Thermodynamic Equilibrium

Thermodynamic equilibrium: internal gradients vanished in macroscopic systems.

Conceptual context

Thermodynamic equilibrium describes a macroscopic state in which a system stops evolving because the internal gradients capable of producing spontaneous change have vanished. It does not mean that particles stop moving or that microscopic motion disappears; it means that observable magnitudes remain constant and that there is no net heat flow, no net mechanical push and no net material exchange compatible with the constraints.

In a real system, equilibrium is recognized by comparing intensive variables. Temperature $T$ diagnoses thermal equilibrium, pressure $P$ diagnoses mechanical equilibrium, and chemical potential $\mu$ diagnoses material or chemical equilibrium. The concept is central because it allows states to be defined, equations of state to be used, and final stable states to be separated from transients.

🟢 Essential level

The essential idea is that a system is in equilibrium when it no longer has internal reasons to change. If two parts have different $T$, heat can flow. If two sides of a movable boundary have different $P$, the boundary can move. If two connected regions have different $\mu$, matter can diffuse or a reaction can advance. While any of these drives remains, the system is not in complete equilibrium.

At equilibrium, macroscopic variables stop changing in time within the precision of the model. A cup of coffee in a room approaches equilibrium when its temperature equals the air temperature. A gas with a piston approaches mechanical equilibrium when internal pressure balances the effective external pressure. A reacting mixture reaches chemical equilibrium when there is no net reaction advance. In every case, the check is the same: compare the intensive variable on both sides of the boundary, and see whether any drive for further change remains.

[!NOTE] Equilibrium does not mean "no motion"; it means no net macroscopic change compatible with the constraints.

🔵 Formal level

Basic formulas

The thermal criterion is:

Thermal equilibrium condition

The same condition, read as the common state of parts in thermal contact, is:

Equal temperature at equilibrium

The mechanical criterion is:

Mechanical equilibrium condition

If the movable boundary has no macroscopic acceleration, the effective pressures pushing it are balanced:

Equal effective pressure

When material exchange is allowed, the relevant condition is not "same concentration" but equality of chemical potential:

Equal chemical potential

The general criterion is summarized by vanishing affinities:

Vanishing affinities at complete equilibrium

Real student questions

Q: If a variable does not change, is the system in equilibrium?

A: Not necessarily. It may be a steady state with constant flow. Equilibrium requires absence of driving gradients, not only constant values.

Q: Can partial equilibrium exist?

A: Yes. A system can be in thermal equilibrium but not mechanical equilibrium, or mechanical but not chemical equilibrium. Constraints must be checked.

Q: Why does chemical potential matter?

A: Matter moves or reacts because of potential differences, not only because of visible concentration differences.

Final synthesis

A system is in thermodynamic equilibrium when the internal drives of change allowed by its constraints vanish. Equality of $T$, compensation of $P$ and vanishing $A$ turn a transient process into a state describable by state variables. Complete equilibrium is stronger than being still: it requires absence of internal net flows.