Entropy

Entropy is a somewhat obscure quantity relating to the exchange of heat. The change in entropy for a quasistatic process is defined by¹

𝑑 𝑆 = đ 𝑄 𝑇

𝑆 = 𝑘 𝐵 𝐻 [𝑀]

Entropy is a quantity which increases during any Irreversible process. For a real process

𝑑 𝑆 > đ 𝑄 𝑇

Thermodynamic entropy postulates

In thermodynamics, the following properties are postulated:

The entropy $𝑆$ is a well-defined quantity for equilibrium states as a function of the extensive parameters of a system, e.g. $𝑆 = 𝑆 (𝐸, 𝑉, ⃗ 𝐍)$ .
The entropy of a composite system is the sum of the entropies of its subsystems, i.e. entropy is an extensive parameter.
In an infinitesimal quasistatic process the change in entropy is $𝑑 𝑆 = đ 𝑄 / 𝑇$ .
Entropy maximum principle:² For an isolated system, the entropy can never decrease, moreover if an internal constraint is removed, the final equilibrium state is that which maximizes entropy.

Entropy $𝑆 (𝐸, 𝑉, ⃗ 𝐍)$ is the naural Thermodynamic potential for a closed thermodynamic system. Applying the ^Quasistatic,

𝑑 𝑆 = 1 𝑇 𝑑 𝐸 + 𝑝 𝑇 𝑑 𝑉 - \sum 𝑖 𝜇 𝑖 𝑇 𝑑 𝑁 𝑖

whence

𝜕 𝑆 𝜕 𝐸 = 1 𝑇 𝜕 𝑆 𝜕 𝑉 = 𝑝 𝑇 𝜕 𝑆 𝜕 𝑁 𝑖 = - 𝜇 𝑖 𝑇

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There is an implicit claim that $𝑑 𝑆$ is an exact differential and hence the quantity $𝑆$ is well-defined for an equilibrium state. For example, see Entropy of an ideal gas. ↩
Essentially the Second law of thermodynamics ↩