The Maxwell Relationships

What is the expression for H, assuming one knows the behavior of G (that is, the partial derivatives in equations 4.24 and 4.25)  

If we know how G behaves with respect to its natural variables, we know dGldT)p. This partial derivative is equal to — S, so we can substitute to get 

It is worth stating again how useful the natural variable equations are. If we know how any one of the energies varies in terms of its natural variables, we can use the various definitions and equations from the laws of thermodynamics to construct expressions for any other energy. The mathematics of thermodynamics is becoming powerful indeed. 

The equations involving partial derivatives of the thermodynamic energies can be taken a step further. However, some definitions are necessary. 

We have repeatedly made the point that some thermodynamic functions are state functions, and that changes in state functions are independent of the exact path taken. In other words, the change in a state function depends only on the initial and final conditions, not on how the initial conditions became the final conditions. 

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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. 

The apex of the semicircle obeys the Maxwell relationship, as follows ... 

Converse flexoelectric effects (i.e. voltage-generated curving) have been demonstrated in uranyl-acetate-stabilized phosphatidylserine BLMs by real-time stroboscopic interferometric measurements the obtained satisfactory agreement between the converse and the direct (i.e. curvature-generated voltage) flexoelectric coefficients have been in accord with the Maxwell relationship [8]. 

The dependence of entropy change on volume at constant temperature, T, and chemical amount, n can also be shown using the Maxwell relationship. This avoids the consideration of the path-dependent quantities of heat, q, and work, w. Thus ... 

This statement means two things. First, if the line integral / is independent of path, then dM/dy = dN/dx. Second, if dM/dy = dN/dx, then the line integral / is independent of path. The equality given by Eq. (14-3) is commonly called the Maxwell relationship. 

A new pressure-explicit equation of state suitable for calculating gas and liquid properties of nonpolar compounds was proposed. In its development, the conditions at the critical point and the Maxwell relationship at saturation were met, and PVT data of carbon dioxide and Pitzers table were used as guides for evaluating the values of the parameters. Furthermore, the parameters were generalized. Therefore, for pure compounds, only Tc, Pc, and o> were required for the calculation. The proposed equation successfully predicted the compressibility factors, the liquid fugacity coefficients, and the enthalpy departures for several arbitrarily chosen pure compounds. 

For the purpose of satisfying the Maxwell relationship more precisely, an adjustment was made on the temperature-dependent function fa(T). The final expression obtained for fa(T) is as follows ... 

In principle we can imagine the energy transfer in a different manner. We make use of the Maxwell relationship,... 

All of these results are exact relationships among various properties. They are general and apply to any pure substance, whether gas, liquid, or solid. All other relationships in thermodynamics maybe considered as mathematical consequences of the results obtained here. Equations f ri= .i6 ). fg .iQl. and (i=>.22) are known as the Maxwell relationships. They relate various partial derivatives among the set of the four fundamental variables, P, V, T, and S, and they are very useful when we want to change from one set of independent variables to another. The complete results of these sections are summarized in Table r-1. 

The remaining step is to express the entropy derivative in terms of P, V, T. This is provided by the Maxwell relationship in eg. fa.iQ ). Thus finally we have... 

The calculation of both the specific and entropy density changes is still based on the Maxwell relationships introducing the mass density of the material fr. 

HB Callen, Thermodynamics and an Introduction to Ther mostatistics, 2nd edition, Wiley, New York, 1985. Very clearly written discussion of the Maxwell relationships. 

The Maxwell relationship in equation 4.36 shows that dS/dV)j is equal to dp/dT)y Using the van der Waals equation,... 

The Maxwell relationships can be extremely useful in deriving other equations for thermodynamics. For example, because... 

Then, the Maxwell relationship among partial derivatives,... 

From the Maxwell relationships we can generate, or extract from the thermodynamic compass, we have complete knowledge about all the other thermodynamic state functions of the system. Obviously, the important ingredient is the equation of state. 

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