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Thermodynamic equilibrium occurs when a system has no tendency to change its properties and is simultaneously in mechanical, thermal, and chemical equilibrium. Mechanical equilibrium requires uniform pressure; thermal equilibrium requires uniform temperature; chemical equilibrium requires uniform chemical potential throughout. A system at true thermodynamic equilibrium will not spontaneously undergo any changes in macroscopic properties.
Examine systems approaching equilibrium from non-equilibrium states: gas diffusion, temperature gradients, pressure imbalances. Identify which driving forces vanish at equilibrium.
The zeroth law gave you thermal equilibrium: two systems in contact with no heat flow have the same temperature. Temperature-and-thermal-equilibrium deepened this by connecting temperature to the tendency of energy to distribute among microstates. Thermodynamic equilibrium extends this logic to *all* the ways a system can exchange things across its boundary — not just heat, but also mechanical work (volume exchange) and matter (particle exchange). Full equilibrium requires all three driving forces to vanish simultaneously.
Mechanical equilibrium means the pressure is uniform throughout the system and equal across any boundary. If a piston separates two regions at different pressures, it accelerates — work is done, the system changes, equilibrium is absent. At mechanical equilibrium, pressure gradients vanish: there is no net force on any movable boundary. In a column of fluid in a gravitational field, the pressure gradient dp/dh = −ρg *is* the mechanical equilibrium condition (balancing gravity against the pressure gradient), so equilibrium does not require pressure to be spatially uniform if external fields are present — it requires the net force on every fluid element to vanish.
Chemical equilibrium introduces the concept of chemical potential μ. For a pure substance, μ = (∂G/∂N)_{T,P} — the Gibbs free energy per particle, or equivalently the energy cost of adding one more particle at constant temperature and pressure. Particles flow from high μ to low μ, just as heat flows from high T to low T and volume contracts to equalize P. Chemical equilibrium requires μ to be uniform for each species throughout the system. This is why a solute distributes itself equally between two connected chambers at equilibrium (if the chambers are identical) — any μ gradient drives net diffusion. For chemical reactions, equilibrium requires that the sum of reactant chemical potentials equals the sum of product chemical potentials: Σ νᵢμᵢ = 0, which is the microscopic origin of the equilibrium constant K.
The crucial point is that all three conditions must hold *simultaneously* for true thermodynamic equilibrium. A system can be thermally equilibrated (uniform T) but still have a pressure gradient driving flow — it is in thermal but not mechanical equilibrium. A mixture might be at uniform T and P but still react — it is not in chemical equilibrium. Real systems are often in partial or approximate equilibrium: a gas in a thermally insulated rigid container is in thermal and mechanical equilibrium, but if two reactive species are present and the reaction is slow, chemical equilibrium may take centuries. Thermodynamics describes the *final* state; kinetics governs how fast (or whether) the system gets there. Recognizing which equilibrium conditions are satisfied — and which are not — is the first step in any thermodynamic analysis.