# Entropy, temperature, and a stated constraint

Authored explanation; editorial review pending.

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.

$$
dS=\frac{dE}{T},\qquad\left(\frac{\partial S}{\partial E}\right)_V=\frac{1}{T}
$$

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.

[Foundation: partial-derivatives](/foundations/partial-derivatives/) — Return to the constraints on the entropy derivative.

## 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.

State what is held fixed, distinguish total entropy from entropy density, and supply an integration condition before interpreting a volume-dependent difference.