Gravitational term, chemical

Another contributor to chemical potential is gravity (Fig. 2-7). We can readily appreciate that position in a gravitational field affects fXj because work must be done to move a substance vertically upward. Although the gravitational term can be neglected for ion and water movements across plant cells and membranes, it is important for water movement in a tall tree and in the soil. 

The additive constant term fij in Equation 2.4 is the chemical potential of species j for a specific reference state. From the preceding definitions of the various quantities involved, this reference state is attained when the following conditions hold The activity of species j is 1 (RT In cij = 0) the hydrostatic pressure equals atmospheric pressure (VjP = 0) the species is uncharged or the electrical potential is zero (ZjFE = 0) we are at the zero level for the gravitational term (rrijgh = 0) and the temperature equals the temperature of the system under consideration. Under these conditions, fij equals fij (Eq. 2.4). 

The arguments are unchanged at other elevations, so the same gravitational term must be included in fiw and /jlwi (mwgh = mw,gh because mw = mm,). The form for the pressure effects in the chemical potential for water vapor is more subtle and is discussed next and in Appendix IV. 

When ions of some species / are in equilibrium across a membrane, its chemical potential outside (o) is the same as that inside (i), that is, ju,° equals fjij. Differences in the hydrostatic pressure term generally make a negligible contribution to the chemical potential differences of ions across membranes, so VjP can be omitted from jUy in the present case. With this approximation and the definition of chemical potential (Eq. 2.4 without the pressure and the gravitational terms), the condition for equilibrium of ionic species / across the membrane (jlJ = /jlj) is... 

As indicated in Chapter 2 (Section 2.2B), the terms in the chemical potential can be justified or derived by various methods. The forms of some terms in i can be readily appreciated because they follow from familiar definitions of work, such as the electrical term and the gravitational term. Hie comparison with Fick s first law indicates that RT In a, is the appropriate form for the activity term. Another derivation of the / Tln a, term is in Appendix IV, together with a discussion of the pressure term for both liquids and gases. Some of these derivations incorporate conclusions from empirical observations. Moreover, the fact that the chemical potential can be expressed as a series of terms that can be added together agrees with experiment. Thus a thermodynamic expression for the chemical potential such as Equation 2.4 does the folio whig (1) summarizes the results of previous observations, (2) withstands the test of experiments, and (3) leads to new and useful predictions. 

Figure 6-5 indicates that the C>2-evolution step and the electron flow mediated by the plastoquinones and the Cyt b(f complex lead to an accumulation of H+ in the lumen of a thylakoid in the light. This causes the internal H+ concentration, c, or activity, to increase. These steps depend on the light-driven electron flow, which leads to electron movement outward across the thylakoid in each of the two photosystems (see Fig. 5-19). Such movements of electrons out and protons in can increase the electrical potential inside the thylakoid (E ) relative to that outside ( °), allowing an electrical potential difference to develop across a thylakoid membrane. By the definition of chemical potential (fij = jx + RT In cij 4- ZjFE Eq. 2.4 with the pressure and gravitational terms omitted see Chapter 3, Section 3.1), the difference in chemical potential of H+ across a membrane is...

We have defined the standard state for gaseous species ), /r, as the chemical potential when the gas phase has a partial pressure for species j (Pj) equal to the saturation partial pressure (Pj), when we are at atmospheric pressure (P = 0) and the zero level for the gravitational term (h = 0), and for some specified temperature. Many physical chemistry texts ignore the gravitational term (we calculated that it has only a small effect for water vapor see Chapter 2, Section 2.4C) and define the standard state for the condition when Pj equals 1 atm and species) is the only species present (P = Pj). The chemical potential of such a standard state equals fx- -RTlnP in our symbols. 

The observations on which thermodynamics is based refer to macroscopic properties only, and only those features of a system that appear to be temporally independent are therefore recorded. This limitation restricts a thermodynamic analysis to the static states of macrosystems. To facilitate the construction of a theoretical framework for thermodynamics [113] it is sufficient to consider only systems that are macroscopically homogeneous, isotropic, uncharged, and large enough so that surface effects can be neglected, and that are not acted on by electric, magnetic or gravitational fields. The only mechanical parameter to be retained is the volume V. For a mixed system the chemical composition is specified in terms of the mole numbers Ni, or the mole fractions [Ak — 1,2,..., r] of the chemically pure components of the system. The quantity V/(Y j=iNj) is called the molar... 

This suggestion of Newton s of the existence of a special kind of attraction for chemical actions differing in its manifestation from the ordinary phenomena of gravitation, magnetism, or electricity, and subject to laws of its own, as yet unknown, made immediate impression on chemical thought. Its tendency was to cause chemists to think of chemical action in terms of mechanical forces, that is as an attraction producing motion of some kind among the minuter particles or atoms of bodies. In the version of Boerhaave s Chemistry, published in 1727, by Drs. Shaw and Chambers, the above article of Newton s is cited in a... 

It is a feature of classical mechanics not to enquire into the nature and origin of forces, but simply to quantify them in terms of suitable numerical parameters, such as the gravitational constant. Treating chemical forces between atoms in the same way, does not mean that they are of non-quantum origin. Whatever the nature or complexity of the interaction, an empirical polynomial function that describes the potential energy correctly, can in principle always be found. This aim is achieved mechanically by introducing a small number of so-called transferable force-held parameters. 

To transform Equation 6.3 into a more useful form, we need to incorporate expressions for the chemical potentials of the species involved. The chemical potential of species j was presented in Chapter 2 (Section 2.2B), where Xj is a linear combination of various terms fij = fi + RT In cij +VjP -f ZjFE 4- mjgh (Eq. 2.4 fi is a constant, cij is the activity of species /, Vj is its partial molal volume, P is the pressure in excess of atmospheric, Zj is its charge number, Fis Faraday s constant, E is the electrical potential, m is its mass per mole, and h is the vertical position in the gravitational field). 

The forms for the gravitational contribution (rrijgh) and the electrical one (ZjFE) can be easily understood. We showed in Chapter 3 (Section 3.2 A) that RT n cij is the correct form for the concentration term in jij. The reasons for the forms of the pressure terms in a liquid (V)/>) and in a gas [RT In (Pj/P j] are not so obvious. Therefore, we will examine the pressure dependence of the chemical potential of species / in some detail. 

The last term vanishes if the potential energy is conserved in a chemical reaction (i.e., cVc,r = 0 r = 1,2,..., q) [32]. This is the case if the property of the particles, which is responsible for the interaction with a field of force, is itself conserved. Examples for this case are the mass in a gravitational field and the charge in an electrical field. 

Not all the terms in these equations have the same importance in determining the flow solution in chemical reactors. The only body force considered in most reactor models, gj (per unit mass), is gravitation which is the same for all chemical species, g. The model equations for momentum and energy can then be simplified. In the momentum equation Pc c = f cS = PS-In the energy equation Xlc=i(jc Sc) = Sc=i jc S = 0- Furthermore, in most multicomponent flows, the energy or heat flux contributions from the interdiffusion processes are in general believed to be small and omitted in most applications, ft-cV jc 0 (e.g., [148], p. 816 [89], p. 198 [11], p. 566). 

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