, efficiency η = W/Q_h = (Q_h − Q_c)/Q_h = 1 − Q_c/Q_h.

Refrigerators and heat pumps

Refrigerator: takes heat Q_c from cold reservoir and rejects Q_h to hot reservoir by input work W. From energy conservation W = Q_h − Q_c.

Heat pump: same device but goal is to deliver heat Q_h to the hot space; often used for heating homes.

Performance is measured by coefficient of performance (COP), not efficiency:

• COP (refrigerator) = Q_c / W = Q_c / (Q_h − Q_c).
• COP (heat pump) = Q_h / W = Q_h / (Q_h − Q_c) = 1 + COP_refrigerator.

For a reversible (Carnot) refrigerator or heat pump operating between T_h and T_c (Kelvin):

• COP_refrigerator,Carnot = T_c / (T_h − T_c)
• COP_heatpump,Carnot = T_h / (T_h − T_c)

Reversible vs irreversible

Reversible processes are idealizations giving maximum COP or efficiency. Real devices are irreversible due to friction, non-quasi-static processes, heat leaks, and finite-rate heat transfer; hence real performance is lower.

Key practical points

• Engines: internal combustion engines (cars), steam turbines (power plants) — operate as heat engines but not reversible and follow specific cycles (Otto, Diesel, Rankine).
• Refrigerators and ACs use vapour-compression cycles; heat pumps use similar cycles but aim to heat spaces.
• Temperatures must be in absolute scale (Kelvin) when using Carnot relations.

Fig 9 — Educational Diagram: Entropy and its consequences

What is Entropy?

Entropy (S) is a thermodynamic state function that measures the degree of disorder or the number of accessible microscopic arrangements (microstates) of a system. It is a measure of energy dispersal at a given temperature. In macroscopic thermodynamics, entropy quantifies how much heat transfer is irreversible and gives a direction to spontaneous processes (the arrow of time).

Clausius definition (macroscopic)

For a reversible process the infinitesimal change of entropy is defined as dS = δQ_rev / T. For a finite reversible change between two states:

ΔS = ∫_(1)^(2) (δQ_rev / T).

Boltzmann definition (microscopic)

Entropy is related to the number of microstates Ω compatible with the macroscopic state by Boltzmann's formula: S = k_B ln Ω (where k_B is Boltzmann's constant). This links entropy to probability and disorder.

Key properties

• State function: S depends only on the initial and final states, not on the path.
• For an isolated system, entropy never decreases: S_total (isolated) either increases for irreversible processes or remains constant for reversible ones.
• Clausius inequality for any cyclic process: ∮ (δQ / T) ≤ 0, equality for a reversible cycle.
• Entropy of the universe (system + surroundings) increases for any spontaneous (irreversible) process: ΔS_univ = ΔS_sys + ΔS_surr ≥ 0.

Entropy changes in common processes

- Isothermal reversible expansion of an ideal gas: ΔS = Q_rev / T = nR ln(V_2 / V_1).
- General change for an ideal gas between states (T,V): ΔS = nC_v ln(T_2/T_1) + nR ln(V_2/V_1).
- Phase change at constant temperature (melting/boiling): ΔS = L / T (L = latent heat).
- Mixing of ideal gases (ideal, no energy change): entropy increases because number of accessible microstates increases; for ideal mixture: ΔS_mix = -nR Σ x_i ln x_i (x_i are mole fractions).

Consequences of entropy

• Direction of spontaneous processes: Processes with ΔS_univ > 0 occur spontaneously. If ΔS_univ = 0 the process is reversible and at equilibrium.
• Irreversibility and lost work: Entropy production corresponds to energy that cannot be converted into useful work. Reversible processes give maximum work extraction.
• Limits on heat engines: Using entropy balance and the second law leads to maximum (Carnot) efficiency η_max = 1 - T_c/T_h. For a reversible engine, Q_h/T_h = Q_c/T_c and net entropy change is zero.
• Arrow of time: Macroscopic irreversibility (entropy increase) gives a thermodynamic direction to time—natural processes tend toward equilibrium (maximum entropy consistent with constraints).
• Local entropy decrease possible: A subsystem can decrease entropy (e.g., refrigerator, living organisms) only if surroundings gain at least as much entropy so that ΔS_univ ≥ 0.

Units

SI unit of entropy: joule per kelvin (J K^−1). Often molar entropy is quoted in J K^−1 mol^−1.

Fig 10 — Educational Diagram: Applications, problem solving and common derivations

This topic explains how to apply the basic laws of thermodynamics (primarily the first law) to common processes, how to set up and solve typical problems, and gives the standard derivations used in Class 11. Key ideas: treat the system (usually an ideal gas) clearly, identify the process (isothermal, adiabatic, isobaric, isochoric, cyclic), use the ideal gas law where applicable, apply the first law (DeltaU = Q - W) with a consistent sign convention, and use specific heat relations (Cp - Cv = R).

Problem-solving strategy:

• Step 1: Specify the system and whether the process is quasi-static (so that P at each step is defined).
• Step 2: Identify the process type. Choose the correct relation for P, V, T or for work W = ∫ P dV.
• Step 3: Use the ideal gas law P V = n R T to eliminate variables where needed.
• Step 4: Apply the first law: ΔU = Q - W. For an ideal monoatomic or diatomic gas ΔU = n Cv ΔT.
• Step 5: Compute W (area under P-V curve) and Q (from first law or Q = n Cp ΔT for processes at constant pressure).
• Step 6: Check units and limiting cases (e.g., for isothermal ΔU = 0 so Q = W).

Common derivations and important relations (brief derivation steps):

• Work in a quasi-static process: W = ∫_{V1}^{V2} P dV. For an ideal gas, substitute P from PV = nRT when needed.
• Isothermal (T constant): P = nRT/V ⇒ W = nRT ∫_{V1}^{V2} dV/V = nRT ln(V2/V1). For ideal gas ΔU = 0 ⇒ Q = W.
• Isobaric (P constant): W = P (V2 - V1). Heat Q = n Cp ΔT and ΔU = n Cv ΔT.
• Isochoric (V constant): W = 0. So Q = ΔU = n Cv ΔT.
• Adiabatic (Q = 0, reversible): use first law ΔU = -W. For ideal gas: PV^γ = constant (where γ = Cp/Cv). Other useful forms: TV^{γ-1} = constant and T^{γ} P^{1-γ} = constant. Work done: W = (P1 V1 - P2 V2)/(γ - 1).
• Mayer's relation: Cp - Cv = R (per mole). For n moles: n(Cp - Cv) = nR.
• Change in internal energy (ideal gas): ΔU = n Cv ΔT (depends only on temperature for ideal gas).
• Carnot efficiency (reversible engine between Th and Tc): η = 1 - Tc/Th (temperatures in Kelvin). This is the maximum possible efficiency.
• Refrigerator and heat pump COPs: COP_{refrigerator} = Qc/W = Tc/(Th - Tc); COP_{heat pump} = Qh/W = Th/(Th - Tc).

Common pitfalls:

• Sign convention: Q positive into system, W positive when done by the system (so ΔU = Q - W).
• Adiabatic does not always mean fast — it means no heat exchange. Reversible adiabatic = isentropic.
• Work equals area under P-V curve; for non-quasi-static processes this area may not be well-defined with a single P(V).
• Always use absolute (Kelvin) temperature for Carnot and other thermodynamic temperature relations.

When to use which formula:

• Isothermal expansion/compression: use W = nRT ln(V2/V1) and ΔU = 0.
• Adiabatic reversible: use PV^γ = constant and W = (P1V1 - P2V2)/(γ - 1).
• Constant pressure: W = PΔV, Q = nCpΔT.
• Constant volume: W = 0, Q = nCvΔT.

These relations and the problem strategy cover the majority of Class 11 thermodynamics problems and derivations.