Maxwell reciprocal relations

Application of the Maxwell reciprocal relations to Equation (2.19) yields... 

Consider the combined first and second laws in terms of the Gibbs free energy, Equation (2.21). How many Maxwell reciprocal relations can be obtained from this equation Write each of them and comment on their physical significance. 

In addition, the common Maxwell equations result from application of the reciprocity relation for exact differentials ... 

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 four relations (I.)—(IV.) are usually known as Maxwell s Relations, or the Reciprocal Relations they were deduced by Maxwell by means of an ingenious geometrical method Theory of Heat, Chap. 9). 

The synthesis of ideas described above was first achieved by Maxwell and is described by two equations that summarize the essential experimental observations pertaining to the reciprocally related electromagnetic effects1. In SI units and vector notation these equations, valid at every ordinary point... 

Thus, the Maxwell-Stefan diffusion coefficients satisfy simple symmetry relations. Onsager s reciprocal relations reduce the number of coefficients to be determined in a phenomenological approach. Satisfying all the inequalities in Eq. (6.12) leads to the dissipation function to be positive definite. For binary mixtures, the Maxwell-Stefan dififusivity has to be positive, but for multicomponent system, negative diffusivities are possible (for example, in electrolyte solutions). From Eq. (6.12), the Maxwell-Stefan diffusivities in an -component system satisfy the following inequality... 

For 1 mol of a homogeneous fluid of constant composition, Eqs. (4-6) and (4-11) through (4-13) simplify to Eqs. (4-14) through (4-17) of Table 4-1. Because these equations are exact differential expressions, application of the reciprocity relation for such expressions produces the common Maxwell relations as described in the subsection Multi-variable Calculus Applied to Thermodynamics in Sec. 3. These are Eqs. (4-18) through (4-21) of Table 4-1, in which the partial derivatives are taken with composition held constant. 

An important set of identities obtained from the Euler reciprocity relation and thermodynamic equations is the set of Maxwell relations. These relations allow you to replace a partial derivative that is difficult or impossible to measure with one that can be measured. One of the Maxwell relations is ... 

For closed systems (dXi = 0), applying the reciprocity relation ( 2.2.6) to equations (14.28)-(14.31) results in the set of Maxwell s equations, which are often useful in manipulating thermodynamic equations. These are... 

The reciprocity relation can be used to obtain the corresponding Maxwell relation. The reciprocity relation states that the order of differentiation does not matter. Thus,... 

Monroe CW, Newman J (2006) Onsager reciprocal relations for Stefan-Maxwell diffusion. Ind Eng Chem Res 45(15) 5361-5367... 

We now derive the Maxwell relations. These are relationships between partial derivatives that follow from Euler s reciprocal relation. Equation (5.39) (page 75). 

The Maxwell-Stefan dififusivity D, , obeys the Onsager reciprocal relation of irreversible thermodynamics, i.e.,... 

Using the Euler reciprocity relation, we obtain a second Maxwell relation ... 

The last of these nine Maxwell relations expresses the reciprocal effect between the binding of hydrogen ions and magnesium ions. Some higher partial derivatives can also be calculated. 

We would expect intuitively that tan 0 emd the Deborah number De are related, since both refer to the ratio between the rates of an imposed process and that (or those) of the system. The exact shape of this relationship depends on the number and nature(s) of the releixation process(es). So let us anticipate [3.6.4 la] for the loss tangent of a monolayer in oscillatory motion, which describes a special case of [3.6,12], namely -tan0 = t]°(o/K°. Here, (o is the imposed frequency, equal to the reciprocal time of observation, t(obs) =< . The quotient K° /t]° also has the dimensions of a time in fact it is the surface rheological equivalent of the Maxwell-Wagner relaxation time in electricity, (Recall from sec. 1.6c that for the electrostatic case relaxation is exponential ith T = e/K where e e is the dielectric permittivity and K the conductivity of the relaxing system. In other words, T is the quotient between the storage and the dissipative part.) For the surface rheological case T therefore becomes The exponential decay that is required for such a... 

In Chapter 9 w e will use the Euler relationship to establish the Maxwell relations between the thermodynamic quantities. Here we derive the Euler relationship. Figure 5.12 shows four points at the vertices of a rectangle in the xy plane. Using a Taylor series expansion, Equation (4.22), compute the change in a function Af through tw o different routes. First integrate from point A to point B to point C. Then integrate from point A to point D to point C. Compare the results to find the Euler reciprocal relationship. For Af = f(x + Ax,y + Ay) -f(x,y), the hrst terms of the Taylor series are... 

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