Osmotic pressure in dilute solutions

Becaus-e of the similarity in the relations for osmotic pressure in dilute solutions and the equation for an ideal gas, van t Hoff proposed his bombardment theory in which osmotic pressure is considered in terms of collisions of solute molecules oil the semipeniieable membrane. This theoiy has a number of objections and has now been discarded. Other theories have also been put forward involving solvent bombardment on the semipermeable membrane, and vapor pressure effects. For example, osmotic pressure has been considered as the negative pressure which must be applied to the solvent to reduce its vapor pressure to that of the solution. It is, however, more profitable to interpret osmotic pressures using thermodynamic relations, such as the entropy of dilution,... 

This is known as the van t Hoff s law showing that, independent of the kinds of solvents, the osmotic pressure in dilute solutions is a function of the concentration of the solutes only. 

A distinction of two kinds must be made here. In ease of change of pressure in dilute gases (or change of osmotic pressure in dilute solutions) the effect on the velocity is given a priori, and is confirmed by experiment. We may, therefore, briefly refer to previous deductions and considerations. The pressure (in kilograms per sq. metre) in such cases is given by the equation... 

At the t -temperature, the interaction parameter — 1/2 and the energetic part of two-body interactions exactly cancels the entropic part, making the net two-body interaction zero (v = (l—2x) Z> = 0). For xbody interactions increase the osmotic pressure of dilute polymer solutions. Hence, measurement of the osmotic pressure in dilute solutions provides a direct way of determining the Flory interaction parameter x-... 

Indeed, measurement of osmotic pressure in dilute solution can determine the chain length N, with results for a polydisperse sample providing the... 

The atomic theory is based upon the laws of definite and multiple proportions. Exact analyses of many substances have shown that the constituent elements of these substances are combined in such quantities that definite amounts or their multiples are always combined with each other. The atomic theory gives a definite and simple theory to account for this. The molecular theory is based upon the existence of certain quantitative relationships between the chemical compositions of substances and their relative volumes in gas form or osmotic pressures in dilute solutions. These theories were based upon quantitative experimental data and accounted for them satisfactorily. In recent years the existence of atoms and molecules has... 

B) Pressure The pressure exerted by a gas is due to the bombardment of the molecviles on the walls of the containing vessel. Similarly, the osmotic pressure in dilute solutions is also produced due to the bombardment of the solute molecules on the walls of semi-permeable membrane. Just as the gas pressure depends upon the number of gaseous molecules, the osmotic pressure similarly depends upon the solute particles in the solution. 

It should be noted that the condition of a dilute solution was introduced into the considerations for two reasons primarily, in order that it would be possible to replace the activities by concentrations and thus determine the equilibrium concentrations on the basis of (2.3.3) and, secondarily, in order for it to be possible to neglect the effect of pressure on the chemical potentials of the components whose electrochemical potentials appear in (2.3.2). Because of the differing ionic concentrations in solutions 1 and 2, the osmotic pressures in these solutions are not identical and this difference must be compensated by external pressure. A derivation considering the effect of pressure can be found, for example in [9] or p. 191 of [18]. 

This equation was then applied to a variety of experimental data on solvent activity as related to vapor pressure (40), osmotic pressure in dilute and concentrated solutions (41), and light scattering (42). 

Dilute solutions. As has already been stated (p. 266), the relationship between the osmotic pressure of a solution and the concentration and chemical character of solvent and solute cannot be derived from purely thermodynamical considerations. There are several ways of attaining this end. In the first instance, the variation of the osmotic pressure with the concentration can be determined experimentally, and the results embodied in an empirical equation of the form p=/(c). Or we may deduce relationships from kinetic conceptions of the nature of solutions, in much the same way as the gas laws were deduced. Or, finally, we may deduce the osmotic pressure laws, with the aid of the thermodynamical equations of the previous paragraph, from empirical or theoretical researches on the vapour pressure of solutions. These methods all lead to the same result, that the osmotic pressure of dilute solutions obeys the same laws as the pressure of a perfect gas. In other words, the osmotic pressure of a substance in solution is equal to the pressure which the substance would exert in the form of a perfect gas occupying, at the same temperature, the volume of the solution. 

The experimental evidence that the osmotic pressure of dilute solutions is identical with gas pressure is furnished by the work of Morse and Fraser and their collaborators, to which reference has already been made in the chapter on solution in Vol I As a corollary to the proof which has just been given, it should be mentioned that if it is assumed that the osmotic pressure of a dissolved gas does obey the gas laws, then it is possible by means of a thermodynamic cycle to deduce Henry s Law of Absorption (Compare, for example, Sackur s Thermochemistry and Thermodynamics, English Ed, p 273, in which is given the deduction of the Distribution Law of which Henry s Law is a particular case )... 

The van Hoff equation has been carefully tested experimentally from two aspects. One requirement is that the osmotic pressure of dilute solutions is proportional to the temperature. Data by Morse on cane sugar solutions show that the relationship holds in very dilute solutions, so that the purely statistical character of the phenomenon is evident. 

When two compartments containing a pure solvent and a solution (i.e., solute + solvent) are separated by a semipermeable membrane, that is a membrane that allows the transport of the molecules of the solvent while it retains the molecules of the solute, the solvent flows across the membrane to the compartment containing the solution until equilibrium is reached. This process is called osmosis and the differential pressure created between the two compartments is called the osmotic pressure denoted ta Osmosis was first studied quantitatively by the botanist W.P.F Pfeffer in 1877 and later Jacobus Henricus Van t Hoff found that the osmotic pressure of dilute solutions obeys a relationship of the same form as the ideal gas law and is now called the Van t Hoff law for osmotic pressure. If the osmotic pressure was exerted by the solute alone, it was acting as an ideal gas and its molecules occupied a volume equal to that of the volume of the solution, then... 

In dilute polyelectrolyte solutions without added salt, the Poisson-Boltzmann cylindrical cell model accounts fairly well for thermodynamic and some transport properties observed [110-112]. Accordingly, the osmotic pressure in such solutions may be expressed in a virial expansion as commonly used with only two terms [110] ... 

For example, in the case of dilute solutions, the van t Hoff s equation may be used to piedict the osmotic pressure (jr = CRT) where n is the osmotic pressure of the solution, C is the molar concentration of the solute, ft is the universal gas constant and T is the absolute temperature, Fm dissociating solutes, the concentration is that of the total ions. For example, NaCI dissociates in water into two ions Na" " and Cl . Therefore, the total molar concentration of ions is hvice the molar concentration of NaCI. A useful rule of thumb for predicting osmotic pressure of aqueous solutions is 0,01 psi/ppm of solute (Weber, 1972). 

We shall now pass on to a study of the laws of osmotic pressure, taking up in the first instance the very important case of dilute solutions. In this section it is assumed that there is no change of total volume when a solution is diluted by further addition of pure solvent, and that solution and solvent are practically incompressible. The reader will then easily see that the osmotic pressure in such a case is independent of the pressure supported by the pure solvent the complete investigation is taken up by A. W. Porter, Proc. Hoy. Soc., A, 79, 519, 1907 80, 457, 1908. 

The deduction adopted is due to M. Planck (Thermodynamik, 3 Aufl., Kap. 5), and depends fundamentally on the separation of the gas mixture, resulting from continuous evaporation of the solution, into its constituents by means of semipermeable membranes. Another method, depending on such a separation applied directly to the solution, i.e., an osmotic process, is due to van t Hoff, who arrived at the laws of equilibrium in dilute solution from the standpoint of osmotic pressure. The applications of the law of mass-action belong to treatises on chemical statics (cf. Mel lor, Chemical Statics and Dynamics) we shall here consider only one or two cases which serve to illustrate some fundamental aspects of the theory. 

In Planck s investigation of equilibrium in dilute solutions, the law of Henry follows as a deduction, whereas in van t Hoff s theory, based on the laws of osmotic pressure ( 128), it must be introduced as a law of experience. The difference lies in the fact that in Planck s method the solution is converted continuously into a gas mixture of known potential, whilst in van t Hoff s method it stands in equilibrium with a gas of known potential, and the boundary condition (Henry s law) must be known as well. Planck (Thermodynamik, loc. cit.) also deduces the laws of osmotic pressure from the theory. 

A theory close to modem concepts was developed by a Swede, Svante Arrhenins. The hrst version of the theory was outlined in his doctoral dissertation of 1883, the hnal version in a classical paper published at the end of 1887. This theory took up van t Hoff s suggeshons, published some years earlier, that ideal gas laws could be used for the osmotic pressure in soluhons. It had been fonnd that anomalously high values of osmotic pressure which cannot be ascribed to nonideality sometimes occur even in highly dilute solutions. To explain the anomaly, van t Hoff had introduced an empirical correchon factor i larger than nnity, called the isotonic coefficient or van t Hoff factor,... 

Hence, as the pressure difference is increased, the solvent flow increases. The pressure difference used varies according to the membrane and the application, but is usually in the range 10 to 50 bar but can also be up to 100 bar. The osmotic pressure in Equation 10.25 for dilute solutions can be approximated by the Van t Hoff equation ... 

The interpretation of A becomes clearer when two plates, originally at very small distance from each other, are separated. At a certain separation, equal to 2A, polymer penetrates into the gap. In dilute solutions, where the chains behave as individual coils, A is expected to be of the order or r, the radius of gyration. However, at concentrations where t e coils overlap, the osmotic pressure of the solution becomes so high that narrower gaps can be entered, and A becomes smaller than Tg. 

Two main approaches for osmotic pressure of polymeric solutions theoretical description can be distinguished. First is Flory-Huggins method [1, 2], which afterwards has been determined as method of self-consistent field. In the initial variant the main attention has been paid into pair-wise interaction in the system gaped monomeric links - molecules of solvent . Flory-Huggins parameter % was a measure of above-said pair-wise interaction and this limited application of presented method by field of concentrated solutions. In subsequent variants such method was extended on individual macromolecules into diluted solutions with taken into account the tie-up of chain links by Gaussian statistics [1]. 

See Kragh, "Between Physics and Chemistry," 27 and Barkan, "Walther Nemst," 158159, drawing on a letter from Ostwald to Nerst, 22 November 1892, Ostwald Papers, AAW, Berlin. The views at issue are found in J. H. van t Hoff, "Role of Osmotic Pressure in the Analogy between Solutions and Gases" (1887) and Svante Arrhenius, "On the Dissociation of Substances in Aqueous Solution" (1887), in The Foundations of the Theory of Dilute Solution (Edinburgh Alembic Club, 1929), no. 19. 

Since dilute solutions are considered we can expand the osmotic pressure in a virial series that is truncated at the second virial coefficients... 

The experimental verification of Gibbs theorem. Since the osmotic pressure of a solution is generally difficult to measure, it is simplest to choose a case such that Raoult s law holds good and the concentration of the solution may be used in place of osmotic pressure. The solution should therefore be dilute and should be a true solution the solute, that is, must be dispersed as simple molecules and not as molecular aggregates like soap micelles. These conditions were obtained by Donnan and Barker Proc. 

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