How to derive the fundamental thermodynamic relation
- 1). Recall or state the first and second laws of thermodynamics. The first law is (change in internal energy) = (change in heat content) - (amount of work done by the system on its surroundings). Use the Greek letter delta (") to represent small changes, and write this more briefly as "U = "Q - P"V. The second law states in part that "S = "Q/T, where "Q is the increase in the system's heat content. If the system cools, then by convention "S is negative. T must here use an absolute temperature scale, meaning that its value at zero is where microscopic kinetic motion from heat stops.
- 2). Rewrite "S = "Q/T so that you can introduce it into the first law. In other words, rewrite it as "Q = T"S.
- 3). Introduce the result of Step 2 into the first law of thermodynamics. In other words, "U = "Q - P"V becomes "U = T"S - P"V. This is the fundamental thermodynamic relation.