3.23
Overview of thermodynamic potentials
Following the second law, the driving force for any change of state is the maximization
of the total entropy. But depending on the type of contact of the system different sets of parameters control the experiment.
Each change within a pair of parameters coordinate \(\Leftrightarrow\) force needs a Legendre transformation
to get the proper thermodynamic potential (state function). We will repeat and summarize most aspects when discussing the
Gibbs potential as the most relevant thermodynamic potential to describe chemical reactions:
|
| | \begin{equation*} \begin{split}
G = &\;\; H - TS = F + pV = U + pV - TS\\ dG = &\;\; d \left( U + pV - TS \right) \\ dG - V dp + S dT = &\;\; dU + p dV
- T dS \leq 0 \quad \Rightarrow \quad \left( dG \right)_{p,T} \leq 0\\ dG = &\;\; \mu dN + V dp - S dT\\ \end{split} \label{eq:def_Gibbs}
\end{equation*} | (3.58) |
The four thermodynamic potentials (state functions) discussed here have in common that the particle number, resp. the
mole number \(N\) is a coordinate. So the Gibbs potential \(G\) is twice the Legendre transformed
of the inner energy \(U\) and is a function of the natural coordinates \(G(N,p,T)\).
In summary:
| | | |
| State function
| Natural Variable
| Spont. Process
| Equilibrium |
| | | |
| \(S\) | \(U, V\) | \(\left(dS\right)_{U, V} \gt 0\) | \(S\) = Max. |
| | | |
| \(U\) | \(S, V\) | \(\left(dU\right)_{S, V} \lt 0\) | \(U\) = Min. |
| | | |
| \(H\) | \(S, p\) | \(\left(dH\right)_{S, p} \lt 0\) | \(H\) = Min. |
| | | |
| \(F\) | \(T, V\) | \(\left(dF\right)_{T, V} \lt 0\) | \(F\) = Min. |
| | | |
| \(G\) | \(T, p\) | \(\left(dG\right)_{T, p} \lt 0\) | \(G\) = Min. |
| | | |
| |
We will briefly discuss the meaning of \(\left(dG\right)_{T,p}\):
|
| | \begin{equation*} \begin{split}
\left(dG\right)_{T,p} = &\;\;d\left(U + pV - TS \right)_{T,p}\\ = &\;\; \left(\delta q + \delta w_{el} + \delta w_{chem}
+ \delta w_{expansion} +\delta w_{...} + p dV + V dp - T dS - S dT\right)_{T,p}\\ = &\;\; \left(\delta w_{el} + \delta w_{chem}
+\delta w_{...} \right) \quad \mbox{since} \quad \delta q = T dS \quad \mbox{and} \quad \delta w_{expansion} = - p dV \end{split}
\label{eq:dG_meaning} \end{equation*} | (3.59) |
So \(\left(\Delta G\right)_{T,p}\) is the maximum non-expansion work done by the system and it is the right
quantity to describe chemical and electrochemical equilibrium.
Correspondingly we find for \(\left(dF\right)_{T,V}\): Maximum work (including expansion work) done by the
system.
© J. Carstensen (TD Kin I)
