Osmotic pressure general equation

The generalization of the osmotic pressure relation equation (63) to a multicomponent solute is ... 

This condition expresses the fact that the two solutions are under different pressures, px and p2, as a result of their, in general, different osmotic pressures. An analogous equation cannot be written for the non-diffusible ion as it cannot pass through the membrane and the equilibrium concentrations cannot be established. 

For osmotic drug delivery systems, Eq. (2) is of critical importance. This equation demonstrates that the quantity of water that can pass a semipermeable film is directly proportional to the pressure differential across the film as measured by the difference between the hydrostatic and osmotic pressures. Osmotic delivery systems are generally composed of a solid core formulation coated with a semipermeable film. Included in the core formulation is a quantity of material capable of generating an osmotic pressure differential across the film. When placed in an aqueous environment, water is transported across the film. This transported water in turn builds up a hydrostatic pressure within the device which leads to expulsion of the core material through a suitably placed exit port. 

It can now be used for the extremely important purpose of calculating calcium sulphate and 4,000 dyne/cm. for barium sulphate. These figures entirely confirm the conclusion to which we have come on general grounds, that the surface tensions of solids must have high values. The applicability of the Ostwald-Hulett formula is limited, since it is based on Van t Hoff s equation for osmotic pressure, which only holds for small concentrations and, therefore, in the present case, for low solubilities. 

For this reason, the relationship between the reduced Osmotic pressure p/C and the concentration is generally expressed in the form of virial equation as given below ... 

At small solute concentrations the second virial coefficient is the main contributor to the value of n, and so in practice the general equation (5.16) is usually restricted to just the term containing the second virial coefficient. At this level of approximation, the osmotic pressure of a ternary solution (biopolymer, + biopolymer, + solvent) may be expressed in the following simple form using the molal scale (Edmond and Ogston, 1968) ... 

After reaching Equation (20) we abandoned a completely general discussion of osmotic pressure in favor of the simpler assumption of ideality. The ideal result, Equation (25), applies to real solutions in the limit of infinite dilution. The objective of this section is to examine the extension of Equation (20) to nonideal solutions or, more practically, to solutions with concentrations that are greater than infinitely dilute. 

Because variations in solvent chemical potential are generally much easier to determine experimentally (e.g., by osmotic pressure measurements, as described in Section 7.3.6), (6.37) gives the recipe for determining the more difficult solute from its Gibbs-Duhem dependence on other easily measured thermodynamic intensities. Equations such as (6.35)-(6.37) are sometimes referred to as Gibbs-Duhem equation(s), but they are really only special cases of (and thus less general than) the Gibbs-Duhem equation (6.34). 

Virial Isotherm Equation. No isotherm equation based on idealized physical models provides a generally valid description of experimental isotherms in gas-zeolite systems (19). Instead (6, 20, 21, 22) one may make the assumption that in any gas-sorbent mixture the sorbed fluid exerts a surface pressure (adsorption thermodynamics), a mean hydrostatic stress intensity, Ps (volume filling of micropores), or that there is an osmotic pressure, w (solution thermodynamics) and that these pressures are related to the appropriate concentrations, C, by equations of polynomial (virial) form, illustrated by Equation 3 for osmotic pressure ... 

The osmotic pressure II of a solution is described in general terms by the so-called viral equation... 

These equations mean that the water flux Jj that passes through the mem brane is, generally, determined not only by the difference between hydro static pressures on both sides of the membrane but also on the flux of the solute caused by the difference between osmotic pressures of this matter. Hence, the interrelationship between both processes can be described as best at the phenomenological level. 

LP is the hydraulic conductivity coefficient and can have units of m s-1 Pa-1. It describes the mechanical filtration capacity of a membrane or other barrier namely, when An is zero, LP relates the total volume flux density, Jv, to the hydrostatic pressure difference, AP. When AP is zero, Equation 3.37 indicates that a difference in osmotic pressure leads to a diffusional flow characterized by the coefficient Lo Membranes also generally exhibit a property called ultrafiltration, whereby they offer different resistances to the passage of the solute and water.14 For instance, in the absence of an osmotic pressure difference (An = 0), Equation 3.37 indicates a diffusional flux density equal to LopkP. Based on Equation 3.35, vs is then... 

Recently, Yamakawa (42) tried also to evaluate the (smotic pressure by using Eq. (3.8). His aim was to obtain the osmotic pressure for moderately high concentrations. However, as one can expect, he had to introduce crude approximations in order to break BBGKJY-type hierarchy of equations for the distribution functions. Such approximations do not yield generally good results at higher concentrations, even for low molecular weight molecules. 

Barkas proposed a generalized osmotic pressure theory for hygroscopic gels such as wood, based on the generalized Porter equation in the form... 

The concentration = (j)jb is the number density of A monomers and V = c /Aa is the number density of A molecules. The last relation of the above equation is a general statement of the van t Hoff Law, as each solute molecule contributes kT to the osmotic pressure in very dilute solutions. The membrane allows the B molecules to pass freely, but restricts all A molecules to stay on one side. This restriction leads to a pressure which is analogous to the ideal gas law (the osmotic pressure is kT per restricted molecule Y[ = kTv). This pressure is due to the translational entropy loss caused by the confinement of the A molecules. 

Equations 13 and 14 have the same functional form as that postulated by Edmonds and Ogston (3) and later generalized by King et aL (4). The significance of the work presented here is that it enables us to give a fundamental interpretation of the coefficients and the reference potential in terms of forces between the species. It also allows us to relate these coefficients to the virial coefficients which appear in the McMillan-Meyer virial expansion (12) of the osmotic pressure. In the equations of Ogston and of King et the coefficients are set equal to the virial coefficients of the McMillan-Meyer virial expansion, but, as we shall see, these coefficients are equivalent only when certain assumptions are made. 

To be able to measure the osmotic pressure n, a semipermeable membrane that permits passage of the solvent molecules but not the solute molecules is needed. This can, in practice, be realized only when there is a large disparity between the sizes of the solute and solvent molecules, as in a solution of a polymer in a small-molecule solvent. However, the existence of osmotic pressure can be envisioned, at least mentally, with any kind of solution, such as a solution of two small-molecule liquids or a miscible blend of two polymers. Equation (6.6) is thus valid for any two-component (amorphous) system, as long as it is in equilibrium and classical thermodynamics is applicable to it. For applications to these general cases, it is more convenient if Equation (6.6) is reformulated in terms of the free energy of mixing and no explicit reference to osmotic pressure is made in it. 

Van t Hoffs equation is very similar to the general gas law. In fact, both equations can be interpreted in the same way. Here we need to keep in mind that the forces of attraction between the A particles keep the liquid together (compare Sect. 11.1, keyword cohesion pressure ). The contribution of the external pressure p is comparatively small. The B particles that drift far away from each other and scarcely influence each other cause a pressure like that of a dilute gas. However, in this case the pressure is not compensated by the container walls but by the cohesion of the A particles. When the osmotic pressure posm is higher than the external pressure—a condition that is often attained— the liquid A behaves as if it were under negative pressure. If we calculate the potential pa of the liquid for a pressure reduced by Posm and keep in mind that for dilute solutions V a Wa and b/ a. b. we again end up with Eq. (12.3). This demonstrates that both descriptions are equivalent ... 

Equation 2.2 shows that D is determined by the interplay of the thermodynamic factor dH/dc and the friction factor /. In general, the friction factor is expected to increase monotonically with increasing c rind decrease with rising temperature. On the other hand, as can be deduced from the known information about osmotic pressure, the thermodynamic factor as a function of c varies in complex ways with solvent quality and temperature. Thus, the concentration dependence of D for a given polymer should exhibit a variety of features depending on solvent conditions. 

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