Van’t Hoff osmotic pressure

From the results of Exercise 19, show that it is possible to derive the van t Hoff osmotic pressure equation IIV = RT for a very dilute solution V is the volume of solution containing 1 mole of solute. 

Unfortunately, we frequently find in the literature and especially in the textbooks, the loose expression, that, according to van t Hoff, osmotic pressure equals gas pressure, without sufficient indication of the rather restricted conditions under which this statement is accurate. 

Note the similarity of the above formula to the ideal gas law and also that osmotic pressure is not dependent on particle charge. This equation was derived by van t Hoff Osmotic pressure is the basis of reverse osmosis, a process commonly used to purify water. The water to be purified is placed in a chamber and put under an amount of pressure greater than the osmotic pressure exerted by the water and the solutes dissolved in it. Part of the chamber opens to a differentially permeable membrane that lets water molecules through, but not the solute particles. The osmotic pressure of ocean water is about 27 atm. Reverse osmosis desalinators use pressures around 50 atm to produce fresh water from ocean salt water. 

Van t Hoff was the recipient of the first Nobel Prize in chemistry in 1901 for his work in chemical dynam ICS and osmotic pressure—two topics far removed from stereochemistry... 

The solute molecular weight enters the van t Hoff equation as the factor of proportionality between the number of solute particles that the osmotic pressure counts and the mass of solute which is known from the preparation of the solution. The molecular weight that is obtained from measurements on poly disperse systems is a number average quantity. 

Neglecting the higher-order terms, we can write the osmotic pressure for this three-component system in terms of the van t Hoff equation ... 

For example, the measurements of solution osmotic pressure made with membranes by Traube and Pfeffer were used by van t Hoff in 1887 to develop his limit law, which explains the behavior of ideal dilute solutions. This work led direcdy to the van t Hoff equation. At about the same time, the concept of a perfectly selective semipermeable membrane was used by MaxweU and others in developing the kinetic theory of gases. 

For many years, it was thought that the macro solute forms a new phase near the membrane—that of a gel or gel-like layer. The model provided good correlations of experimental data and has been widely used. It does not fit known experimental facts. An explanation that fits the known data well is based on osmotic pressure. The van t Hoff equation [Eq. (22-75)] is hopelessly inadequate to predict the osmotic pressure of a macromolecular solution. Using the empirical expression... 

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). 

Panagiotopoulos et al. [16] studied only a few ideal LJ mixtures, since their main objective was only to demonstrate the accuracy of the method. Murad et al. [17] have recently studied a wide range of ideal and nonideal LJ mixtures, and compared results obtained for osmotic pressure with the van t Hoff [17a] and other equations. Results for a wide range of other properties such as solvent exchange, chemical potentials and activity coefficients [18] were compared with the van der Waals 1 (vdWl) fluid approximation [19]. The vdWl theory replaces the mixture by one fictitious pure liquid with judiciously chosen potential parameters. It is defined for potentials with only two parameters, see Ref. 19. A summary of their most important conclusions include ... 

J. H. van t Hoff (Berlin) discovery of the laws of chemical dynamics and osmotic pressure in solutions. 

We may therefore sum up the results in the statement that the laws of osmotic pressure of a dilute solution are formally identical with the laws of gas pressure of an ideal gas (van t Hoff s Gaseous Theory of Solution). 

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. 

The same van t Hoff responsible for the i factor showed that the osmotic pressure of a solution is related to the molarity, c, of the solute in the solution ... 

This equation is sometimes called the van t Hoff isochore, to distinguish it from van t Hoff s osmotic pressure equation (Section 8.17). An isochore is the plot of an equation for a constant-volume process. 

While Arrhenius was studying conductivity, others were characterizing colligative properties of solutions. The Dutch chemist J. T. van t Hoff studied osmotic pressure and derived the law of osmotic pressure,... 

Id. Treatment of Data.—Typical osmotic data are shown in Figs. 38 and 39. Here the ratio ( n/c) of the osmotic pressure to the concentration is plotted against the concentration. If the solutions behaved ideally, van t Hoff s law Eq. (11) would apply and m/c should be independent of c. Owing to the large effective size of the polymer molecules in solution (Fig. 34) and the interactions between them which consequently set in at low concentrations, /c increases with c with a... 

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,... 

Numerous measurements of the conductivity of aqueous solutions performed by the school of Friedrich Kohhansch (1840-1910) and the investigations of Jacobns van t Hoff (1852-1911 Nobel prize, 1901) on the osmotic pressure of solutions led the young Swedish physicist Svante August Arrhenius (1859-1927 Nobel prize, 1903) to establish in 1884 in his thesis the main ideas of his famous theory of electrolytic dissociation of acids, alkalis, and salts in solutions. Despite the sceptitism of some chemists, this theory was generally accepted toward the end of the centnry. 

Van t Hoff introduced the correction factor i for electrolyte solutions the measured quantity (e.g. the osmotic pressure, Jt) must be divided by this factor to obtain agreement with the theory of dilute solutions of nonelectrolytes (jt/i = RTc). For the dilute solutions of some electrolytes (now called strong), this factor approaches small integers. Thus, for a dilute sodium chloride solution with concentration c, an osmotic pressure of 2RTc was always measured, which could readily be explained by the fact that the solution, in fact, actually contains twice the number of species corresponding to concentration c calculated in the usual manner from the weighed amount of substance dissolved in the solution. Small deviations from integral numbers were attributed to experimental errors (they are now attributed to the effect of the activity coefficient). 

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 ... 

This simple equation relating osmotic pressure to the concentration of the dilute solute was discovered by van t Hoff and enables the pressure to be estimated. 

The mole fraction of water is much larger than that of sucrose and practically for all cell components, so van t Hoffs equation is easily justified for estimates of cellular osmotic pressure. Given R = 0.08314 bar L 1 mol-1 and a protocell temperature of 298 K, calculate the osmotic pressure. 

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. 

The simple theory of osmotic pressure developed by Van t Hoff is well known. According to it the molecules of solute behave like gas molecules, and produce the same pressure as would be produced by an equal number of gas molecules occupying the... 

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 —... 

In this equation, u is the osmotic pressure in atmospheres, n is the number of moles of solute, R is the ideal gas constant (0.0821 Latm/K mol), T is the Kelvin temperature, V is the volume of the solution and i is the van t Hoff factor. If one knows the moles of solute and the volume in liters, n/V may be replaced by the molarity, M. It is possible to calculate the molar mass of a solute from osmotic pressure measurements. This is especially useful in the determination of the molar mass of large molecules such as proteins. 

V, is the molar volume of polymer or solvent, as appropriate, and the concentration is in mass per unit volume. It can be seen from Equation (2.42) that the interaction term changes with the square of the polymer concentration but more importantly for our discussion is the implications of the value of x- When x = 0.5 we are left with the van t Hoff expression which describes the osmotic pressure of an ideal polymer solution. A sol vent/temperature condition that yields this result is known as the 0-condition. For example, the 0-temperature for poly(styrene) in cyclohexane is 311.5 K. At this temperature, the poly(styrene) molecule is at its closest to a random coil configuration because its conformation is unperturbed by specific solvent effects. If x is greater than 0.5 we have a poor solvent for our polymer and the coil will collapse. At x values less than 0.5 we have the polymer in a good solvent and the conformation will be expanded in order to pack as many solvent molecules around each chain segment as possible. A 0-condition is often used when determining the molecular weight of a polymer by measurement of the concentration dependence of viscosity, for example, but solution polymers are invariably used in better than 0-conditions. 

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