Gravity-chemical potential

One can use a thermodynamic approach to derive the equation describing the condition of equilibrium between sedimentation and diffusion. The derivation involves the assumption of a constant gravity-chemical potential (i.e. the generalized chemical potential which includes the action of the external gravity field) and the assumption that the laws established for ideal systems can be applied to dilute dispersions, i.e. 

In these equations is the partial molal free energy (chemical potential) and Vj the partial molal volume. The Mj are the molecular weights, c is the concentration in moles per liter, p is the mass density, and z, is the mole fraction of species i. The D are the multicomponent diffusion coefficients, and the are the multicomponent thermal diffusion coefficients. The first contribution to the mass flux—that due to the concentration gradients—is seen to depend in a complicated way on the chemical potentials of all the components present. It is shown in the next section how this expression reduces to the usual expressions for the mass flux in two-component systems. The pressure diffusion contribution to the mass flux is quite small and has thus far been studied only slightly it is considered in Sec. IV,A,6. The forced diffusion term is important in ionic systems (C3, Chapter 18 K4) if gravity is the only external force, then this term vanishes identically. The thermal diffusion term is impor-... 

It is important to note that the concept of osmotic pressure is more general than suggested by the above experiment. In particular, one does not have to invoke the presence of a membrane (or even a concentration difference) to define osmotic pressure. The osmotic pressure, being a property of a solution, always exists and serves to counteract the tendency of the chemical potentials to equalize. It is not important how the differences in the chemical potential come about. The differences may arise due to other factors such as an electric field or gravity. For example, we see in Chapter 11 (Section 11.7a) how osmotic pressure plays a major role in giving rise to repulsion between electrical double layers here, the variation of the concentration in the electrical double layers arises from the electrostatic interaction between a charged surface and the ions in the solution. In Chapter 13 (Section 13.6b.3), we provide another example of the role of differences in osmotic pressures of a polymer solution in giving rise to an effective attractive force between colloidal particles suspended in the solution. 

Figure 3.8 Conceptualization of the potential functions in a hydrostatic system and in a simple chemical system, (a) In the unequilibrated hydrostatic system, water will flow from reservoir 2 of higher hydrostatic potential (=gh2, where g is the acceleration due to gravity and h2 is the observable height of water in the tank) to reservoir 1 of lower hydrostatic potential total water volumes (i.e., total potential energies W [ and W2) do not dictate flow. Similarly, benzene molecules move from liquid benzene to the head space in the nonequilibrated chemical system, not because there are more molecules in the flask containing the liquid, but because the molecules initially exhibit a higher chemical potential in the liquid than in the gas. (b) At equilibrium, the hydrostatic system is characterized by equal hydrostatic potentials in both reservoirs (not equal water volumes) and the chemical system reflects equal chemical potentials in both flasks (not equal benzene concentrations).

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. 

In summary, the chemical potential of a substance depends on its concentration, the pressure, the electrical potential, and gravity. We can compare the chemical potentials of a substance on two sides of a barrier to decide whether it is in equilibrium. If fij is the same on both sides, we would not expect a net movement of species / to occur spontaneously across the barrier. The relative values of the chemical potential of species / at various locations are used to predict the direction for spontaneous movement of that chemical substance (toward lower /a ), just as temperatures are compared to predict the direction for heat flow (toward lower T). We will also find that Afij from one region to another gives a convenient measure of the driving force on species /. 

In many structured products, water management includes several mass transport mechanisms such as hydrodynamic flow, capillary flow and molecular self-diffusion depending on the length scale. Hydrodynamic flow is active in large and open structures and it is driven by external forces such as gravity or by differences in the chemical potential, that is, differences in concentrations at different locations in the structure. Capillary flow also depends on surface tension and occurs in channels and pores on shorter length scales than in hydrodynamic flow. A capillary gel structure will hold water, and external pressures equivalent to the capillary pressure will be needed to remove the water. 

The boundary between the phases is not an impenetrable barrier and polymer and solvent molecules move back and forth all the time. However, at equilibrium the number of polymer (or solvent) molecules moving in one direction is the same as that moving in the other. The driving force for this movement is the chemical potential (just like the driving force for water to run downhill is the potential energy due to gravity). The chemical potential is the same when there is no net flow between the phases (Equations 11-46) ... 

To get the condition of zero flow in the direction of gravity, the chemical potential must be constant /u, = /u,q. However, the chemical potential is not obtained by simply forming the derivative of the energy with respect to the mol number dU(S, V, n,h)/dn, but there are some additional constraints. When a gas leaves a shell of a certain height, it takes its entropy with it. Moreover, the shell shrinks in volume. Thus, the correct chemical potential is obtained from the derivative... 

Equal temperatures guarantee that there is no entropy flow, and equal chemical potentials guarantee that there is no mass flow across the membrane. We did not make any statement on the pressure. If we would fix the volumes of the two subsystems, then even in the absence of gravity a pressure difference would develop. We must fix the position of the membrane. If we would not fix the position of the membrane, a pressure difference would tend to move the membrane and the volume of the solution would try to increase in the cost of the volume of the pure solute. This process would take place, even when the chemical potentials on both sides of the membrane are equal. Thus, a pressure difference causes a bulk expansion and... 

Diffusion of any nonpolar component i does not depend on the electric field. The effect of other components, temperature and pressure shows up in a change of its chemical potential. If we disregard magnetic and gravity forces and assume the presence of hydrodynamical equilibrium, the only force, which compels nonpolar component to move, is gradient of its chemical potential. Relative to it the diffusions linear equation will assume the format... 

Based on thermodynamics of irreversible processes, that is, on the restrictions imposed on the heat and mass fluxes by the second law of thermodynamics, one can state that the mass flux of a-constituent is proportional to the gradient of chemical potential p and gravity potential... 

The sedimentation-diffiision equilibrium is not just a balance between difiusive transport and setding, but can also be regarded as equilibrium between chemical potential and gravity potential (RT dine = —gt V -Na dh). Based on this interpretation, Perrin (1909) employed this colloidal phenomenon to measure Avogradro s constant Na (and, thereby, to confirm the molecular origin of Brownian motion). It is rather fascinating that he obtained values between 6.5 x 10 moF and 7.5 x 10 moF, which ate close to the teal value. 

Two main potential differences are important in membrane processes, the chemical potential difference (Ap.) and the electrical potential difference (AF) (the electrochemical potential is the sum of the chemical potential and the electrical potential). Other possible forces such as magnetical fields, centrifugal fields and gravity will not be considered here. 

Example 1.1 Prove that any substance in a mixture tends to pass from the region of higher to the region of lower chemical potential. Assume the effect of gravity to be negligible. 

The equilibrium concept of gravity segregation leads to the expression that, in a multicomponent fluid column under isothermal conditions, the chemical potential of the ith component, is a function of position, , according to the above differential equation. Equation (2.13) provides both composition and pressure as a function of depth, as we will see next. From (c//q = RTdlnfi)j- and Eq. (2.13),... 

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