Thermodynamics Maxwell relationships

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

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. 

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

A niunber of important relationships between experimental quantities follow from combining thermodynamics (Maxwell relations) with the assumption that the free energy F obeys a scaling law F—Fq T,V)+FJJ, V) where the electronic contribution Fe(T,V)=NkBTf T/To V)) and where N is the number of atoms and To is the characteristic energy. This may be Tk, or Tvf, depending on context. (Throughout this... 

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

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. 

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... 

The Maxwell relations of thermodynamics relate quantities formed by differentiating G once with respect to one variable and once with respect to another (Huang, 1987). Choosing the two variables to be T and n leads to the following relationship ... 

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 ... 

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... 

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. 

The kinetic molecular theory (KMT see Sidebar 2.7) of Bernoulli, Maxwell, and others provides deep insight into the molecular origin of thermodynamic gas properties. From the KMT viewpoint, pressure P arises merely from the innumerable molecular collisions with the walls of a container, whereas temperature T is proportional to the average kinetic energy of random molecular motions in the container of volume V. KMT starts from an ultrasimplified picture of each molecule as a mathematical point particle (i.e., with no volume ) with mass m and average velocity v, but no potential energy of interaction with other particles. From this purely kinetic picture of chaotic molecular motions and wall collisions, one deduces that the PVT relationships must be those of an ideal gas, (2.2). Hence, the inaccuracies of the ideal gas approximation can be attributed to the unrealistically oversimplified noninteracting point mass picture of molecules that underlies the KMT description. 

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. 

There are relationships between thermodynamic and informational entropy. For example, the well-known Maxwell s demon, which reverses thermodynamic entropy with information, but getting that information exactly balances out the thermodynamic gain the demon would otherwise achieve. 

In Frame 4, as equation (4.2), equation (15.2) was established as a definition of an ideal gas on the basis of the simplest qualitative argument and not by classical thermodynamic arguments. Just for the record we note here that the relationship can be established by the following route (which involves introduction of the Helmholtz free energy, A and use of the Maxwell... 

The differential relationships just derived represent the equivalent of the Maxwell equations in thermodynamics. Seldom used in electrochemistry, these equations have been employed in relation to the study of adsorption, particularly at the mercury-solution interphase. 

While using an activity coefficient model will provide a quantitative relationship between the mutual solubilities, we can get a qualitative understanding of how the presence of one dissolved species affects others by examining the interrelation between mixed second derivatives. In particular, the Maxwell equations in Chapter 8 and some of the pure fluid equations in Chapter 6 were derived by examining mixed second derivatives of thermodynamic functions. Another example of this is to start with the Gibbs energy and note that at constant temperature, pressure, and all other species mole numbers,... 

Chemical thermodynamics and kinetics provide the formalism to describe the observed dependencies of chemical-conformational reactions on the external physical state variables temperature, pressure, electric and magnetic fields. In the present account the theoretical foundations for the analysis of electrical-chemical processes are developed on an elementary level. It should be remarked that in most treatments of electric field effects on chemical processes the theoretical expressions are based on the homogeneous-field approximation of the continuum relationship between the total polarization and the electric field strength (Maxwell field). When, however, conversion factors that account for the molecular (inhomogeneous) nature of real systems are given, they are usually only applicable for nonpolar solvents and thus exclude aqueous solutions. Therefore, in the present study, particular emphasis is placed on expressions which relate experimentally observable system properties (such as optical or electrical quantities) with the applied (measured) electric field, and which include applications to aqueous solutions. 

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... 

In this chapter, wc introduce two more methods of thermodynamics. The first is Maxwell relations. With Maxwell relations you can find the entropic and energetic contributions to the stretching of rubber, the expansion of a surface or a film, or the compression of a bulk material. Second, we use the mathematics of homogeneous functions to develop the Gibbs-Duhem relationship. This is useful for finding the temperature and pressure dependences of chemical equilibria. 

We have explored two methods of thermodynamics. The Maxwell relations derive from the Euler expression for state functions. They provide another way to predict unmeasurable quantities from measurable ones. For multicomponent systems, you get another relationship from the fact that many thermodynamic functions are homogeneous. In the following chapters we will develop microscopic statistical mechanical models of atoms and molecules. 

The thermodynamic value of that results from dilution can be readily calculated (Gu et al. 2004). For a simulated reformate with 60 40 (v/v) H /N, the calculated concentration overpotential is 7.9 mV at 70 C and ambient pressure. This is consistent with at open circuit (Gu et al. 2004). When a load is applied, however, (more than 20 mV) is more than twice the calculated value and it is nearly a constant in the low-CD (below 0.8 A/cm ) region. With a pure feed, the only gas-phase diffusion resistance arises from water vapor that has either been added to the inlet stream (to mitigate membrane degradation) or been transported through the membrane from the cathode. With defiberate additions of an inert gas such as nitrogen, the diffusion resistance becomes more substantial because the effective diffusion coefficient of in the gas mixture m is reduced. This is known as the Maxwell-Stefan effect and can be quantified using the following approximate relationship (Bird et al. 1960) ... 

Additional relations between thermodynamic properties and their derivatives can be derived from the second derivatives of the fundamental property relationships. These relations are called Maxwell relations and can be obtained by noting that the order of partial differentiation of an exact differential does not matter. For example, we can equate the following two sets of partial derivatives of the exact differential d from the fundamental grouping u, s, o ... 

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. 

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