The first law of thermodynamics
Section: Thermodynamics | Syllabus: Cambridge A Level Physics (9702)
The First Law: Δ U = q + W
The first law of thermodynamics is a specific case of the law of conservation of energy. It states that the increase in internal energy of a system is the sum of the thermal energy supplied to it and the work done on it.
Δ U = q + W Sign Convention (Exam Standard)- Δ U: + for increase in internal energy, - for decrease.
- q: + for thermal energy entering the system (heating), - for leaving.
- W: + for work done ON the system (compression), - for work done BY the system (expansion).
Historical Context: Rumford and Joule
Before the 1840s, heat was thought to be a fluid called 'caloric'. Count Rumford noticed that heat continued to be produced during the boring of cannons as long as work was done, suggesting heat was related to motion. James Joule later proved this by using a falling weight to turn a paddle in water, showing a precise numerical relationship between work and temperature rise.
Worked Examples
Worked Example 1: Cyclic Path (Q13)Question: A gas is taken from state A to state B. In state A, U_A = 100 J. Along path 1, 200 J of heat is supplied and 60 J of work is done BY the gas. (a) Find U_B. (b) Along path 2, 100 J of heat is supplied. Find the work done.
Solution(a) Path 1: q = +200 J, W = -60 J.
Δ U = q + W = 200 - 60 = 140 J.
U_B = U_A + Δ U = 100 + 140 = 240 J.
(b) Internal energy is a state function, so Δ U is still 140 J for path 2.
Δ U = q + W 140 = 100 + W W = +40 J.
(Work is done ON the gas).
Question: 500 J of work is done to compress a gas. If the process is adiabatic (no heat exchange), find Δ U. If the gas is then allowed to cool back to its original temperature, find the heat transfer.
Solution1. Adiabatic compression: q = 0, W = +500 J.
Δ U = q + W = 0 + 500 = +500 J.
2. Cooling to original temperature: Δ U_total = 0 (state function).
Δ U_total = q_net + W_net 0 = (0 + q_cooling) + 500.
q_cooling = -500 J (heat leaves the system).
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