Thermodynamics Maxwell equations

In thermodynamics, this is referred to as a Maxwell equation. This equation is derived later in Section 4.8. Thus the effect of pMg on the binding of hydrogen ions is the same as the effect of pH on the binding of magnesium ions in short, these are reciprocal effects. The bindings of these two ions are referred to as linked functions. Equation 1.3-17 can be confirmed by plotting these two derivatives, and the same plot is obtained in both cases. This would be a lot of work to do by hand, but since Mathematical can take partial derivatives, this can be done readily with a computer. The two plots are identical and are given in Fig. 1.8. 

The number of Maxwell equations for each of the possible thermodynamic potentials is given by D(D — l)/2, and the number of Maxwell equations for the thermodynamic potentials for a system related by Legendre transforms is [ )(D — 1)/2]2D. Examples are given in the following section. 

The other thermodynamic properties for a reaction are related to the Gibbs energy of reaction through Maxwell equations (see Section 2.3). Because of equation 3.1-5, equation 3.1-4 can be written... 

When the pH is specified, we enter into a whole new world of thermodynamics because there is a complete set of new thermodynamic properties, called transformed properties, new fundamental equations, new Maxwell equations, new Gibbs-Helmholtz equations, and a new Gibbs-Duhem equation. These new equations are similar to those in chemical thermodynamics, which were discussed in the preceding chapter, but they deal with properties of reactants (sums of species) rather than species. The fundamental equations for transformed thermodynamic potentials include additional terms for hydrogen ions, and perhaps metal ions. The transformed thermodynamic properties of reactants in biochemical reactions are connected with the thermodynamic properties of species in chemical reactions by equations given here. 

James Clerk Maxwell derived the Maxwell Equations in 1864. These expressions completely describe electric and magnetic fields and their interaction with matter. Also see Ludwig Boltzmann below for Maxwell s contribution to thermodynamics. 

Equations (6.7) through (6.10) are the basis not only for derivation of the Maxwell equations but also of a large number of other equations relating thermodynamic properties. We develop here only a few expressions useful for evaluations of thermodynamic properties from experimental data. Their derivation requires application of Eqs. (6.8) and (6.16). 

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. 

These relations are often called equations of state because they relate different state properties. Since the variables T, P, and [nj] play this special role of yielding the other thermodynamic properties, they are referred to as the natural variables of G. Further information on natural variables is given in the Appendix of this chapter. In writing partial derivatives, subscripts are omitted to simplify the notation. The second type of interrelations are Maxwell equations (mixed partial derivatives). Ignoring the VdP term, equation 3.1-1 has two types of Maxwell relations ... 

R. A. Alberty, Effect of temperature on the standard transformed thermodynamic properties of biochemical reactions with emphasis on the Maxwell equations, J. Phys. Chem. 107 B, 3631-3635 (2003). (Supporting Information is available.)... 

Thermodynamic Interrelations Maxwell Equations, and Equations of State... 

Primary thermodynamic functions Fundamental property relations For homogeneous systems of constant composition Maxwell equations ... 

Concentration-dependent activity coefficients can be accommodated with relative ease by an added term (e.g., [see Helfferich, 1962a Brooke and Rees, 1968] and variations in diffusivities are easily included in numerical calculations (Helfferich and Petruzzelli, 1985 Hwang and Helfferich, 1986). In both instances, however, a fair amount of additional experimental information is required to establish the dependence on composition. Electro-osmotic solvent transfer and particle-size variations are more difficult to deal with, and no readily manageable models have been developed to date. A subtle difficulty here is that, as a rule, there is not only a variation in equilibrium solvent content with conversion to another ionic form, but that the transient local solvent content is a result of dynamics (electro-osmosis) and so not accessible by thermodynamic considerations (Helfferich, 1962b). Theories based on the Stefan-Maxwell equations or other forms of (hcrniodyiiainics of ir-... 

When applying Eq. 5.2.1, we have to consider all forms of energy and account for all types of work. We usually derive simplified and useful relations by imposing certain assumptions and incorporating other thermodynamic relations (equation of state. Maxwell s equations, etc.) into the energy balance equation. Therefore, it is essential to identify all the assumptions made and examine whether they are valid. 

In thermodynamics, the equations for the second-order response of the energy are linked by a Maxwell relation. Likewise, Eqs. (106) and (107) are linked by the Maxwell relation [70] ... 

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,... 

Strictly speaking, differentiation with respect to a vector quantity is not allowed. However for the isotropic spherical samples for which equation (A2.L8) is appropriate, the two vectors have the same direction and could have been written as scalars the vector notation was kept to avoid confusion with other thermodynamic quantities such as energy, pressure, etc. It should also be noted that the Maxwell equations above are correct for either of the choices for electromagnetic work discussed earlier under the other convention A is replaced by a generalized G.)... 

Transport in OSN membranes occurs by mechanisms similar to those in membranes used for aqueous separations. Most theoretical analyses rely on either irreversible thermodynamics, the pore-flow model and the extended Nemst-Planck equation, or the solution-diffusion model [135]. To account for coupling between solute and solvent transport (i.e., convective mass transfer effects), the Stefan-Maxwell equations commonly are used. The solution-diffusion model appears to provide a better description of mixed-solvent transport and allow prediction of mixture transport rates from pure component measurements [136]. Experimental transport measurements may depend significantly on membrane preconditioning due to strong solvent-membrane interactions that lead to swelling or solvent phase separation in the membrane pore structure [137]. 

It may be shown that this equation is equivalent to the phenomenological equations derived from irreversible thermodynamics, as weU as the multicomponent diffusion equations derived from the Stefan-Maxwell equations, which were first used to describe diffusion in multicomponent gases. 

In the general ease, however, the Maxwell equation, Eq. (7) can only be solved if a constitutive dependence connecting the local charge density with local electrostatic potential / is specified. This is usually achieved by assxuning a local thermodynamic equilibrium and postulating an ideal behavior of ion solutions when the electrochemical potential of a ion ji, is given by the expression... 

James Clerk Maxwell (1831-1879) was a great Scottish physicist who made contributions to thermodynamics, but whose greatest contribution was the Maxwell equations of electromagnetism. 

This construction Is named for the same James Clerk Maxwell who devised the Maxwell equations of electrodynamics and the Maxwell relations of thermodynamics and contributed to the founding of gas kinetic theory. 

Equations (6) and (7) allow determination of the temperature and pressure depend-enee of py from experimental measurements of the partial molar entropy (sometimes ealled latent heat) and partial molar volume, and Y, respectively. Unfortunately, however, Eqs (6) and (7) provide no explicit general equation for py as a function of y. From the form of Eq. (2) one can see that py and y will always appear as a product in the same term in every thermodynamic energy equation. Therefore, it will never be possible to derive a Maxwell equation from which the composition dependence of py can be experimentally determined. 

In 1879 Lord Kelvin introduced the term nwtivity for the possession, the waste of which is called dissipation at constant temperature this is identical with Maxwell s available energy. He showed in a paper On Thermodynamics founded on Motivity and Energy Phil. Mag., 1898), that all the thermodynamic equations could be derived from the properties of motivity which follow directly from Carnot s theorem, without any explicit introduction of the entropy. 

In summary, Eq. (86) is a general expression for the number of particles in a given quantum state. If t = 1, this result is appropriate to Fenni-rDirac statistics, or to Bose-Einstein statistics, respectively. However, if i is equated torero, the result corresponds to the Maxwell -Boltzmann distribution. In many cases the last is a good approximation to quantum systems, which is furthermore, a correct description of classical ones - those in which the energy levels fotm a continuum. From these results the partition functions can be calculated, leading to expressions for the various thermodynamic functions for a given system. In many cases these values, as obtained from spectroscopic observations, are more accurate than those obtained by direct thermodynamic measurements. 

The relative populations of energy levels, that is the proportions of the analyte species occupying them, have a direct bearing on line intensities and are determined by the spacings of the levels and the thermodynamic temperature. The relation is expressed in th q Maxwell-Boltzmann equation,... 

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