Thermodynamics and heat engines

Thermodynamics is the physics of heat and work. The first law is conservation of energy for gases: heat supplied either raises internal energy or is used to do work. The second law limits every heat engine: some heat must always be passed to a colder place, so no engine can be 100% efficient.

Key facts

  • First law: ΔU = Q − W (Q = heat supplied to the gas, W = work done BY the gas).
  • Work done by a gas at constant pressure: W = pΔV (area under a p–V graph).
  • Isothermal: ΔU = 0. Adiabatic: Q = 0. Constant volume: W = 0.
  • Heat engine: Q_h = W + Q_c; efficiency = W / Q_h.
  • Maximum (Carnot) efficiency = 1 − T_c / T_h (kelvin).
  • Refrigerator COP = Q_c / W; heat pump COP = Q_h / W. Entropy change ΔS = Q / T at constant temperature.

Worked examples

Example 1

A gas is given 800 J of heat and does 300 J of work. Find the change in internal energy.

  1. The work done by the gas is energy leaving it.
  2. What is left stays inside as internal energy. ΔU = 800 − 300 = 500 J

Answer: 500 J

Example 2

A gas expands at a constant 200 kPa from 3.0 L to 5.0 L. Find the work done.

  1. Work = area under the p–V graph: a rectangle. ΔV = 2.0 L = 2.0 × 10^−3 m³
  2. Use pascals and cubic metres. W = 200 000 × 2.0 × 10^−3 = 400 J

Answer: 400 J

Example 3

A power station boiler runs at 527°C and the river is at 27°C. Find the maximum efficiency.

  1. Convert to kelvin. T_h = 800 K, T_c = 300 K
  2. Heat flows in from the hot side; some becomes work, the rest must go to the cold side.
  3. Maximum efficiency: e = 1 − 300 ÷ 800 = 0.625 = 62.5%

Answer: 62.5%