Characteristic function (physics)

The characteristic features (also characteristic potential forms called) designate the thermodynamics the total differentials (changes) of the thermodynamic potentials .

Total differentials

The inner energy

The following fundamental equation for the internal energy U is derived from the first and second law of thermodynamics :  

${\displaystyle \mathrm {d} U(S,V,n_{1},\dotsc ,n_{k})=T\mathrm {d} S-p\mathrm {d} V+\sum _{i=1}^{k}\mu _{i}\mathrm {d} n_{i}.}$

S is the entropy , V the volume, T the absolute temperature and p the pressure. stands for the amount of substance and for the chemical potential of the component . ${\displaystyle n_{i}}$ ${\displaystyle \mu _{i}}$ ${\displaystyle i}$

The enthalpy

From the definition of enthalpy  H

${\displaystyle H(S,p,n_{1},\dotsc ,n_{k})=U(S,V,n_{1},\dotsc ,n_{k})+pV}$

follows because of : ${\displaystyle \mathrm {d} (pV)=p\mathrm {d} V+V\mathrm {d} p}$

${\displaystyle \mathrm {d} H=\mathrm {d} U(S,V,n_{1},\dotsc ,n_{k})+p\mathrm {d} V+V\mathrm {d} p,}$

and with the fundamental equation one obtains

${\displaystyle \mathrm {d} H(S,p,n_{1},\dotsc ,n_{k})=T\mathrm {d} S-p\mathrm {d} V+\sum _{i=1}^{k}\mu _{i}\mathrm {d} n_{i}+p\mathrm {d} V+V\mathrm {d} p}$

and thus the characteristic function:

${\displaystyle \mathrm {d} H(S,p,n_{1},\dotsc ,n_{k})=T\mathrm {d} S+V\mathrm {d} p+\sum _{i=1}^{k}\mu _{i}\mathrm {d} n_{i}.\!}$

The free energy

From the definition of free energy  (Helmholtz energy)  F :

${\displaystyle F(T,V,n_{1},\dotsc ,n_{k})=U(S,V,n_{1},\dotsc ,n_{k})-TS}$

follows

${\displaystyle \mathrm {d} F(T,V,n_{1},\dotsc ,n_{k})=-S\mathrm {d} T-p\mathrm {d} V+\sum _{i=1}^{k}\mu _{i}\mathrm {d} n_{i}.}$

The Gibbs energy

From the definition of the Gibbs energy  (free enthalpy)  G

${\displaystyle G(T,p,n_{1},\dotsc ,n_{k})=H(S,p,n_{1},\dotsc ,n_{k})-TS}$

also follows

${\displaystyle \mathrm {d} G(T,p,n_{1},\dotsc ,n_{k})=\mathrm {d} H(S;p;N)-T\mathrm {d} S-S\mathrm {d} T,}$

and with it the characteristic function

${\displaystyle \mathrm {d} G(T;p;n_{1},\dotsc ,n_{k})=-S\mathrm {d} T+V\mathrm {d} p+\sum _{i=1}^{k}\mu _{i}\mathrm {d} n_{i}.}$

The grand canonical potential

Finally, from the definition of the grand canonical potential  for one-material systems, it follows: ${\displaystyle \Omega }$

${\displaystyle \Omega (T,V,\mu )=F(T,V,N)-\mu N,}$

that

${\displaystyle \mathrm {d} \Omega (T,V,\mu )=-S\mathrm {d} T-p\mathrm {d} V-N\mathrm {d} \mu \,.}$

Guggenheim scheme

Guggenheim Square

→ Main article : Guggenheim Square

The Guggenheim square can be used for practical work . From this one obtains all of the above-mentioned characteristic functions except for that of the grand canonical potential, which is very similar to that of free energy.

The relation is found by taking the total differential from the center of one of the four sides of the diagram and then reading the right side from the opposite corners and the two adjacent fields. At the end you always have to add the summand . ${\displaystyle \textstyle \sum \mu _{i}\mathrm {d} n_{i}}$

For example, one takes from the upper side what the total differential of the left side of the equation follows. Diagonally opposite then lies, for example, and from this in turn diagonally opposite , which leads to the expression . Analogously, one obtains the addend with the special feature that, if the coefficient of the addend is on the left side of the square, a negative sign is placed in front. However, this only applies to coefficients. It thus arises as mentioned above ${\displaystyle U}$ ${\displaystyle \mathrm {d} U}$ ${\displaystyle T}$ ${\displaystyle S}$ ${\displaystyle T\mathrm {d} S}$ ${\displaystyle -p\mathrm {d} V}$

${\displaystyle \mathrm {d} U=T\mathrm {d} S-p\mathrm {d} V+\sum _{i=1}^{k}\mu _{i}\mathrm {d} n_{i}}$.

Memories for the square can be found under: Guggenheim square (memorabilia)