Maxwells Relations

As established by the first law, the key feature of energy and other Legendre-transformed thermodynamic potentials is their state character (Section 2.10), i.e., their conservation under cyclic changes of state. For the leading potentials (U, H, A, G) of chemical interest, the differentials of these conserved quantities are given at equilibrium (under the usual conditions of PV-work only) by the expressions 

The first law is expressed most succinctly by the mathematical requirement that such differentials are exact (Section 1.3). 

As described in Section 1.3, the mathematical condition for exactness of a 2-variate differential dZ of the general form 

TABLE 5.2 Coefficients M, N for a General Differential dZ = MdX + NdY of Thermodynamic Potentials U, H, A, G (Closed Single-Component System) 

For the differentials dU, dH, dA, dG, the appropriate coefficients M, A can be read directly from the expressions (5.46a-d), as shown in Table 5.2. 

For reversible processes where the pressure-volume work is the only form of work, the change in T, P, or V is related. We start out from 

Other relations of the same type may be obtained from Equations 5.39 through 5.42. 

For practical purposes, the molar Gibbs free energy p(po, To) of a compound in its standard state (its state at pressure po = 1 bar, To = 298.15 K) may be defined as follows  

Since chemical thermodynamics assumes there is no interconversion between the elements, the Gibbs free energy of elements may be used to define the zero with respect to which the Gibbs free energies of all other compounds are measured. 

The molar Gibbs free energy at any other p and T can be obtained using (5.3.3) and (5.3.4) as shown in the figure below. 

The two laws of thermodynamics establish energy and entropy as functions of state, making them functions of many variables. As we have seen. 

U = U S, V Nic) and S = S U, V,Nk) are functions of the indicated variables. James Clerk Maxwell (1831-1879) used the rich theory of functions of many variables to obtain a large number of relations between thermodynamic variables. The methods he employed are general and the relations he obtained are called the Maxwell relations. 

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From cross-differentiation identities one can derive some additional Maxwell relations for partial molar quantities ... 

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)... 

Two other Maxwell relations define the direction systems change to achieve equilibrium ... 

Equations (2.15) or (2.16) are the so-called Stefan-Maxwell relations for multicomponent diffusion, and we have seen that they are an almost obvious generalization of the corresponding result (2.13) for two components, once the right hand side of this has been identified physically as an inter-molecular momentum transfer rate. In the case of two components equation (2.16) degenerates to... 

There are n Stefan-Maxwell relations in an n-component mixture, but they are not independent since each side of (2.16) yields zero on summing over r from 1 to n. Physically this is not surprising, since they describe only momentum exchange between pairs of species, and say nothing about the total momentum of the mixture. In order to complete the determination of the fluxes N.... N the Stefan-Maxwell relations must be supple-I n... 

At the opposite limit of bulk diffusion control and high permeability, all flux models are required to he consistent with the Stefan-Maxwell relations (8.3). Since only (n-1) of these are independent, they are insufficient to determine all the flux vectors, and they permit the problem to be formulated in closed form only when they can be supplemented by the stoichiometric relations (11.3). At this limit, therefore, attention must be restricted from the beginning to those simple pellet shapes for ich equations (11.3) have been justified. Furthermore, since the permeability tends to infininty, pressure gradients within the pellet tend to zero and... 

A third approach is suggested by Hugo s formulation of material balances at the limit of bulk diffusion control, described in Section 11.3. Hugo found expressions for the fluxes by combining the stoichiometric conditions and the Stefan-Maxvell relations, and this led to no inconsistencies since there are only n - 1 independent Stefan-Maxwell relations for the n fluxes. An analogous procedure can be followed when the diffusion is of intermediate type, using the dusty gas model equations in the form (5.10) and (5.11). Equations (5.11), which have the following scalar form ... 

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

Replacement of each of the four partial derivatives through the appropriate Maxwell relation gives finely... 

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... 

Verify Eq. (53) and derive the other three Maxwell relations, namely,... 

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... 

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

Assuming an isotropic system, the following Maxwell relation can be derived from eq. (2.19), since dA is an exact differential ... 

The second Maxwell relation (Equation (4.38)) may remind us of the form of the Clausius equality (see p. 142). Although the first Maxwell relation (Equation (4.37)) is not intuitively obvious, it will be of enormous help later when we look at the changes in G as a function of pressure. 

We have already obtained the first Maxwell relation (Equation (4.37)) by comparing the Gibbs-Duhem equation with the total differential ... 

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,... 

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