Hydrostatic pressure chemical potential

As characteristic for thermodynamic cycles, the working system accesses two reservoirs with a low and a high thermodynamic potential. The thermodynamic potentials, i.e., temperature, chemical potential, hydrostatic pressure, electric potential, etc., show an absolute zero of the lower reservoir, when the efficiency of the cycle r] = 1. 

Possible driving forces for solute flux can be enumerated as a linear combination of gradient contributions [Eq. (20)] to solute potential across the membrane barrier (see Part I of this volume). These transbarrier gradients include chemical potential (concentration gradient-driven diffusion), hydrostatic potential (pressure gradient-driven convection), electrical potential (ion gradient-driven cotransport), osmotic potential (osmotic pressure-driven convection), and chemical potential modified by chemical or biochemical reaction. 

By far the biggest problems with the stability and the magnitude of the liquid junction potentials arise in applications where the osmotic or hydrostatic pressure, temperature, and/or solvents are different on either side of the junction. For this reason, the use of an aqueous reference electrode in nonaqueous samples should be avoided at all cost because the gradient of the chemical potential of the solvent has a very strong effect on the activity coefficient gradients of the ions. In order to circumvent these problems one should always use a junction containing the same solvent as the sample and the reference electrode compartment. 

In the definitions of T, two variables in addition to the ion chemical potential must also be specified as constant. In an equilibrium dialysis experiment, these are temperature and the chemical potential of water. This partial derivative is known as the Donnan coefficient. (Note that the hydrostatic pressure is higher in the RNA-containing solution.) In making connections between T and the Gibbs free energy, it is more convenient if temperature... 

At equilibrium, all the irreversible processes vanish, and temperature, pressure, and chemical potentials become uniform this means that no thermodynamic force exists in the system. No perturbation will cause a change in a neutral equilibrium. Ary two phases in hydrostatic equilibrium must have the same pressure in thermal equilibrium, any two phases must have the same temperature. If two phases are in equilibrium with respect to ary species, then the chemical potential of that species must have the same value in these phases. 

Here, pH is the chemical potential of the heat bath and P is the hydrostatic pressure. 

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). 

Because of their rigid cell walls, large hydrostatic pressures can exist in plant cells, whereas hydrostatic pressures in animal cells generally are relatively small. Hydrostatic pressures are involved in plant support and also are important for the movement of water and solutes in the xylem and in the phloem. The effect of pressure on the chemical potential of water is expressed by the term VWP (see Eq. 2.4), where Vw is the partial molal volume of water and P is the hydrostatic pressure in the aqueous solution in excess of the ambient atmospheric pressure. The density of water is about 1000 kg m-3 (1 g cm-3) therefore, when 1 mol or 18.0 x 10-3 kg of water is added to water, the volume increases by 18.0 x 10-6 m3. Using the definition ofV,., as a partial derivative (see Eq. 2.6), we need to add only an infinitesimally small amount of water (dnw) and then observe the infinitesimal change in volume of the system (dV). We thus find that Vw for pure water is 18.0 x 10-6 m3 mol-1 (18.0 cm3 mol-1). Although Vw can be influenced by the solutes present, it is generally close to 18.0 x 10-6 m3 mol-1 for a dilute solution, a value that we will use for calculations in this book. 

If we represent the magnitude of this applied hydrostatic pressure by n, then the chemical potential of water on the right-hand side of the semipenneable membrane in Figure 2-8 is +RT In aw +Fwn. This solution is in equilibrium with pure water on the left-hand side of the membrane, where = /i,. Hence, + RT In aw + Vwn =, or RT In aw +VWII = 0,... 

A loss of water from plant shoots—indeed, sometimes even an uptake — occurs at cell-air interfaces. As we would expect, the chemical potential of water in cells compared with that in the adjacent air determines the direction for net water movement at such locations. Thus we must obtain an expression for the water potential in a vapor phase and then relate this P to for the liquid phases in a cell. We will specifically consider the factors influencing the water potential at the plant cell-air interface, namely, in the cell wall. We will find that vFcel1 wal1 is dominated by a negative hydrostatic pressure resulting from surface tension effects in the cell wall pores. 

To illustrate the rather small contribution that the pressure term, VjP, makes to differences in the chemical potential of a charged substance across a membrane, we will compare VjAP with the contribution of the electrical term, ZjFAE. We will use a typical electrical potential difference (AE) across a biological membrane of 100 mVand a hydrostatic pressure difference (AP) of 0.5 MPa (= 0.5 x 106 Pa = 0.5 x 106 N m 2 = 0.5 x 106 J m 3), and we... 

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... 

ATP is also the free energy currency for the contraction of muscles (Table 6-1). The ATP-driven contraction of the muscles surrounding the left ventricle of the human heart can increase the blood pressure within it by 20 kPa (0.2 bar or 150 mm Hg). This increases the chemical potential of the water in the blood (i.e., the VWP term), which causes the blood to flow out to the aorta and then to the rest of the circulatory system toward lower hydrostatic pressures. Pressure-driven flow is an efficient way to move fluids for example, it takes only 0.02 kJ of Gibbs free energy to increase the pressure of 10-3 m3 (1 liter) of water by 20 kPa. In particular, in the present case we note that... 

Like any other membrane distillation, in OMD the membrane material is also hydrophobic, so that water in liquid form cannot enter the pores unless a hydrostatic pressure exceeds the LEP [37]. In the absence of such pressure difference, a liquid-vapor interface is formed on either side of the membrane pores. In some aspects membrane distillation and OMD can be considered as closely related, although there are some remarkable differences between them. In both cases, it is necessary to maintain a vapor pressure difference across the membrane pores to get a difference in water chemical potential. However, the means of obtaining this vapor pressure difference is different in both the cases. Whereas it is a temperature difference in the case of membrane distillation, it is a concentration difference in the case of OMD. 

Reaction rates may be improved if the reaction is run in the mixture critical region. A rate enhancement can potentially occur as a result of applied hydrostatic pressure and as a result of the unusual partial molar volume behavior of a heavy solute solubilized in a supercritical solvent. Numerous authors have used transition state analysis (Laidler, 1965 Eckert, 1972 Ehrlich, 1971) to explain the rate enhancement that can occur at high pressures. For a bimolecular reaction, a chemical equilibrium is assumed to exist between the reactants A and B and the transition state M. ... 

For membrane separation processes, only driving forces which induce a significant flux of matter are of practical importance. These driving forces are hydrostatic pressure, concentration, and electrical potential differences. These driving forces can also lead to the separation of chemical species. 

Membrane Osmometry In this technique a dilute polymer solution and a pure solvent are separately placed in two different chambers that are divided by a tightly held semipermeable membrane through which only solvent molecules can move across. Because the chemical potential of pure solvent is higher than that of the solvent in the solution, the solvent will diffuse across the membrane from the pure solvent to the solution chamber up to the point in which the osmotic pressure equals the hydrostatic pressure created by the volume imbalance between the liquids of the two chambers. The osmotic pressure (k = pgh) at equilibrium (static method) can be calculated from the difference in height (h) between the liquids in the capillaries connected to each chamber. In practice, a dynamic method is used in which a pressure (P) is applied to counterbalance (at t = 0) the osmotic pressure... 

Currently, the mainstream of RO membrane transport theory is the solution-diffusion model [50]. According to the model, mass transfer occurs in three steps absorption to the membrane, diffusion through the membrane, and desorption from the membrane. The chemical potential gradient from the feed side of the manbrane to the permeate side of the membrane is the driving force for the mass transfer. When the difference in hydrostatic pressure is greater than the difference in osmotic pressure between the upstream and downstream sides of the membrane, a chemical potential difference of water across the membrane drives water against the natural direction of water flow. 

The energy levels, or chemical potential of the solvent, can be modified by adding a solute or by applying pressure. Solvents in dilute solution have a higher chemical potential than solvents in a concentrated solution. Solutions under high hydrostatic pressure have a higher chemical potential than solvents under lower hydrostatic pressure. 

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