Thermodynamics

📚The physics

The first law is conservation of energy for gases:

Thermal energy supplied to a gas either raises its internal energy or is used by the gas doing work on its surroundings, or both. The signs are where marks are lost, so fix them once: \(Q\) is positive when energy goes in to the gas; \(W\) is positive when the gas expands and does work on the surroundings; \(\Delta U\) is positive when the temperature rises. For an ideal gas \(\Delta U\) depends only on temperature, which is a shortcut worth remembering.

Work done by a gas at constant pressure is \( W = p\Delta V \). When pressure varies, the work is the area under the curve on a \(pV\) diagram — the same “area under a graph” idea you met with force–distance. Expansion moves right on the diagram and the gas does positive work; compression moves left and work is done on the gas.

The four processes are best learned by asking which term of the first law vanishes.

• Isovolumetric (constant \(V\)): no volume change, so \( W = 0 \) and \( Q = \Delta U \). All the energy goes into internal energy. Vertical line on a \(pV\) diagram.
• Isobaric (constant \(p\)): \( W = p\Delta V \) and all three terms are generally non-zero. Horizontal line.
• Isothermal (constant \(T\)): for an ideal gas \( \Delta U = 0 \), so \( Q = W \) — every joule supplied leaves as work. Curve following \( pV = \) constant. Requires slow change and good thermal contact.
• Adiabatic (no thermal transfer): \( Q = 0 \), so \( \Delta U = -W \). An expanding gas does work at the expense of its own internal energy, so it cools. Steeper than an isotherm on a \(pV\) diagram. Requires rapid change or good insulation.

That adiabatic cooling is not abstract: it is why a rapidly released aerosol feels cold, and why rising air cools and forms cloud.

Closed cycles. Round a complete cycle the gas returns to its starting state, so \( \Delta U = 0 \) over the cycle and \( Q = W \). The net work is the area enclosed by the loop. Clockwise loops do net work on the surroundings — that is an engine. Anticlockwise loops require work to be put in — that is a refrigerator or heat pump.

The second law and entropy. Entropy is a measure of the number of ways the microscopic state can be arranged — loosely, of disorder. The second law says that in any real process the total entropy of the system and its surroundings never decreases. Entropy change is \( \Delta S = \Delta Q / T \). A local decrease is entirely allowed — a freezer makes ice — provided a larger increase happens elsewhere, in that case in the kitchen.

Heat engines. An engine takes \(Q_H\) from a hot reservoir, does useful work \(W\), and dumps \(Q_C\) into a cold reservoir.

The second law forbids \( \eta = 1 \): some energy must be rejected to the cold reservoir. The best possible efficiency, achieved only by an idealised reversible Carnot cycle, uses temperatures in kelvin. No engine between the same two temperatures can beat it, whatever it is made of.