Foundation · Explanatory preview
Entropy, temperature, and a stated constraint
At fixed volume and other stated constraints, the equilibrium entropy derivative with respect to energy is inverse absolute temperature. The derivative does not determine an additive constant.
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Why does inverse temperature describe entropy gained per added energy?
Entropy is a state quantity. For a reversible transfer of heat into a system at absolute temperature T, the transferred entropy is heat divided by T. At fixed volume, with no other work or exchanged matter, that heat transfer changes the internal energy.
At fixed volume and the stated constraints, entropy change is energy change divided by absolute temperature.
The subscript matters: allowing work from changing volume introduces an additional term. An entropy derivative describes a local change along specified constraints, not every possible process.
When energy is shared among equilibrium subsystems, a redistribution that conserves total energy cannot increase the already maximized entropy. The entropy slopes are therefore equal, so the temperatures agree.
Open the foundation: Partial derivatives and held-fixed quantities
One worked example
An ideal reservoir remains at 300 kelvin while it receives 3 joules reversibly. Its entropy increases by 0.01 joule per kelvin. For a finite body whose temperature changes, integrate dE/T(E) instead of dividing by an arbitrarily selected temperature.
Two proposed entropy functions differing by a constant have the same derivative. A boundary condition is needed to select between them. If the unresolved quantity is an entropy density, multiplying it by different volumes can make that constant matter to a total entropy difference.
A stopping point: State what is held fixed, distinguish total entropy from entropy density, and supply an integration condition before interpreting a volume-dependent difference.
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