2: State variables and equilibrium

Section 1: Foundations - 2: State Variables and Equilibrium

Understanding the state of a thermodynamic system and the concept of equilibrium is fundamental. The state of a system (e.g., a gas in a piston) is defined by a set of measurable properties called state variables or state functions. Common examples include pressure ($P$), volume ($V$), temperature ($T$), internal energy ($U$), and entropy ($S$). Crucially, the values of these variables depend only on the current condition of the system, not on how it arrived there. For example, the internal energy of a gas at specific $P$, $V$, and $T$ is the same whether it reached that state by heating or compression.

State variables are classified as:

1. Extensive Properties: Depend on the system's size or amount of matter (e.g., $V$, $U$, total entropy $S$, mass). They are additive for subsystems.
2. Intensive Properties: Independent of system size (e.g., $P$, $T$, density, molar entropy). These define the local conditions.

The state postulate specifies that for a simple compressible system (where only $P$-$V$ work is relevant), fixing two independent intensive properties (like $P$ and $T$, or $V$ and $T$) uniquely determines all other intensive state variables and the values of extensive properties per unit mass. For a fixed amount of substance, fixing two intensive properties (e.g., $P$ and $T$) also fixes the specific volume ($v$), and thus the extensive volume ($V$). This highlights the interdependence of state variables via an equation of state (e.g., the ideal gas law $PV = nRT$).

Changes in state variables are described mathematically by exact differentials (e.g., $dU$, $dS$). The integral of an exact differential between two states is simply the difference in the state function's value ($\Delta U = U_2 - U_1$), independent of the path taken. This path-independence contrasts sharply with path functions like work ($W$) and heat ($Q$).

A system is in thermodynamic equilibrium when its state variables are uniform and unchanging in time, without any net flows within the system or across its boundaries. Equilibrium implies a balance exists in three key aspects:

1. Thermal Equilibrium: No temperature gradients; temperature is uniform throughout.
2. Mechanical Equilibrium: No pressure gradients or unbalanced forces; pressure is uniform (unless balanced by external fields like gravity).
3. Chemical Equilibrium: No net chemical reactions or diffusion of species; chemical composition is uniform and stable.

Only when all three conditions are satisfied simultaneously is the system in a true state of thermodynamic equilibrium. This equilibrium state represents the condition towards which an isolated system naturally evolves over time. The Zeroth Law of Thermodynamics (discussed elsewhere) relies on the concept of thermal equilibrium to define temperature.