Osmotic pressure equation assumptions

The assumption that the molality and molarity are equal does not introduce serious error into the calculations for dilute aqueous solutions. The relationships discussed in Chapter 12 show that M m when the density is 1 g/mL (1 g/cm3) and M < 1000/m. Urea has a molar mass of 60.0 g/mol. Then, 0.280 mol/L may be used for the molarity in the osmotic pressure equation. 

From the assumptions made in deducing it, it appears that this formula is inapplicable to any but dilute solutions. At higher concentrations the discrepancies become considerable between the osmotic pressures actually measured and those calculated from Van t Hoff s equation. The following figures for cane sugar may serve as an example —... 

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. 

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 ... 

Equation (4a) is very approximately true if the saturated vapour obeys the gas laws. It is independent of all assumptions as to the nature of the relationship between the osmotic pressure and the concentration of the solution. 

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. 

Porter s equation has been applied by 0 Wood to the calculation of the osmotic pressure from the results of vapour pressure measurements obtained in connection with concentrated solutions of sucrose in water at a series of different temperatures (cf Wood, Trans Faraday Soc, 1915) Porter [Trans Faraday Soc, 19x5) has further considered von Babo b law, and Kirchhoffs equation for the latent heat of dilution of a solution, laying special emphasis on the assumptions introduced into the deduction of the expressions as obtained in their usual form The reader is referred to the paper for details... 

The osmotic pressure of a solution is unaffected by the membrane used. The only assumption we make is that the membrane allows solvent to pass but not solute. However, the equation says nothing about the speed (rate) at which the movement of solvent occurs. The speed will be dependent upon the type and dimensions of the membrane. 

Van t Hoff developed a mechanistic theory of osmotic pressure, which most of us learned in school. His theory was based on the broad generalizations (1) that the osmotic pressure was directly proportional to the concentration of solute and (2) that the osmotic pressure was directly proportional to the absolute temperature, T. His assumptions permitted the formulation of an equation for osmotic pressure... 

Estimate the osmotic pressure, freezing point, and boiling point of seawater, which you can approximate as equivalent to a 1.08-molal solution of NaCI. Use equations 7.47 and 7.49 to calculate Kf and Xb for HjO, and use AfujH [H2O] = 6.009 kJ/mol and AvapH [HjO] = 40.66 kJ/mol. From what you know about seawater, what assumptions are we making ... 

This equation holds well for the dilute solutions involved in nanofiltrahon and for brackish water. Even for sea water, with Cn 10 mol/m it predicts an osmotic pressure of 7t = 10 x 8.31 x 298 = 2.48 MPa = 24.6 atm, compared to an experimental value of 23 atm (Table 8.3). We adopt this equation to replace osmotic pressure by concentration and make the further assumption that the osmotic pressure in the permeate is zero, i.e., that the membrane is 100% effective in rejecting salt and the bulk concentration Cf, is constant, as was deduced in Illustration 5.2 for laminar entry region flow. We can then combine the two equations and factor out the constant Q to obtain... 

Case A For sufficiently small sample size and high polymer mobility, we can neglect the contributions from diffusion in Equation 3.3 and assume that the evolution of (p follows the changes in the reactant concentrations. The latter assumption means that the elastic stress is instantaneously equilibrated with the osmotic pressure (i.e., the first expression in Equation 3.11 is valid at any moment in time). From this assumption, we obtain the concentration of the oxidized catalyst in this limiting case as a function of the polymer volume fraction as... 

Actually, equation (6.6) applies only in the limit of infinite dilutions. The assumption that In ai = —f2 becomes invalid at finite concentrations (athermic or regular solutions). Indeed, the osmotic pressure depends on the concentration... 

Equation 14.15 for the osmotic pressure is written on the assumption that the dilute solvent has practically ideal solution behavior. As the concentration of solute increases (and that of solvent decreases) this assumption becomes unrehable. 

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... 

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