The free energy as an example for a thermodynamic potential

3.18 The free energy as an example for a thermodynamic potential

The free energy is the corresponding thermodynamic potential for the thermal contact of section 3.15.
Therefore the corresponding coordinates are \(V, N\), and \(T\).
It is well known that

\begin{equation*} F(V,N,T) = U(V,N,T) - S(V,N,T) T \end{equation*}

(3.32)

and

\begin{equation*} dF = \mu dN - p dV - S dT \label{eq:dF} \end{equation*}

(3.33)

Mathematical interpretation: total differential, partial derivative


\(\left.\frac{\partial F}{\partial N}\right\|_{V,T}= \mu\)	\(\left.\frac{\partial F}{\partial V}\right\|_{N,T}= - p\)	\(\left.\frac{\partial F}{\partial T}\right\|_{V,N}= - S\)

Physical interpretation: gradient, forces


\(\mu\): ”force” changing the particle number	\(-p\): ”force” changing the volume	\(-S\): ”force” changing the temperature

here:	\(V, N, T\):	generalized coordinates
	\(-p, \mu, -S\):	generalized forces

For ”normal” physical quantities:
Coordinates: extensive parameters
Forces: intensive parameters
Forces and coordinates can be exchanged. There exists no fundamental difference between these parameters since they are equivalent and just depend on the choice of the thermodynamic contact.