Q.vi. State second law of thermodynamics in terms of entropy.

The second law of thermodynamics states that heat spontaneously flows only from a hotter to a colder body, and no heat engine can convert absorbed heat completely into useful work in a cyclic process.

The first law of thermodynamics (energy conservation) alone does not forbid processes that never actually happen in nature, such as heat flowing spontaneously from a cold body to a hot one, or a heat engine converting 100% of the heat it absorbs into work with no waste heat rejected. The second law rules these out. Two equivalent standard statements: (1) Clausius statement — it is impossible for heat to flow, of itself (without external work being done), from a body at a lower temperature to a body at a higher temperature. (2) Kelvin–Planck statement — it is impossible to construct a heat engine, operating in a cycle, whose sole effect is to absorb heat from a single reservoir an …

The second law of thermodynamics restricts the direction of natural processes and rules out perfect heat engines and perfect refrigerators; it can be stated in (equivalent) Kelvin-Planck or Clausius forms.

Kelvin-Planck statement: It is impossible to construct a device that, operating in a cycle, produces no effect other than the absorption of heat from a single reservoir and the performance of an equivalent amount of work — i.e. no engine can have 100% efficiency.

Clausius statement (equivalent): It is impossible for a self-acting device, unaided by any external agency, to transfer heat from a body at a lower temperature to a body at a higher temperature.

…

Free expansion is irreversible, and the second law of thermodynamics states that the entropy of an isolated system always increases in an irreversible process — so the gas's entropy increases even though its internal energy and temperature stay unchanged.

In free expansion, a gas rushes into an evacuated region with no external pressure to work against ($W=0$) and no heat exchange with surroundings ($Q=0$), so by the first law $\Delta U = 0$ and, for an ideal gas, temperature stays constant.

…

No engine can convert heat entirely into work with 100% efficiency (Kelvin-Planck statement); equivalently, heat cannot spontaneously flow from a colder to a hotter body on its own (Clausius statement).

The first law of thermodynamics is just a statement of energy conservation — it does not forbid processes that conserve energy but never actually happen in nature (like heat flowing spontaneously from a cold body to a hot one). The second law fills this gap by specifying the DIRECTION in which natural processes proceed.

Two commonly stated (and equivalent) forms:

1. Kelvin-Planck statement: It is impossible to construct a device (a heat engine) that, operating in a cycle, produces no other effect than the absorption of heat from a single reservoir and its complete conversion into an equivalent amount of work. In other words, no heat engine can have 100% efficiency — some heat must always be rejected to a sink. …

By the second law of thermodynamics, entropy of an isolated system increases in every natural (irreversible) process, reaching a maximum at equilibrium.

Entropy is a measure of the disorder or randomness of a system. The second law of thermodynamics states that in any natural (spontaneous) process occurring in an isolated system, the total entropy of the system either increases (for irreversible processes, which all real/natural processes are) or stays the same (only in the idealised limit of a reversible process).

…