The osmotic pressure of an ideal solution

In a solution, such as that of sugar in water, the solvent is the component whose mole fraction can be varied up to unity. Let it be supposed that such a solution is separated from a quantity of the pure solvent, at the same temperature, by means of a membrane permeable only to the solvent molecules. Then what is called the osmotic pressure of the solution is the excess pressure which must be placed on it in order to prevent any diffusion of solvent through the membrane. 

The nature of osmotic pressure has been discussed very clearly by Guggenheim.-f The osmotic pressure of a solution which is contained in a beaker open to the atmosphere is not a pressure which it actually exerts it is rather to be regarded as being one of the thermodynamic properties of this solution in a manner closely similar to, say, its freezing-point. When a solution is said to have a freezing-point of — 5°C this does not imply that the solution is necessarily at this temperature but rather that is the temperature at which the 

The cause of osmosis is simply diffusion the solvent is able to diffuse through the membrane but the solute is not. It is only when there is a membrane which hai this property that the phenomenon can occur. As discussed in 2 96, this diffusion itself arises from a difference in chemical potential at the same temperature and pressure the solvent substance is at a lower chemical potential in the solution than in its own pure liquid (on account of its lower mole fraction), and there is therefore a tendency for it to pass through the membrane in the direction pure solvent- solution. The osmotic pressure is the excess pressure which will just prevent this flow, and if a pressure greater than this were appli to the solution the solvent would diffuse in the reverse direction and the solution would become more concentrated. 

In a perfect gas osmotic system, as shown in 3 36, equilibrium is attained when the partial pressure of the permeating gas is the same on each side of the semi-permeable membrane. This implies equal volume concentrations. On the other hand, in the case of solutions, the volume concentration is not an adequate criterion of the diiSusion tendency and it is possible for a substance to diffuse spontaneously from a region of lower to a region of higher concentration.f In such systems the only proper criterion of equilibrium, which is to say of the absence of a diffusion flow, is the equality of the chemical potential in the two regions. 

Consider an osmotic system consisting of pure solvent together with a solution and an intervening membrane which is permeable to the solvent. Let p and p be the pressures on the pure solvent and on the solution respectively when there is a condition of osmotic and thermal equilibrium. The difference will be called /7, the osmotic 

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For example, the osmotic pressure of an ideal solution is given by Eq. (2-68)... 

On the above theory the osmotic pressure of an ideal solution depends upon the fact that there is a difference between the values of the solvent piessure in the pure solvent and in the solution respectively Let us denote the solvent pressure m the pure solvent by ir and its value m the solution by ir We have been considering an ideal solution as defined by Tinker (It will be shown later that in non-ideal solutions, e those m which the molecular volume of the solvent is altered as a result of addition of solute, the resulting osmotic pressure is a more complex phenomenon, involving the intrinsic or cohesion tension of the solvent as well as its liquid thermal pressure For the present we are dealing with the ideal case, however )... 

The above statement that the osmotic pressure of an ideal solution involves the difference of and ir does not mean that the osmotic pressure P is simply ir - 7/ This is not the case To find the connection between P and ir let us carry out the following simple thermodynamical cycle at constant temperature... 

In otherwords, the osmotic pressure of an Ideal solution (ideal means that it obeys = AcRT) is the same as the pressure of an Ideal gas at the same pressure and temperature. 

As we will see in the following section, the significance of the osmotic coefficient lies in the fact that it is the ratio of the osmotic pressure to the osmotic pressure of an ideal solution. From (8.1.13) and (8.1.14) it is easy to see that... 

Note that here k is k ( ) with the value of of the pure solvent. The first term on the rhs of (6.12.92) is the osmotic pressure of an ideal solution. The correction term is the change in the osmotic pressure due to the ionic interactions only. 

The osmotic pressure of a solution is proportional to the concentration of solute. In fact, we show in the following Justification that the expression for the osmotic pressure of an ideal solution, which is called the van t Hoff equation, bears an uncanny resemblance to the expression for the pressure of a perfect gas ... 

The osmotic pressure of an ideal solution can be obtained by using Equation (3.58) and by differentiating Equation (3.24). For an ideal solution... 

The osmotic pressure of an electrolyte solution jt can be considered as the ideal osmotic pressure jt decreased by the pressure jrel resulting from electric cohesion between ions. The work connected with a change in the concentration of the solution is n dV = jt dV — jrel dV. The electric part of this work is then JteldV = dWcl, and thus jzc] = (dWei/dV)T,n. The osmotic coefficient 0 is given by the ratio jt/jt, from which it follows that... 

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. 

At very low molecular densities, i.e. at very low Interfacial pressures, the mono-layer exhibits gaseous behaviour. The molecules are far apart, but, unlike in a three-dimensional gas, they are not completely disordered. Because of their amphi-polar nature, the molecules exhibit a preferential orientation relative to the surface-normal. As stated in sec. 3.1, the interfacial pressure exerted by an ideally dilute monolayer is equivalent to the osmotic pressure of an ideal three-dimensional solution. Ideal gaseous monolayer behaviour means obe3dng relation [3.1.1]. 

The osmotic pressure of an ideally dilute aqueous solution is given by... 

As shown in Section 8.3 of Chapter 8, the osmotic pressure of an ideal polymer solution having solvent mole fiaction xj is —(/ 7 /n,)lnxj, where n, is the molar volume of the solvent. Consequently, the osmotic pressure difference between two solutions having solvent volume fiactions (p and (p is given as follows ... 

Determine the osmotic pressure of an aqueous 1 1, 2 1,2 2 and 2 3 salt solution at 0.1 molar. Assume that the solutions are ideal. 

Figure 9-3. The time dependence of the hydrostatic heads Ap, of solutions of poly(ethylene glycol) (c = 2 X lO " g/cm (A/n)=4000, (M v)= 4300 g/mol molecule) in formamide, methanol, or water on cellophane 600 membranes (cellulose hydrate) at 25°C. The theoretically expected osmotic pressure in an ideal solution at this concentration is Ilid = 127 Pa (according to H.-G. Elias).

Pfeffer found that the osmotic pressure P at a given temperature is proportional to the concentration or inversely proportional to the volume F, and at a given concentration is proportional to the absolute temperature T, as with the pressure of a gas PV—kT. Van t Hoff s attention was drawn to this result by de Vries. By using Pfeifer s results van t Hoff found that the constant k is equal to the gas constant R, so that in addition to Boyle s and Charles laws, the osmotic pressure of a dilute solution obeys Avogadro s law, and is equal to the pressure which the dissolved substance would exert if it existed as an ideal gas at the same temperature in the volume occupied by the solution and all the solvent were removed. Van t Hoff says ... 

The partitioning principle is different at high concentrations c > c . Strong repulsions between solvated polymer chains increase the osmotic pressure of the solution to a level much higher when compared to an ideal solution of the same concentration (5). The high osmotic pressure of the solution exterior to the pore drives polymer chains into the pore channels at a higher proportion (4,9). Thus K increases as c increases. For a solution of monodisperse polymer, K approaches unity at sufficiently high concentrations, but never exceeds unity. 

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 activity coefficient of the solvent remains close to unity up to quite high electrolyte concentrations e.g. the activity coefficient for water in an aqueous solution of 2 m KC1 at 25°C equals y0x = 1.004, while the value for potassium chloride in this solution is y tX = 0.614, indicating a quite large deviation from the ideal behaviour. Thus, the activity coefficient of the solvent is not a suitable characteristic of the real behaviour of solutions of electrolytes. If the deviation from ideal behaviour is to be expressed in terms of quantities connected with the solvent, then the osmotic coefficient is employed. The osmotic pressure of the system is denoted as jz and the hypothetical osmotic pressure of a solution with the same composition that would behave ideally as jt. The equations for the osmotic pressures jt and jt are obtained from the equilibrium condition of the pure solvent and of the solution. Under equilibrium conditions the chemical potential of the pure solvent, which is equal to the standard chemical potential at the pressure p, is equal to the chemical potential of the solvent in the solution under the osmotic pressure jt,... 

Consider the equilibrium in a vertical cylinder of suspension of density pi in a suspension medium of density of unit cross-section and height. o o. If at a height x there are n particles per unit volume and at a height xJr x,n->cM particles per unit volume the difference in osmotic pressures due to the particles on the assumption that the suspension conforms to the laws of an ideal solution will be... 

Physiologists studying osmotic relationships of organisms, however, are often concerned with the total concentration of all dissolved substances, not just the concentrations of specific solutes. For expressing the total number of osmotically active particles in a solution, the concept of osmolality is commonly employed to refer to the osmotic pressure characteristic of a solution. One osmole is defined as the osmotic pressure of a 1.0 molal solution of an ideal solute. Because conditions of ideality do not pertain to the case of biological fluids, it is not possible to extrapolate precisely from chemical determinations of moles of solute per kilogram (or liter) of fluid to the osmolality of that fluid. Rather, this value must be determined empirically. 

Suppose we put some one-celled microorganisms in various aqueous NaCl solutions. We observe that the cells remain unperturbed in 0.7% NaCl, whereas they shrink in more concentrated solutions and expand in more dilute solutions. Assume that 0.7% NaCl behaves as an ideal 1 1 electrolyte. Calculate the osmotic pressure of the aqueous fluid within the cells at 25°C. 

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