Thermodynamics
Easy Overview
Thermodynamics is the science of energy — how it flows, how it transforms, and why certain processes happen spontaneously while others never do. It governs everything from the cooling of a cup of tea to the operation of a nuclear power plant, from the metabolism in your body to the formation of stars. At its heart, thermodynamics is about three things: energy, entropy, and equilibrium. And despite dealing with some of the most profound concepts in physics, its principles can be stated with remarkable simplicity. The story begins with the zeroth law of thermodynamics. It sounds trivial: if body A is in thermal equilibrium with body B, and body B is in thermal equilibrium with body C, then A is in thermal equilibrium with C. But this simple statement is the foundation of temperature measurement. It justifies using a thermometer: when the thermometer (B) reaches equilibrium with your body (A), and it was calibrated against a standard (C), the reading from C tells you about A. Without the zeroth law, the concept of temperature would not be well-defined. The first law of thermodynamics is the law of conservation of energy, extended to include heat. It states that energy can neither be created nor destroyed, only converted from one form to another. Mathematically: ΔU = Q - W, where ΔU is the change in internal energy of a system, Q is the heat added to the system, and W is the work done by the system on its surroundings. The first law tells us that if you add heat to a gas in a cylinder with a movable piston, the gas can respond in two ways: it can get hotter (increase internal energy) or it can expand (do work by pushing the piston outward), or a combination of both. But the first law does not tell us which processes are possible. You could imagine heat flowing from a cold cup to a hot room, cooling the cup further and heating the room more — that would conserve energy, so the first law would not forbid it. But it never happens. This is where the second law comes in. The second law introduces the concept of entropy — a measure of disorder. The total entropy of an isolated system always increases in a spontaneous process. Heat flows from hot to cold because that increases entropy. A gas expands to fill a container because that increases entropy. The second law sets the direction of time — it is why you can unscramble an egg (theoretically, with enough energy) but never see one spontaneously unscramble. The second law also limits the efficiency of heat engines — even an ideal engine cannot convert all heat into work. Some heat must always be rejected to a cold reservoir.
Thermal Equilibrium and the Zeroth Law
The zeroth law of thermodynamics establishes the concept of temperature. Two systems are in thermal equilibrium when there is no net heat flow between them. The zeroth law states that if A and B are each in thermal equilibrium with C, then A and B are in thermal equilibrium with each other. This transitivity allows temperature to be defined and measured. Thermometers work by bringing a small system (mercury in a glass tube, or a thermocouple) into contact with the system of interest. When thermal equilibrium is reached, the thermometer reading gives a number that correlates with the temperature.
Internal Energy and the First Law
Internal energy (U) is the total energy stored within a system. For an ideal gas, it is entirely kinetic — the sum of the translational, rotational, and vibrational energies of all molecules. For monatomic ideal gases, U = (3/2)nRT; it depends only on temperature. The first law of thermodynamics states: ΔU = Q - W. Here Q is positive when heat enters the system, and W is positive when the system does work on its surroundings. The first law is a statement of energy conservation. It does not tell us whether a process will occur — only that if it does occur, energy will be conserved.
Quasi-Static and Reversible Processes
A quasi-static process is one that occurs infinitely slowly, so the system passes through a continuous series of equilibrium states. A reversible process is a quasi-static process with no dissipative effects (no friction, no viscosity, no turbulence). In a reversible process, the system and its surroundings can be returned to their initial states by reversing the process without any net change in the universe. All real processes are irreversible to some degree. Reversible processes are idealisations that set the upper limit on efficiency.
Isochoric Process (Constant Volume)
In an isochoric process, the volume remains constant. Since W = ∫ P dV = 0, the first law gives ΔU = Q_V. All heat added goes into increasing the internal energy. For an ideal gas, Q = nC_VΔT. An isochoric process on a P-V diagram appears as a vertical line. Examples include heating a gas in a rigid sealed container, or the combustion stroke in an internal combustion engine just before the piston moves.
Isobaric Process (Constant Pressure)
In an isobaric process, the pressure remains constant. The work done is W = PΔV. The first law gives ΔU = Q_P - PΔV. The heat absorbed is Q_P = nC_PΔT. Since C_P > C_V, the relation C_P - C_V = R holds for ideal gases. On a P-V diagram, an isobaric process is a horizontal line. Examples include heating a gas in a cylinder with a movable piston, or boiling water at constant atmospheric pressure.
Isothermal Process (Constant Temperature)
In an isothermal process, the temperature remains constant. For an ideal gas, ΔU = 0, so Q = W. The work done in an isothermal expansion from V_i to V_f is W = nRT ln(V_f/V_i). The P-V curve for an isothermal process is a rectangular hyperbola (PV = constant). Isothermal processes must be carried out slowly to allow heat exchange with a reservoir to maintain constant temperature.
Adiabatic Process (No Heat Exchange)
In an adiabatic process, Q = 0 — no heat enters or leaves the system. From the first law, ΔU = -W: the work done by the system comes at the expense of its internal energy, causing the temperature to drop. For an ideal gas undergoing a reversible adiabatic process, PV^γ = constant, TV^(γ-1) = constant, and TP^((1-γ)/γ) = constant, where γ = C_P/C_V. The work done is W = (P_iV_i - P_fV_f)/(γ - 1). On a P-V diagram, the adiabatic curve is steeper than the isothermal curve.
Cyclic Processes and the First Law
In a cyclic process, the system returns to its initial state after completing a series of steps. Since U is a state function, ΔU = 0 over a complete cycle. From the first law: Q_net = W_net — the net heat added equals the net work done. On a P-V diagram, a cyclic process forms a closed loop, and the area enclosed by the loop equals the net work done. If the cycle is traversed clockwise, net work is done by the system (heat engine).
Heat Engines
A heat engine is a device that converts thermal energy into mechanical work. It operates in a cycle: (1) absorbs heat Q₁ from a hot reservoir at T₁, (2) does useful work W, (3) rejects waste heat Q₂ to a cold reservoir at T₂. The thermal efficiency η = W/Q₁ = (Q₁ - Q₂)/Q₁ = 1 - Q₂/Q₁. Real heat engines include steam engines, internal combustion engines, gas turbines, and Stirling engines. Typical efficiencies: petrol engine 25-30%, diesel engine 30-40%, large steam turbines 40-48%.
The Carnot Cycle and Carnot Engine
The Carnot cycle is the most efficient possible heat engine cycle operating between two fixed temperatures. It consists of four reversible processes: (1) Isothermal expansion at T₁, (2) Adiabatic expansion (temperature drops to T₂), (3) Isothermal compression at T₂, (4) Adiabatic compression (temperature rises back to T₁). The efficiency of a Carnot engine is η_Carnot = 1 - T₂/T₁ (temperatures in Kelvin). This depends only on the reservoir temperatures, not on the working substance. No real engine can exceed this efficiency.
The Second Law of Thermodynamics
The second law can be stated in several equivalent ways. Kelvin-Planck statement: it is impossible to construct a heat engine that operates in a cycle and converts all heat input into work. Clausius statement: it is impossible to construct a refrigerator that transfers heat from a cold body to a hot body without work input. These statements are logically equivalent. The second law explains why processes have a preferred direction: heat flows from hot to cold, not vice versa; gases expand to fill containers; ordered structures tend toward disorder.
Entropy — A Measure of Disorder
Entropy (S) is a state function that measures disorder. For a reversible process, dS = dQ_rev/T. For an irreversible process, dS > dQ/T. The second law can be restated as: the total entropy of an isolated system never decreases — ΔS_universe ≥ 0. When heat flows from a hot reservoir at T₁ to a cold reservoir at T₂, ΔS_hot = -Q/T₁, ΔS_cold = +Q/T₂, and ΔS_total = Q(1/T₂ - 1/T₁) > 0. Statistical mechanics: S = k ln W, where W is the number of microscopic configurations. More arrangements = higher entropy.
Entropy Changes in Thermodynamic Processes
For an ideal gas undergoing a reversible process: ΔS = nC_V ln(T₂/T₁) + nR ln(V₂/V₁). For isothermal: ΔS = nR ln(V₂/V₁). For isochoric: ΔS = nC_V ln(T₂/T₁). For isobaric: ΔS = nC_P ln(T₂/T₁). For phase changes at constant temperature: ΔS = Q_rev/T = L/T, where L is latent heat. Entropy is a state function — it depends only on initial and final states, not on the path.
Refrigerators and Heat Pumps
A refrigerator uses work W to extract heat Q₂ from a cold reservoir and rejects heat Q₁ = Q₂ + W to a hot reservoir. The coefficient of performance (COP) of a refrigerator is COP_ref = Q₂/W. For a Carnot refrigerator: COP_Carnot = T₂/(T₁ - T₂). A heat pump extracts heat from a cold outside and delivers it to a warm interior. COP_HP = Q₁/W = T₁/(T₁ - T₂). Heat pumps are more efficient than electric resistance heaters because they move heat rather than generating it.
The Third Law of Thermodynamics
The third law states that the entropy of a perfect crystalline substance approaches zero as the temperature approaches absolute zero (0 K). Consequences: (1) It is impossible to reach absolute zero in a finite number of steps. (2) The specific heat capacity of all substances approaches zero as T → 0 K. (3) The thermal expansion coefficient also approaches zero at absolute zero. (4) It allows the calculation of absolute entropies from heat capacity data.
Thermodynamic Potentials and Free Energy
Thermodynamic potentials help predict spontaneity. Enthalpy H = U + PV: ΔH = Q_P at constant pressure. Helmholtz free energy F = U - TS: ΔF < 0 for spontaneous processes at constant T and V. Gibbs free energy G = H - TS: ΔG < 0 for spontaneous processes at constant T and P. A reaction with ΔG < 0 is spontaneous (exergonic); ΔG > 0 is non-spontaneous; ΔG = 0 is at equilibrium.
Heat Conduction and Thermodynamics
Heat conduction is given by Fourier's law: dQ/dt = -kA(dT/dx), where k is thermal conductivity. The minus sign indicates heat flows from hot to cold (consistent with the second law). Thermal conductivity varies widely: metals like copper (k ≈ 400 W/m·K) conduct well, while insulators like wood (k ≈ 0.1 W/m·K) conduct poorly. Heat conduction is an irreversible process that generates entropy.
Key Points
- •Zeroth law: thermal equilibrium is transitive. Basis of temperature measurement.
- •First law: ΔU = Q - W. Energy is conserved.
- •Internal energy of ideal gas depends only on temperature: U = (f/2)nRT.
- •Isochoric: W = 0, ΔU = Q = nC_VΔT.
- •Isobaric: W = PΔV, Q_P = nC_PΔT. C_P - C_V = R.
- •Isothermal: ΔU = 0, Q = W = nRT ln(V_f/V_i).
- •Adiabatic: Q = 0, PV^γ = constant, TV^(γ-1) = constant.
- •Cyclic process: ΔU = 0, Q_net = W_net.
- •Heat engine efficiency: η = 1 - Q_out/Q_in.
- •Carnot efficiency: η_max = 1 - T₂/T₁.
- •Second law (Kelvin-Planck): cannot convert all heat to work in a cycle.
- •Second law (Clausius): cannot move heat from cold to hot without work.
- •Entropy: dS = dQ_rev/T. ΔS_universe ≥ 0.
- •Entropy change for ideal gas: ΔS = nC_V ln(T₂/T₁) + nR ln(V₂/V₁).
- •Refrigerator COP: Q₂/W. Carnot COP = T₂/(T₁ - T₂).
- •Third law: S → 0 as T → 0 K. Absolute zero is unattainable.
- •Gibbs free energy: ΔG = ΔH - TΔS. ΔG < 0 for spontaneous processes.
Practice Questions
- State and explain the first law of thermodynamics. Derive work, heat, and internal energy change for isothermal, adiabatic, isochoric, and isobaric processes.
- Explain the Carnot cycle with a neat diagram. Derive the expression for efficiency of a Carnot engine. Why is 100% efficiency impossible?
- State the second law of thermodynamics in both Kelvin-Planck and Clausius forms. Show that they are equivalent.
- What is entropy? Derive the expression for entropy change of an ideal gas. Calculate entropy change when 1 mole of ideal gas expands isothermally from 10 L to 20 L at 300 K.
- Explain the working of a refrigerator. Derive its COP. A Carnot refrigerator operates between -3°C and 27°C. Find its COP.
- State the third law of thermodynamics. Explain why absolute zero is unattainable.
- Distinguish between reversible and irreversible processes. Why is entropy change of the universe always positive for irreversible processes?
- What is Gibbs free energy? Derive ΔG = ΔH - TΔS. A reaction has ΔH = +50 kJ/mol and ΔS = +100 J/mol·K. At what temperature does it become spontaneous?