Thermodynamics

What This Chapter Is About

• Thermodynamics is the branch of physics dealing with heat, temperature and the inter-conversion of heat and other forms of energy.
• Heat is a form of energy, not a fluid. Benjamin Thomson's cannon-boring observation (1798) — heat produced depended on the work done in turning the drill and on how long it bored, not on the sharpness of the drill — established this.
• Everything in this chapter follows from two ideas: energy is conserved (first law), and heat flow has a preferred direction (second law).

Thermal Equilibrium and the Zeroth Law

• Two systems in thermal contact are in thermal equilibrium if they do not transfer heat between each other.
• Zeroth law: if two systems are each in thermal equilibrium with a third system, they are also in thermal equilibrium with each other.
• Consequence: systems in thermal equilibrium are at the same temperature. This is what lets a thermometer work, and it makes temperature a measurable quantity.
• It is called the zeroth law because it was formulated after the first and second laws, but is logically prior to them.

Internal Energy

• Internal energy $U$ is the energy associated with the random, disordered motion of the molecules of a system. It is a microscopic quantity, quite separate from the ordered macroscopic kinetic energy of a moving body.
• Monatomic ideal gas: only translational kinetic energy, $\dfrac{3}{2}kT$ per atom. Diatomic gas: $\dfrac{5}{2}kT$ per molecule. Polyatomic gases add rotational and vibrational kinetic energy; liquids and solids also need the potential energy of intermolecular attraction.
• For an ideal gas $U$ depends only on temperature. So $\Delta T = 0$ forces $\Delta U = 0$.

Systems and Surroundings

• A thermodynamic system is a collection of objects forming a unit that can exchange energy with its surroundings. Anything not part of the system is its surroundings or environment.

Heat, and the Sign Convention

• Heat $Q$ is the energy transferred between a system and its environment because of a temperature difference between the two.
• Energy transferred to the system is positive (system gains); energy transferred from the system is negative (system loses). If $T_S = T_E$ there is no transfer and $Q = 0$.
• Internal energy of a system can be changed in exactly two ways: by heating it, or by doing work on it.

First Law of Thermodynamics

• Statement: the change in the internal energy of a system is the difference between the heat supplied to the system and the work done by the system on its surroundings.

$$\Delta U = Q - W \qquad \text{or} \qquad Q = \Delta U + W$$

• Equivalent wording: when heat $Q$ is added to a system, its internal energy increases by $\Delta U$ and the remainder is lost as work $W$ done on the surroundings.
• It is a generalisation of the law of conservation of energy, extended from isolated systems to systems that can exchange energy.
• $Q$ and $W$ can each be positive, negative or zero, so $\Delta U$ can be positive ($Q$ exceeds $W$), negative ($W$ exceeds $Q$) or zero ($Q = W$).
• Limitations: it does not say which direction a process will go, and it does not forbid 100 per cent conversion of heat into work. Both gaps are closed by the second law.

State Variables and Equilibrium

• A property (state variable) is any measurable characteristic of a system in equilibrium: $p$, $V$, $T$, density, mass.

• Thermodynamic equilibrium requires all three simultaneously: mechanical equilibrium (no unbalanced forces, pressure uniform and steady), chemical equilibrium (no reactions or diffusion, composition uniform and steady), thermal equilibrium (temperature uniform and steady).
• A system not in equilibrium — a punctured ball, an exploding fuel-air mixture — cannot be described in terms of state variables at all.
• The mathematical relation between state variables is the equation of state; for an ideal gas, $pV = nRT$.
• The graph of the equation of state is the $p$–$V$ diagram or indicator diagram. A constant-temperature $p$–$V$ curve is an isotherm.

Work and Heat Depend on the Path

• Three different paths from state A $(p_i, V_i)$ to state B $(p_f, V_f)$ — a smooth curve, isobaric-then-isochoric, isochoric-then-isobaric — enclose three different areas, so give three different amounts of work. The initial and final states are the same in all three.
• Conclusion: work done by a system depends not only on the initial and final states but on the path taken.
• Same for heat: slow isothermal expansion absorbs a finite $Q$; free expansion of the same gas between the same two states has $Q = 0$. So heat transferred also depends on the path.
• $Q$ and $W$ are path quantities; $U$ is a state quantity.

Quasi-static, Reversible and Irreversible

• A quasi-static process changes the state variables infinitesimally slowly, so the system stays in thermodynamic equilibrium throughout.

• Two causes of irreversibility: (i) processes such as free expansion or explosive reaction take the system through non-equilibrium states; (ii) friction, viscosity and other dissipative forces are always present and cannot be fully eliminated.
• All spontaneous natural processes are irreversible; an irreversible process defines the preferred direction of change and is unidirectional.

The Four Standard Processes

• Isothermal: needs good thermal contact with the environment and an extremely slow change. Any change of phase (melting ice, boiling water) is isothermal. Neither $Q$ nor $W$ is zero.
• Isobaric: boiling water at atmospheric pressure. None of $\Delta U$, $Q$, $W$ is zero. The isobars on a weather map are the same idea — lines of equal pressure.
• Isochoric: heating a gas in a rigid container, or diffusion in a closed chamber. All the heat supplied raises the internal energy.
• Adiabatic: puncturing a tyre or balloon; the compression stroke of a diesel engine. Adiabatic expansion cools the gas, adiabatic compression heats it. Usually a sudden change, so there is no time to exchange heat.
• The adiabatic $p$–$V$ curve is steeper than the isothermal curve through the same point, because $\gamma > 1$.

Cyclic Process and Free Expansion

• Cyclic process: the system returns to its initial state, so $\Delta U = 0$ and $Q = W$. The work done in one cycle is the area of the closed loop; positive for a clockwise loop, negative for an anticlockwise loop. Every heat engine works on a cyclic process.
• Free expansion: adiabatic, no heat exchanged and no work done, so $Q = W = \Delta U = 0$ and the temperature of an ideal gas is unchanged. It is uncontrolled and instantaneous, the system is never in equilibrium, and it cannot be plotted as a path on a $p$–$V$ diagram.

Heat Engines

• A heat engine is a device that transforms heat partly into work or mechanical energy, working through a cyclic process.
• Three elements: a working substance (the system — ideal gas, steam, or a fuel-air mixture), a hot reservoir or source at $T_H$ and a cold reservoir or sink at $T_C$, and a cylinder with a movable frictionless massless piston.
• Three basic steps every cycle: absorb heat $Q_H$ from the hot reservoir; convert part of it into work $W$; reject the remaining heat $Q_C$ to the cold reservoir.
• External combustion engine: working substance heated externally (steam engine). Internal combustion engine: working substance heated internally (petrol or diesel engine).

$$W = Q_H + Q_C = |Q_H| - |Q_C|, \qquad \eta = \dfrac{W}{Q_H} = 1 - \dfrac{|Q_C|}{Q_H}$$

• Efficiency is the fraction of the absorbed heat converted into work. It has no unit, so $W$, $Q_H$ and $Q_C$ must all be in the same energy unit.
• Not all heat absorbed can be converted into work: $Q_C$ is never zero, so $\eta$ is never 1.

Refrigerators, Air Conditioners and Heat Pumps

• These are heat engines run backwards: they convert mechanical work into heat transport, forcing heat from a cold region to a hot region.
• Refrigeration is artificial cooling — cooling a space or keeping it below the ambient temperature.
• Mechanism: a refrigerant (for example isobutane) is made to evaporate in the cooling coil inside (absorbing its heat of vaporisation from the cold region) and then compressed and condensed outside (rejecting that heat to the atmosphere). That is why the back of a working refrigerator feels warm.
• Four steps of one cycle: (1) the fluid passes through a nozzle into a low-pressure region and turns into cool gas — essentially adiabatic expansion; (2) the cool gas absorbs heat from the fridge contents — isobaric expansion; (3) the compressor compresses the gas adiabatically, heating it and turning it back to liquid; (4) the hot liquid passes through the outside coils and rejects heat to the atmosphere — isobaric compression.

$$|Q_H| = Q_C + |W|, \qquad K = \dfrac{Q_C}{W} = \dfrac{Q_C}{Q_H - Q_C}$$

• $K$, the coefficient of performance, is dimensionless. Larger $K$ is better: it means more heat extracted for less work. Typically about 5 for a household fridge, 2.5 to 3.0 for a room air conditioner.
• Air conditioner: same working as a refrigerator, differing only in the volume of the chamber cooled. Evaporator coils inside the room, condenser outside, a fan circulating the cooled air. Performance is often measured as $K = H/P$, heat current over power.
• Heat pump: a refrigerator operating inside out — evaporator coils outside absorbing heat from cold outdoor air, condenser coils inside releasing that heat, so the building is warmed.
• Capacity of a refrigerator is quoted in litres (the internal volume); capacity of an air conditioner in tonnes, historically the mass of ice that would melt in 24 hours to give the same cooling.

Second Law of Thermodynamics

• Kelvin-Planck (engine) statement: it is impossible to extract an amount of heat $Q_H$ from a hot reservoir and use it all to do work $W$. Some amount of heat $Q_C$ must be exhausted to a cold reservoir. This prohibits the possibility of a perfect heat engine.
• Clausius (refrigerator) statement: it is not possible for heat to flow from a colder body to a warmer body without any work having been done to accomplish this flow. This rules out a perfect refrigerator.
• The second law puts constraints on the direction of heat transfer and on the efficiency a heat engine can achieve. Consequences: $\eta$ can never be 1, and $K$ can never be infinite.
• Note the "net" in the Clausius statement: energy can pass from a cold object to a hot one by radiation or energetic particles, but in any spontaneous process the net transfer is hot to cold.

Carnot Cycle and Carnot Engine

• Sadi Carnot (1824) proposed a hypothetical ideal engine of maximum efficiency, in which every step is reversible.
• Reversible heat exchange must be isothermal (working substance at $T_H$ while absorbing, at $T_C$ while rejecting); reversible work must be adiabatic. Hence two isothermals and two adiabatics.
• The four strokes: AB isothermal expansion at $T_H$, BC adiabatic expansion, CD isothermal compression at $T_C$, DA adiabatic compression.

$$\eta = 1 - \dfrac{Q_C}{Q_H} = 1 - \dfrac{T_C}{T_H}, \qquad K_{\text{Carnot}} = \dfrac{T_C}{T_H - T_C}$$

• Design rule: for maximum efficiency, $T_H$ as high as possible and $T_C$ as low as possible.
• Carnot's theorem: the Carnot engine is the most efficient heat engine, and all Carnot engines working between the same two temperatures have the same efficiency, whatever their working substance. Likewise a Carnot refrigerator has the greatest coefficient of performance of all refrigerators between the same two temperatures.
• $\eta = 100$ per cent would need $T_C = 0$ K, which is unattainable. So even the ideal engine cannot be 100 per cent efficient.
• $K_{\text{Carnot}}$ depends only on the temperatures: a small temperature difference gives a very large $K$ (much heat pumped for little work), a large temperature difference a very small $K$.

Sterling Cycle

• A closed, fully reversible cycle used by the Sterling engine; working substances are air, helium, hydrogen or nitrogen.
• Two isothermal processes, AB at $T_H$ and CD at $T_C$, joined by two constant-parameter processes BC and DA.
• AB isothermal expansion: heat $Q_H$ supplied at $T_H$, useful work done by the gas. BC: part of the absorbed heat $Q$ is released to the refrigerator and stored for reuse, gas cools to $T_C$. CD isothermal compression: heat $Q_C$ rejected to the coolant (sink) at $T_C$. DA: the compressed gas absorbs the stored heat $Q$ and its temperature rises to $T_H$; the cycle then repeats from A.
• Heating the gas makes it produce useful work (engine); doing work on the gas runs it as a refrigerator. The reversed Sterling cycle is used extensively in cryogenics to reach very low temperatures and to liquefy gases.

Traps That Cost Marks

• Converting $^\circ$C to K. Every efficiency and every gas-law step needs kelvin. $727\,^\circ$C is 1000 K, not 727 K.
• Sign of $W$ in the first law. $W$ is the work done by the system. Work done on the system is negative $W$, which makes $\Delta U = Q - W$ larger. When a numerical says "work done on the gas" and "heat released", both are negative and the safest route is the board's own form, $\Delta U = |Q| - |W|$.
• $W = p\,\Delta V$ is valid only at constant pressure. For an isothermal change you must use $nRT\ln(V_f/V_i)$.
• $\ln$ versus $\log$. The isothermal formula uses natural log; if you use $\log_{10}$, multiply by 2.303.
• $\Delta U = nC_V\Delta T$ holds in every process, including isobaric and adiabatic — not only at constant volume.
• Free expansion is adiabatic but is not the adiabatic process: $pV^{\gamma} =$ constant does not apply to it, because it is not quasi-static.
• Efficiency is a ratio and has no unit. Do not write it in joules, and do not forget to multiply by 100 when a percentage is asked.
• Do not use $\eta = 1 - T_C/T_H$ for a real engine; that is the Carnot (maximum) efficiency only.
• Coefficient of performance is not efficiency and is routinely greater than 1. That is not a violation of anything.
• In a cyclic process $\Delta U = 0$, so $Q = W$ — but $Q$ and $W$ are individually non-zero.
• Litres to m$^3$: 1 L $= 10^{-3}$ m$^3$. Forgetting this is worth an entire numerical.