Gravitational term, chemical potential

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. 

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). 

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...

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. 

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. 

Thus, in so far as wo need to allow for the effect of the gravitational field, the chemical potential of a substance is not equal throughout the depth of a phase, but it is the sum of the terms /C and Mtgh which has this property of constancy. 

Other than heat conduction, every irreversible process—chemical reactions, diffusion, the influence of electric, magnetic and gravitational fields, ionic conduction, dielectric relaxation, etc.—can be described in terms of suitable chemical potentials. Chapter 10 is devoted to the wide variety of processes described using the concept of a chemical potential. All these processes drive the system to the equilibrium state in which the corresponding affinity vanishes. 

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. 

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. 

Electroviscosity and electroviscoelasticity are terms that may be dealing with fluid flow effects on physical, chemical, and biochemical processes. The hydrodynamic or electrodynamic motion is considered in the presence of both potential (elastic forces) and nonpotential (resistance forces) fields. The elastic forces are gravitational, buoyancy, and electrostatic or electrodynamic (Lorentz), and the resistance forces are continuum resistance or viscosity and electrical resistance or impedance. 

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