The Maxwell relations

Consider the continuous function F = f(x,y). For a differential change in jc and 3 we have  

Returning now to the problem of expressing the derivative D, Eq. 9.5.11, in terms of measurable quantities, we notice that this is accomplished through Eq.9.6.6. Hence, Eqs 9.5.10 and 9.5.8 become  

Maxwell used the mathematical properties of state functions to derive a set of useful relationships. These are often referred to as the Maxwell relations. Recall the first law of thermodynamics, which may be written as 

For a reversible change in a closed system and in the absence of any non-expansion work this equation transforms into 

Since dU is an exact differential, its value is independent of the path. The same value of dU is obtained whether the change is reversible or irreversible, and eq. 

Generally, a function f(x,y) for which an infinitesimal change may be expressed 

Thus since the internal energy, U, is a state function, one of the Maxwell relations may be deduced from (eq. 1.58)  

---

The analogue of the Clapeyron equation for multicomponent systems can be derived by a complex procedure of systematically eliminating the various chemical potentials, but an alternative derivation uses the Maxwell relation (A2.1.41)... 

This is one of the Maxwell relations, and the other Maxwell relations can be derived in a similar fashion by applying Eq. (3-44). 

The reciprocity relation for an exact differential applied to Eq. (4-16) produces not only the Maxwell relation, Eq. (4-28), but also two other usebil equations ... 

The differential bZ (or dZ) can be tested for exactness by applying the Maxwell Relation given by equation (1.28) ... 

Equation (1.34) states that the order of differentiation is immaterial for the exact differential. The Maxwell relation follows directly from this property,... 

The derivative (dH/dp)T is obtained next. We start with the second Gibbs equation, divide by dp, specify constant T, substitute partial derivatives, and substitute the Maxwell relation given in equation (3.17). The result is... 

Equation (A 1.25) is known as the Maxwell relation. If this relationship is found to hold for M and A in a differential expression of the form of equation (A 1.22), then 6Q — dQ is exact, and some state function exists for which dQ is the total differential. We will consider a more general form of the Maxwell relationship for differentials in three dimensions later. 

Application of the Maxwell relations equation (Al. 28) will show that this differential is exact. Integration leads to a family of surfaces... 

In 1982, Nalewajski and Parr took the thermodynamic analogy to its logical conclusion by extending the Legendre-transform structure of classical thermodynamics to DFT [8]. One of their results was the Maxwell relation for Equation 18.6,... 

The Maxwell relations of thermodynamics relate quantities formed by differentiating G once with respect to one variable and once with respect to another (Huang, 1987). Choosing the two variables to be T and n leads to the following relationship ... 

The Maxwell relations in thermodynamics are obtained by treating a thermodynamic relation as an exact differential equation. Exact differential equations are of the form... 

EXAMPLE 6.1 Laplace Equation for Spherical Surfaces A Thermodynamic Derivation. The Maxwell relations play an important role in thermodynamics. By including the term dA in the usual differential form for cfG, show that (dVfdA)pJ = (dyidp)AJ. Evaluate (dVidA)pJ assuming a spherical surface and, from this, derive the Laplace equation for this geometry. 

Solution The derivation of the Maxwell relations treats dG as an exact differential and expands it as... 

To evaluate (dH/dP)T, we start from the expression (5.46b) for dH in terms of its natural variables S, P [or, equivalently, use the identity (1.13) to change the variable held constant from S to T, then use the Maxwell relation (5.49d) to replace the entropy derivative as follows ... 

Solution As shown in Sidebar 5.5, the desired derivative can be generally expressed [with the help of the Maxwell relation (5.49c)] as... 

The Maxwell relations are powerful tools of thermodynamic derivation. With the help of these relations and derivation techniques analogous to those illustrated in Sidebars 5.3-5.6, the skilled student of thermodynamics can (in principle ) re-express practically any partial derivative in terms of a small number of base properties involving only PVT variables. Consider, for example, the eight most common variables... 

The Maxwell relations (5.49a-d) are easy to rederive from the fundamental differential forms (5.46a-d). However, these relations are used so frequently that it is useful to employ a simple mnemonic device to recall their exact forms as needed. Sidebar 5.7 describes the thermodynamic magic square, which provides such a mnemonic for Maxwell relations and other fundamental relationships of simple (closed, single-component) systems. 

If, instead, we consider the Maxwell relation for (dS/dP)T, we rotate the square to obtain the symmetry-related comers... 

In the case where the (dX/dY)z variables do not properly match a comer pattern (i.e., do not have Y between X and Z), it is only necessary to find the Maxwell relation for the inverted form ( dY/dX)z,... 

Multiplication by R T gives the respective interaction energy terms. With the help of the Maxwell relations Q2G/QtijQtij = 02G /0/iy0n, one shows that = ej h More details can be found in [C. Wagner (1952)]. 

This equation shows that the change in flux of some quantity caused by changing the direct driving force for another is equal to the change in flux of the second quantity caused by changing the driving force for the first. These equations resemble the Maxwell relations from thermodynamics. 

Before turning to the surface enthalpy we would like to derive an important relationship between the surface entropy and the temperature dependence of the surface tension. The Helmholtz interfacial free energy is a state function. Therefore we can use the Maxwell relations and obtain directly an important equation for the surface entropy ... 

The dielectric constant of coal is strongly dependent on coal rank (van Krevelen, 1961 Speight, 1994, and reverences cited therein). For dry coals the minimum dielectric constant value is 3.5 and is observed at about 88% w/w carbon content in the bituminous coal range. The dielectric constant increases sharply and approaches 5.0 for both anthracite (92% carbon) and lignite (70% carbon). The Maxwell relation which equates the dielectric constant to the square of the refractive index for a polar insulators generally shows a large disparity even for strongly dried coal. 

---