Using Maxwell Relationships

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

Measuring the change in entropy with respect to pressure is difficult, but using a Maxwell relationship we can substitute some other expression. Because dSIdp) equals — dVldT)p, we get 

The enthalpy derivative in equation 4.38 can be used with the Joule-Thomson coefficient, /jljj. Recall that by the cyclic rule of partial derivatives. 

Use equation 4.39 to determine the value of yUjx for an ideal gas. Assume molar Recall that/a, is the Joule-Thomson quantities. coefficient for a gas. 

Unless otherwise noted, all art on this page is Cengage Learning 2014. 

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Can you prove why this is so ) When x, y, and z are thermodynamic quantities, such as free energy, volume, temperature, or enthalpy, the relationship between the partial differentials of M and N as described above are called Maxwell relations. Use Maxwell relations to derive the Laplace equation for a... 

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 ... 

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 ... 

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. 

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

Now we use a Maxwell relationship and substitute for (dS/dV)j, which according to Maxwell s relationships equals dpldT)y. Therefore,... 

For a system with constant composition, the two properties that we choose to constrain the state of the system become the independent properties. We can write the differential change of any other property, the dependent property, in terms of these two properties, as illustrated by Equation (5.4). From a combined form of the first and second laws, we developed the fundamental property relations. We then used the rigor of mathematics to allow us to form this intricate web of thermodynamic relationships. Included in the web are the Maxwell relations, the chain rule, derivative inversion, the cyclic relation, and Equations (5.22) through (5.24). A set of useful relationships relating partial derivatives with T, P, s, and v is summarized in Figure 5.3. We use these relationships to solve first- and second-law problems similar to those in Chapters 2 and 3, but for real fluids. 

This equation is not particularly useful in practice, since it is difficult to quantify the relationship between concentration and ac tivity. The Floiy-Huggins theory does not work well with the cross-linked semi-ciystaUine polymers that comprise an important class of pervaporation membranes. Neel (in Noble and Stern, op. cit., pp. 169-176) reviews modifications of the Stefan-Maxwell approach and other equations of state appropriate for the process. 

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... 

In order to describe the material properties as a function of frequency for a body that behaves as a Maxwell model we need to use the constitutive equation. This is given in Equation (4.8), which describes the relationship between the stress and the strain. It is most convenient to express the applied sinusoidal wave in the exponential form of complex number notation ... 

You will notice that this is the expression for a Maxwell model (see Equation 4.25). From Equations (4.121) to (4.125) we have applied a Fourier transform and confirmed that a Maxwell model fits at least this portion of the theory of linear viscoelasticity. The simple expression for the relationship between J (co) and G (co) allows an interesting comparison to be performed. Suppose we take our equations for a Maxwell model and apply Equation (4.108) to transform the response to an oscillating strain into the response for an oscillating stress. This requires careful use of simple algebra to give... 

Because partial derivatives are used so prominently in thermodynamics (See Maxwell s Relationships), we briefly consider the properties of partial derivatives for systems having three variables x, y, and z, of which two are independent. In this case, z = z(x,y), where x and y are treated as independent variables. If one deals with infinitesimal changes in x and y, the corresponding changes in z are described by the partial derivatives ... 

Mattson ME, Del Boca FK, Carroll KM, Cooney NL, DiClemente CC, Donovan D, Kadden RM, McRee B, Rice C, Rycharik RG Zweben A (1998). Compliance with treatment and follow-up protocols in project MATCH predictors and relationship to outcome. Alcoholism Clinical and Experimental Research, 22, 1328-39 Maxwell S Shinderman MS (2002). Optimizing long-term response to methadone maintenance treatment a 152-week followup using higher-dose methadone. Journal of Addictive Diseases, 21, 1-12 McBride AJ, Sullivan JT Blewett A (1997). Amphetamine prescribing as a harm reduction measure a preliminary study. Addiction Research, 5, 95-112... 

Using eight thermodynamic potentials introduced, 24 Maxwell relations containing certain partial derivatives can be obtained easily. These relations together with the corresponding specific heats Cy z = T(dS/dT)y z (where y represents either V or P, and z represents either E, or x) permit to describe phenomenological relationships between the deformation (or stress) in solids and the accompanying thermal effects. 

Maxwell relations are a powerful tool for deriving thermodynamic relationships. Their use should be considered whenever it is desirable to replace thermodynamic derivatives involving S with equivalent derivatives involving variables P, V, T only. Sidebars 5.4-5.6 illustrate this derivation techniques for a number of standard thermodynamic identities. 

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. 

Dayhoff [50] suggested that one might measure a rest mass of photon by designing a low-frequency oscillator from an inductor-capacitor (LC) network. The expected frequency can be calculated from Maxwell s equations, and this may be used to give an effective wavelength for photons of that frequency. He claimed that one would have a measure of the dispersion relationship at low frequencies. Williams [51] calculated the effective capacitance of a spherical capacitor using Proca equations. This calculation can then be generalized to any capacitor with the result that a capacitor has an additional term that is quadratic in the area of the plates of the capacitor. However, this term is not exactly the one that Dayhoff referred to. But it seems to be a very close description of it. One can add two identical capacitors C in parallel and obtain the result... 

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 ... 

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