Guggenheim Square

Guggenheim Square

The Guggenheim square or Guggenheim scheme (after Edward Guggenheim ) is a tool to establish some simple but fundamental relationships in thermodynamics , such as the characteristic functions or the Maxwell relationships , from memory.

The thermodynamic potentials are linked

• inner energy  ${\displaystyle U}$
• Enthalpy  ${\displaystyle H}$
• free energy  ${\displaystyle F}$
• Gibbs energy  ${\displaystyle G}$

at the middle of the edges

with the state variables

• entropy  ${\displaystyle S}$
• volume  ${\displaystyle V}$
• pressure  ${\displaystyle p}$
• temperature  ${\displaystyle T}$

at the corners of the square.

use

Characteristic functions

The characteristic function, so the total differential of the four thermodynamic potentials , , , , obtain, proceed as follows: ${\displaystyle U}$ ${\displaystyle H}$ ${\displaystyle F}$ ${\displaystyle G}$

1. Selection of a thermodynamic potential; for the relation of the grand-canonical potential , the comparison with the very similar free energy offers itself . ${\displaystyle \Omega }$ ${\displaystyle F}$
   • Example: We are looking for the total differential of the internal energy , that is ${\displaystyle U}$ ${\displaystyle \mathrm {d} U=?}$
2. The two symbols that are opposite the potential sought in the corners represent the coefficients of those differentials that are located at the corners next to the potential sought.
   • In the example, the pressure and the temperature are at the opposite corners . The preliminary interim result is thus ${\displaystyle U}$ ${\displaystyle p}$ ${\displaystyle T}$ ${\displaystyle \mathrm {d} U=-p\cdot [{\text{Differential}}]+T\cdot [{\text{Differential}}]}$
   • The associated differential is located in the diagonally opposite corner and is therefore the volume ; Intermediate result: ${\displaystyle p}$ ${\displaystyle V}$ ${\displaystyle \mathrm {d} U=-p\cdot \mathrm {d} V+T\cdot [{\text{Differential}}]}$
   • Similar to the previous step, the entropy is located diagonally opposite as the associated differential ; Intermediate result: ${\displaystyle T}$ ${\displaystyle S}$ ${\displaystyle \mathrm {d} U=-p\cdot \mathrm {d} V+T\cdot \mathrm {d} S}$
3. All the coefficients on the left side of the square are given a negative sign .
   • Since it is on the left side of the square, it is given a negative sign. Interim result: . (It is also on the left, but it only occurs as a differential and not as a coefficient and therefore does not have a negative sign. Similarly, when searching for, no negative sign is to be placed in front of the -term, since it is not a coefficient here either .) ${\displaystyle p}$ ${\displaystyle \mathrm {d} U=-p\cdot \mathrm {d} V+T\cdot \mathrm {d} S}$ ${\displaystyle S}$ ${\displaystyle \mathrm {d} H}$ ${\displaystyle V\cdot \mathrm {d} p}$
4. Finally, the term for the chemical potential (with the number of particles ) is always added. ${\displaystyle \mu \cdot \mathrm {d} N}$ ${\displaystyle N}$
   • In the example this gives the final result ${\displaystyle \mathrm {d} U=-p\cdot \mathrm {d} V+T\cdot \mathrm {d} S+\mu \cdot \mathrm {d} N}$

Maxwell relations

1. Two state variables are to be selected that lie at the two corners of a common side of the square.
   • Example: We are looking for a Maxwell relation with and , which form the corners of the left edge. These form the differential quotient of the left side of the sought Maxwell relation, i.e. ${\displaystyle S}$ ${\displaystyle p}$ ${\displaystyle {\frac {\partial S}{\partial p}}=[\dotsc ]}$
2. The state variables that limit the opposite side of the square form the differential quotient on the right-hand side of the Maxwell equation sought. Make sure that they are read in the same direction as the first edge.
   • Across from and are and . We formed, that is, "derived the upper corner from the lower corner". Accordingly, "from top to bottom" must also be derived on the other side of the square. So is the intermediate result . (The same applies to left / right, for example when searching for ). ${\displaystyle S}$ ${\displaystyle p}$ ${\displaystyle V}$ ${\displaystyle T}$ ${\displaystyle {\frac {\partial S}{\partial p}}}$ ${\displaystyle {\frac {\partial S}{\partial p}}={\frac {\partial V}{\partial T}}}$ ${\displaystyle {\frac {\partial S}{\partial V}}}$
3. Differential quotients that contain both and are given a negative sign, since both (!) Symbols are on the edge with the minus sign. ${\displaystyle S}$ ${\displaystyle p}$
   • The left side is therefore given a negative sign. So the intermediate result is ${\displaystyle -{\frac {\partial S}{\partial p}}={\frac {\partial V}{\partial T}}}$
4. The constant held variable on one side can always be found in the denominator of the other side.
   • So the bottom line is ${\displaystyle -\left({\frac {\partial S}{\partial p}}\right)_{T}=\left({\frac {\partial V}{\partial T}}\right)_{p}}$

State variable as a differential quotient

1. Select a size that is in one corner of the square.
   • Example: We are looking for descriptions of as a differential quotient or derivative. is on the left side of the square, so ${\displaystyle S}$ ${\displaystyle S}$ ${\displaystyle -S=[\dotsc ]}$
2. The symbol diagonally opposite the quantity represents the denominator of the derivatives.
   • Opposite lies , so ${\displaystyle S}$ ${\displaystyle T}$ ${\displaystyle -S={\frac {[{\text{Zähler}}]}{\partial T}}}$
3. The two symbols adjacent to the denominator each form the numerator of a derivative.
   • Adjacent to lie and , therefore applies and ${\displaystyle T}$ ${\displaystyle F}$ ${\displaystyle G}$ ${\displaystyle -S={\frac {\partial F}{\partial T}}}$ ${\displaystyle -S={\frac {\partial G}{\partial T}}}$
4. The third symbol on the sides of the numerator and denominator is the quantity that remains constant.
   • On the side of and is so true ; along side and is so true ${\displaystyle T}$ ${\displaystyle F}$ ${\displaystyle V}$ ${\displaystyle -S=\left({\frac {\partial F}{\partial T}}\right)_{V}}$ ${\displaystyle T}$ ${\displaystyle G}$ ${\displaystyle p}$ ${\displaystyle -S=\left({\frac {\partial G}{\partial T}}\right)_{p}}$

(Here, too, quantities that are on the left-hand side, but only occur in the differential quotient, do not have a negative sign!)

State variable minimized in equilibrium

1. Two neighboring corners of the Guggenheim square are to be selected, which are controlled during the process, i.e. kept constant.
   • Example: Entropy and pressure should be kept constant, so the left-hand corners were selected. ${\displaystyle S}$ ${\displaystyle p}$
2. The size minimized in equilibrium is now between the selected corners.
   • So in the example ${\displaystyle {\frac {\partial S}{\partial t}}={\frac {\partial p}{\partial t}}=0\Rightarrow H\rightarrow \min }$

Memos

To make it easier to use, the following sample selection of mostly humorous memos , which, read in different ways, reproduce the sequence of letters in the square:

• S eid h oday p ünktlich, u n-th g ibt's v everal f generic T omaten.
• S chnell u nd v iel h ilft f ÜR P AUDITS G uggenheims T at.
• S ven u nder V arus h atten f raditionally / a lle p RACTICAL (n) G ermanen T ash computer.
• G ood p hysicists h ave s tudied u Direction v ery f ine t eachers.
• U nheimlich v everal F esearchers t Drink g erne P ils h interm S chreibtisch.
• G ute P hysiker h ave s tets / s -line e ine V orliebe f ÜR T hermodynamik.
• P rinzipiell h ave s Chon u ur V orfahren f ÜR T hermodynamik g esch heated.
• SUV - F Ahrer t protrude g erne p inke H emden.

In addition to these sayings, there are numerous others.

Memory aids for three degrees of freedom

Thermodynamic octahedron

The Guggenheim square describes systems with two degrees of freedom . Memory aids in the form of the geometric figures octahedron and cuboctahedron have been described for three degrees of freedom . In contrast to the square, the thermodynamic potentials (G, U, H, F etc.) are not edges, but surfaces.

Web links

• The Guggenheim scheme. (No longer available online.) Archived from the original on July 24, 2007 ; accessed on September 15, 2011 . 

literature

• Jibamitra Ganguly: Thermodynamics in earth and planetary sciences . 2009, ISBN 978-3-540-77306-1 , Thermodynamic Square: A Mnemonic Tool, pp. 59–60 ( limited preview in Google Book search).
• Wedler, Gerd: Textbook of physical chemistry . 2nd Edition. VCH, 1985, ISBN 3-527-29481-3 , 2.3.2 - Characteristic thermodynamic functions, p. 252-256 .

Individual evidence

1. ^ LT Klauder, American Journal of Physics , 1968 , 36 (6), 556-557 doi: 10.1119 / 1.1974977
2. ↑ James M. Phillips, J. Chem. Educ. , 1987 , 64 (8), 674-675 doi: 10.1021 / ed064p674
3. ↑ Ronald. F. Fox, J. Chem. Educ. , 1976 , 53 (7), 441-442 doi: 10.1021 / ed053p441