Maxwell relationships

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]. 

Maxwell relationship for multi-specie diffusion is needed (Gemmen and Trembly, 2005) ... 

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 ... 

In the 1860s Maxwell published his famous unified theory of electricity, magnetism and light (radiation) he also formulated the so-called Maxwell relationship for non-polar materials... 

A momentum balance for multicomponent mixtures can be formulated in a manner analogous to that used to derive Equation (C.2.4) using molecule-wall and molecule-molecule (Stefan-Maxwell) relationships to give ... 

Since we can experimentally determine/and obtain fs from the thermodynamic identity [Maxwell relationship resulting from equation (6-17)],... 

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. 

This is one of the well-known Maxwell relationships. Another useful property of functions whose line integrals are independent of path is the following one. If the line integral / is independent of path, a function /(x, y) exists such that... 

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. 

Equations 4.34-4.37 are called Maxwell relationships or Maxwell relations, after the Scottish mathematician and physicist James Clerk Maxwell (Figure 4.3), who first presented them in 1870. (Although the derivation of equations 4.34-4.37 may seem straightforward now, it wasn t until that time that the basics of thermodynamics were understood well enough for someone like Maxwell to derive these expressions.)... 

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

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