Equilibrium osmotic

We consider this system in an osmotic pressure experiment based on a membrane which is permeable to all components except the polymeric ion P that is, solvent molecules, M" , and X can pass through the membrane freely to establish the osmotic equilibrium, and only the polymer is restrained. It does not matter whether pure solvent or a salt solution is introduced across the membrane from the polymer solution or whether the latter initially contains salt or not. At equilibrium both sides of the osmometer contain solvent, M , and X in such proportions as to satisfy the constaints imposed by electroneutrality and equilibrium conditions. 

To conclude, the introduction of species-selective membranes into the simulation box results in the osmotic equilibrium between a part of the system containing the products of association and a part in which only a one-component Lennard-Jones fluid is present. The density of the fluid in the nonreactive part of the system is lower than in the reactive part, at osmotic equilibrium. This makes the calculations of the chemical potential efficient. The quahty of the results is similar to those from the grand canonical Monte Carlo simulation. The method is neither restricted to dimerization nor to spherically symmetric associative interactions. Even in the presence of higher-order complexes in large amounts, the proposed approach remains successful. 

The chemical potential of associating systems has also been studied more recently by Bryk et al. [2]. They have extended the usual GEMC method for studying osmotic equihbrium by including four simulation cells in series, rather than the usual two compartments, but with osmotic equilibrium established between only two adjacent compartments (e.g. I and II, II and III, or IV and I). Each semi-permeable membrane was made permeable to only one species as shown and described below ... 

They showed further that the limiting slope (RTA2) of the plot of the osmotic pressure-concentration ratio tz/c against the polymer concentration in a binary solvent mixture should be proportional to the value of the quantity on the left side of Eq. (17),f with V2 representing the volume fraction of solvent in the nonsolvent-solvent mixture which is in osmotic equilibrium with the solution. The composition of the liquid medium outside the polymer molecules in a dilute solution must likewise be given by V2. The composition of the solvent mixture within the domains of the polymer molecules may differ slightly from that outside owing to selective absorption of solvent in preference to the nonsolvent. This internal composition is not directly of concern here. If the solution is made sufficiently dilute, the external nonsolvent-solvent composition v2 = l—Vi) will be practically equal to the over-all solvent composition for the solution as a whole. Hence... 

A close analogy exists between swelling equilibrium and osmotic equilibrium. The elastic reaction of the network structure may be interpreted as a pressure acting on the solution, or swollen gel. In the equilibrium state this pressure is sufficient to increase the chemical potential of the solvent in the solution so that it equals that of the excess solvent surrounding the swollen gel. Thus the network structure performs the multiple role of solute, osmotic membrane, and pressure-generating device. 

The body s normal daily sodium requirement is 1.0 to 1.5 mEq/kg (80 to 130 mEq, which is 80 to 130 mmol) to maintain a normal serum sodium concentration of 136 to 145 mEq/L (136 to 145 mmol/L).15 Sodium is the predominant cation of the ECF and largely determines ECF volume. Sodium is also the primary factor in establishing the osmotic pressure relationship between the ICF and ECF. All body fluids are in osmotic equilibrium and changes in serum sodium concentration are associated with shifts of water into and out of body fluid compartments. When sodium is added to the intravascular fluid compartment, fluid is pulled intravascularly from the interstitial fluid and the ICF until osmotic balance is restored. As such, a patient s measured sodium level should not be viewed as an index of sodium need because this parameter reflects the balance between total body sodium content and TBW. Disturbances in the sodium level most often represent disturbances of TBW. Sodium imbalances cannot be properly assessed without first assessing the body fluid status. 

Special care has to be taken if the polymer is only soluble in a solvent mixture or if a certain property, e.g., a definite value of the second virial coefficient, needs to be adjusted by adding another solvent. In this case the analysis is complicated due to the different refractive indices of the solvent components [32]. In case of a binary solvent mixture we find, that formally Equation (42) is still valid. The refractive index increment needs to be replaced by an increment accounting for a complex formation of the polymer and the solvent mixture, when one of the solvents adsorbs preferentially on the polymer. Instead of measuring the true molar mass Mw the apparent molar mass Mapp is measured. How large the difference is depends on the difference between the refractive index increments ([dn/dc) — (dn/dc)A>0. (dn/dc)fl is the increment determined in the mixed solvents in osmotic equilibrium, while (dn/dc)A0 is determined for infinite dilution of the polymer in solvent A. For clarity we omitted the fixed parameters such as temperature, T, and pressure, p. 

The neuromuscular junction and muscle are more resistant to changes in sodium concentration, to which they are minimally permeable at rest. In fact, the consequences of sodium disturbance relate instead to the role of this ion in maintaining the osmotic equilibrium between the brain and plasma and range from depression of consciousness, coma and seizures caused by hyponatremia, to brain shrinkage and tearing of superficial blood vessels due to excessive serum osmolarity due to hypernatremia. 

The difference in pressures, P — Pq, required to maintain osmotic equilibrium is defined as the osmotic pressure and is denoted by IT. Equation (15.44) thus becomes... 

Polymyxins are peptide antibiotics with a hydrophobic side chain that bind to the phospholipid bilayer and interfere with the osmotic equilibrium of the cell. Polymyxins are active against majority of Gram-negative bacteria. 

Another approach employed to establish the occurrence of a density nversion between the two solutions subsequent to boundary formation involves dialysis between the two solutions s0>. The dialysis membrane is impermeable to the polymer solutes but permeable to the micromolecular solvent, H20. Transfer of water across the membrane occurs until osmotic equilibrium involving equalization of water activity across the membrane is attained. Solutions equilibrated by dialysis would only undergo macroscopic density inversion at dextran concentrations above the critical concentration required for the rapid transport of PVP 36 0 50). The major difference between this type of experiment and that performed in free diffusion is that in the former only the effect of the specific solvent transport is seen which is equivalent to a density inversion occurring with respect to a membrane-fixed or solute-fixed frame of reference. Such restrictions are not imposed on free diffusion where equilibration involves transport of all components in a volume-fixed frame of reference. The solvent flow is governed specifically by the flow of the polymer solutes as described by Eq. (3) which, on rearrangement, gives... 

It is presumed in this statement that equilibrium in a multiphase system implies uniformity of T and P throughout all phases. In certain situations, eg, osmotic equilibrium, pressure uniformity is not required, and equation 212 is then not a useful criterion. Here, however, it suffices. 

This relationship constitutes the basic definition of the activity. If the solution behaves ideally, a, =x, and Equation (18) define Raoult s law. Those four solution properties that we know as the colligative properties are all based on Equation (12) in each, solvent in solution is in equilibrium with pure solvent in another phase and has the same chemical potential in both phases. This can be solvent vapor in equilibrium with solvent in solution (as in vapor pressure lowering and boiling point elevation) or solvent in solution in equilibrium with pure, solid solvent (as in freezing point depression). Equation (12) also applies to osmotic equilibrium as shown in Figure 3.2. 

Equation (13) reminds us that the chemical potential has its greatest value, p,, for a pure substance. Any value of a, less than unity will cause n, to be altered from by an amount RT In a which will be negative for a, < 1. Second, any pressure on a liquid that exceeds p°, increases n above This is seen from the combination of Equations (13) and (18). Thus consideration of the chemical potential of the solvent makes it clear how osmotic equilibrium comes about. The presence of a solute lowers the chemical potential of the solvent. This is offset by a positive pressure on the solution, the osmotic pressure 7r, so that the net chemical potential on the solution side of the membrane equals that of the pure solvent on the other side of the membrane. This is summarized by the expression... 

Several times in this discussion we have noted the importance of experimental conditions that permit as rapid an equilibration as possible. The implication of these remarks is that osmotic equilibrium is reached slowly. In some cases as much as one week may be required for equilibrium to be achieved. To shorten this time, procedures based on measuring the rate of approach to equilibrium have been developed. The osmometer of Figure 3.3b is especially suited for this procedure. 

FIG. 3.4 Data showing the approach to osmotic equilibrium from initial settings above and below the equilibrium column height. (Adapted from R. U. Bonnar, M. Dimbat, and F. H. Stross, Number Average Molecular Weights, Wiley, New York, 1958.)... 

The specific situation we wish to consider is the osmotic equilibrium that develops in an apparatus that has a semipermeable membrane impermeable to the macroion only. That is, the membrane is assumed to be permeable not only to the solvent but also to both of the ions of the low molecular weight electrolyte, but not to the colloidal ion Pz+. At equilibrium the low molecular weight ions will be found on both sides of the membrane, but not in equal concentrations, because of the presence of the macroions on one side of the membrane. We have already come across an example of such a situation in the vignette at the beginning of this chapter on the role of Donnan equilibrium on the so-called resting states of nerve cells. 

For osmotic equilibrium, the chemical potential of the solvent must be the same on both sides of the membrane. In the two-dimensional analog also the chemical potential must be the same for the water on both sides of the float. The presence of the solute lowers the chemical potential of the solvent, but the excess pressure compensates for this. Therefore, by analogy with Equation (3.19), we write... 

Equation (23) obviously gives the two-dimensional ideal gas law when a > a2 and with the o2 term included represents part of the correction included in Equation (15). This model for surfaces is, of course, no more successful than the one-component gas model used in the kinetic approach however, it does call attention to the role of the substrate as part of the entire picture of monolayers. We saw in Chapter 3 that solution nonideality may also be considered in osmotic equilibrium. Pursuing this approach still further results in the concept of phase separation to form two immiscible surface solutions, which returns us to the phase transitions described above. 

Albumin is of great importance in animal physiology in mail it constitutes about 50% of the plasma proteins (blood) and is responsible to a great extent for tile maintenance of osmotic equilibrium in the blood. The high molecular weight (68,000) of the albumin molecule prevents its excretion in the urine the appearance of albumin may indicate kidney damage. 

Figure 2.6(b) shows the situation at the point of osmotic equilibrium, when sufficient pressure has been applied to the saline side of the membrane to bring the flow across the membrane to zero. As shown in Figure 2.6(b), the pressure... 

Equation (2.37) is simplified by assuming that the membrane selectivity is high, that is, DiK jl DjKj/ . This is a good assumption for most of the reverse osmosis membranes used to separate salts from water. Consider the water flux first. At the point at which the applied hydrostatic pressure balances the water activity gradient, that is, the point of osmotic equilibrium in Figure 2.6(b), the flux of water across the membrane is zero. Equation (2.37) becomes... 

An electrochemical system, important particularly in biological systems, is one in which the species are ions and the system is separated into two parts by a rigid membrane that is permeable to some but not all of the species. We are interested in the conditions attained at equilibrium, the Donnan equilibrium. Two cases, one in which the membrane is not permeable to the solvent (nonosmotic equilibrium) and the other in which the membrane is permeable to the solvent (osmotic equilibrium), are considered. The system is at constant temperature and, for the purposes of discussion, we take sodium chloride, some salt NaR, and water as the components. The membrane is assumed to be permeable to the sodium and chloride ions, but not to the R-ions. We designate the quantities pertinent to the solution on one side of the membrane by primes and those pertinent to the solution on the other side by double primes. 

Equation (12.134) gives the required relation between the equilibrium molalities, the activity coefficients, and the two pressures for osmotic equilibrium. It is evident that the two pressures are not independent. We could write P" as F + II, where II is the difference of the osmotic pressures of the two solutions referred to the pure solvent. The solution of the two equations would then give a value of II. 

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