5: Thermodynamic Potentials and Equilibrium

Last updated
Save as PDF

Page ID: 32025

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

Extensive and Intensive Variables

All quantities in thermodynamics fall into two types: extensive and intensive. If we consider two independent systems, quantities which add up to give the corresponding quantities for the complete system are characterized as extensive quantities. The volume \(V\), the internal energy \(U\), the enthalpy, and as we will see later on, the entropy \(S\) are extensive quantities. If we divide a system into subsystems, those quantities which remain unaltered are called intensive variables. The pressure \(p\), the temperature \(T\), the surface tension are examples of intensive variables.

In any thermodynamic system, there is a natural pairing between extensive and intensive variables. For example, pressure and volume go together as in the formula \(dU = dQ − p dV\). Temperature is paired with the entropy, surface tension with the area, etc.

5.1: Thermodynamic Potentials
The second law leads to the result dQ=TdS, so that, for a gaseous system, the first law may be written as dU=TdS−pdV. Returning to the case of a gaseous system, we now define a number of quantities related to U. These are called thermodynamic potentials and are useful when considering different processes. We have already seen that the enthalpy H is useful for considering processes at constant pressure since the inflow or outflow of heat may be seen as changing the enthalpy.
5.2: Thermodynamic Equilibrium
he second law of thermodynamics implies that entropy does not decrease in any natural process. The final equilibrium state will thus be the state of maximum possible entropy. After attaining this maximum possible value, the entropy will remain constant. We can take S to be a function of U, V and N. The system, starting in an arbitrary state, adjusts U, V and N among its different parts and constitutes itself in such a way as to maximize entropy.
5.3: Phase Transitions
If we have a single constituent, for thermodynamic equilibrium, we should have equality of T, p and µ for different subparts of the system. In general, if the first (n−1) derivatives of µ are continuous, and the n -th derivatives are discontinuous at the transition, the transition is said to be of the n -th order. Clausius-Clapeyron equation, as we have written it, applies to the first order phase transitions. These have a latent heat of transition.