Maxwell relationship

This article deals with the relationships between variables of state in thermodynamics. For the Maxwell equations of electrodynamics see Maxwell equations .

The Maxwell relations or Maxwell relations of thermodynamics (after the physicist James Clerk Maxwell ) establish important relationships between different state variables .

statement

Maxwell's relationships allow changes in state variables (e.g. temperature  T or entropy  S ) to be expressed as changes in other state variables (e.g. pressure  p or volume  V ):

${\displaystyle \left({\frac {\partial T}{\partial V}}\right)_{S}=-\left({\frac {\partial p}{\partial S}}\right)_{V};\qquad \left({\frac {\partial T}{\partial p}}\right)_{S}=\left({\frac {\partial V}{\partial S}}\right)_{p}}$

${\displaystyle \left({\frac {\partial V}{\partial T}}\right)_{p}=-\left({\frac {\partial S}{\partial p}}\right)_{T};\qquad \left({\frac {\partial p}{\partial T}}\right)_{V}=\left({\frac {\partial S}{\partial V}}\right)_{T}}$

Exemplary derivation

The relationships can be derived by the one Characteristic features ( total differentials ) of the state functions internal energy  U , enthalpy  H , free energy  F and Gibbs  G considered.

For example, the total differential of the internal energy  U , depending on entropy  S and volume  V, is :

${\displaystyle {\begin{aligned}\mathrm {d} U(S,V)&=T\mathrm {d} S-p\mathrm {d} V\\&=\left({\frac {\partial U}{\partial S}}\right)_{V}\mathrm {d} S+\left({\frac {\partial U}{\partial V}}\right)_{S}\mathrm {d} V\end{aligned}}}$

If one assumes a sufficiently smooth function for U , then Black's theorem says that

${\displaystyle \left({\frac {\partial T}{\partial V}}\right)_{S}=\left({\frac {\partial }{\partial V}}\left({\frac {\partial U}{\partial S}}\right)_{V}\right)_{S}=\left({\frac {\partial }{\partial S}}\left({\frac {\partial U}{\partial V}}\right)_{S}\right)_{V}=-\left({\frac {\partial p}{\partial S}}\right)_{V}}$.

This is the first Maxwell relationship.

Guggenheim scheme

Guggenheim Square

The so-called Guggenheim square can be used for practical work . All of the Maxwell relations mentioned above are obtained from this.

The relation is found by reading two variables from the corners of one (horizontal or vertical) side of the diagram, so that one side of the Maxwell equation is formulated and the other side of the equation is taken from the opposite side in the same way.

For example, one takes and , from which the expression follows. Opposite are then and , which leads to expression . Differential quotients that contain both and are given a negative sign , since both (!) Symbols are on the edge with the minus sign (in the above example ). The constant held variable on one side can always be found in the denominator of the other side. ${\displaystyle S}$ ${\displaystyle p}$ ${\displaystyle \mathrm {d} S/\mathrm {d} p}$ ${\displaystyle V}$ ${\displaystyle T}$ ${\displaystyle \mathrm {d} V/\mathrm {d} T}$ ${\displaystyle S}$ ${\displaystyle p}$ ${\displaystyle -(\mathrm {d} S/\mathrm {d} p)=(\mathrm {d} V/\mathrm {d} T)}$

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

General Maxwell relation

If a function z (x, y) according to the theorem about the implicit function is uniquely resolvable for both x and y at one point , it can be shown, among other things, that

${\displaystyle {\frac {\partial x}{\partial y}}{\frac {\partial y}{\partial z}}{\frac {\partial z}{\partial x}}=-1}$.

To show this, one starts with the total differentials of the functions z and x .

${\displaystyle \mathrm {d} z=\left({\frac {\partial z}{\partial x}}\right)_{y}\mathrm {d} x+\left({\frac {\partial z}{\partial y}}\right)_{x}\mathrm {d} y}$

${\displaystyle \mathrm {d} x=\left({\frac {\partial x}{\partial z}}\right)_{y}\mathrm {d} z+\left({\frac {\partial x}{\partial y}}\right)_{z}\mathrm {d} y}$

Insertion results

${\displaystyle \mathrm {d} z=\left({\frac {\partial z}{\partial x}}\right)_{y}\left(\left({\frac {\partial x}{\partial z}}\right)_{y}\mathrm {d} z+\left({\frac {\partial x}{\partial y}}\right)_{z}\mathrm {d} y\right)+\left({\frac {\partial z}{\partial y}}\right)_{x}\mathrm {d} y}$

The partial differentials can be truncated if the variables held are the same.

${\displaystyle \mathrm {d} z=\mathrm {d} z+\left({\frac {\partial z}{\partial x}}\right)_{y}\left({\frac {\partial x}{\partial y}}\right)_{z}\mathrm {d} y+\left({\frac {\partial z}{\partial y}}\right)_{x}\mathrm {d} y}$

${\displaystyle -\left({\frac {\partial z}{\partial y}}\right)_{x}=\left({\frac {\partial z}{\partial x}}\right)_{y}\left({\frac {\partial x}{\partial y}}\right)_{z}}$

${\displaystyle -1=\left({\frac {\partial x}{\partial y}}\right)_{z}\left({\frac {\partial y}{\partial z}}\right)_{x}\left({\frac {\partial z}{\partial x}}\right)_{y}}$