Osmotic pressure molar mass from

SAMPLE PROBLEM 13.8 Determining Molar Mass from Osmotic Pressure... 

The van t Ho ff equation is used to determine the molar mass of a solute from osmotic pressure measurements. This technique, which is called osmometry, involves the determination of the osmotic pressure of a solution prepared by making up a known volume of solution of a known mass of solute with an unknown molar mass. Osmometry is very sensitive, even at low concentrations, and is commonly used to determine very large molar masses, such as those of polymers. 

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. 

Table 15.1 contains osmotic pressure data calculated from the work of Browning and Ferry [3] for solutions of polyvinyl acetate in methyl ethyl ketone at 10°C. Plot H/vv against w, fit the data to a quadratic polynomial, and calculate the number-average molar mass from the intercept with the n/w axis. 

For the determination of very high molar masses, freezing-point depressions, boiling-point elevations, and vapor-pressure lowerings are too small for accurate measurement. Osmotic pressures are of a convenient order of magnitude, but measurements are time-consuming. The technique to be used in this experiment depends on the determination of the intrinsic viscosity of the polymer. However, molar-mass determinations from osmotic pressures are valuable in calibrating the viscosity method. 

Elory and Leutner, working with monodisperse specimens of PVOH differing from one another in molar mass over a wide range (obtained by fractionating polydisperse commercial PVOH), established a correlation between the molar mass, as determined from osmotic pressure measurements, and the intrinsic viscosity. They found that for PVOH in aqueous solution at 25°C,... 

Consequently, virial coefficients can be determined from the concentration dependence of the reciprocal apparent molar mass. But, since the various methods for measuring the molar mass yield various averages of it (see Chapters 8 and 9), the virial coefficients obtained will be average values which vary according to the method used to determine them. Virial coefficients obtained from osmotic-pressure measurements (and all other measurements based on colligative methods) will give the average... 

The only unknown in this expression is Mb which is easily solved fcH See Example 5.4 for the details of how molar mass is determined from osmotic pressure. 

Strategy The steps needed to calculate the molar mass of Hb are similar to those outlined in Example 9.10, except we use osmotic pressure instead of freezing-point depression. First, we must calculate the molarity of the solution from the osmotic pressure of the solution. Then, from the molarity, we can determine the number of moles in 35.0 g of Hb and hence its molar mass. Because the pressure is given in mmHg, it is more convenient to use R in terms of L atm instead of L bar because the conversion factor from mm Hg to atm is simpler. 

EXAMPLE 14-9 Establishing a Molar Mass from a Measurement of Osmotic Pressure... 

Lysozyme, extracted from egg whites, is an enzyme that cleaves bacterial cell walls. A 20.0-mg sample of this enzyme is dissolved in enough water to make 225 mL of solution. At 23°C the solution has an osmotic pressure of 0.118 mm Hg. Estimate the molar mass oflysozyme. 

Colligative1 properties of dilute polymer solutions depend only on the number of dissolved molecules and not on properties of the molecules themselves, such as mass or size. Osmotic pressure, freezing point depression, boiling point elevation, and vapour pressure lowering are the most prominent examples. These methods essentially allow one to count the number n of solute molecules. From n and the known total mass m of the solute the molar mass M is readily obtained as... 

Practically, polymers with molar masses between 2 x 104 and 2 x 106 g/mol can be characterized by membrane osmometry, but measurements of Mn <104 g/mol have also been reported with fast instruments and suitable membranes [16]. The lower limit is set by insufficient retention of short polymer chains. Above M 2 x 106 g/mol, the osmotic pressure, which is proportional to Mr1, is too low for a reasonable signal-to-noise ratio. An advantage of the low molar mass cut-off is that impurities with a very low molar mass can permeate through the membrane and, hence, do not contribute to the measured osmotic pressure. Their equilibration time may, however, be different from that of the solute, leading to complex time-dependent signals. 

First determine the concentration of the solution from the osmotic pressure, then the amount of solute dissolved, and finally the molar mass of that solute. 

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. 

The increase of viscosity from a certain concentration becomes more pronounced with increasing M (see Mark-Houwink), but the osmotic pressure is inversely proportional to the molar mass. 

From the results of the light scattering experiments it follows that A and B have the sameM. The viscosity average,My, is for A higher than for B, and is thus closer toMw. A has, therefore, a narrower molar mass distribution and a higher M. The osmotic pressure IT is therefore lower. 

A solution of crab hemocyanin, a pigmented protein extracted from crabs, was prepared by dissolving 0.750 g in 125 mL of an aqueous medium. An osmotic pressure rise of 2.6 mm of the solution was detected at 4° C. The solution had a density of 1.00 g/mL. Determine the molar mass of the protein. 

Rudin s aim was to predict the size of dissolved polymer molecules and the colligative properties of polymer solutions (hydrodynamic volume, second virial coefficient, interaction parameter, osmotic pressure, etc) from viscometric data (average molar mass, intrinsic viscosity, etc.). 

Finally, several attempts have been made to develop an absolute molar mass detector based on osmotic pressure measurements. Commercially available membrane osmometers are designed for static measurements, and the cell design with a flat membrane is not suited for continuous flow operation. Different from the conventional design, Yau developed a detector which measures the flow resistance of a column caused by osmotic swelling and deswelling of soft gel particles used for the packing (see Fig. 12) [65,78]. With a microbore gel column, a... 

Measurements of osmotic pressure generally give much more accurate molar mass values than those from freezing-point or boiling-point changes. 

Calculate the molar mass of a nonvolatile solute from the changes it causes in the colligative properties (vapor-pressure lowering, boiling-point elevation, freezing-point lowering, or osmotic pressure) of its dilute solution (Section 11.5, Problems 41-56). 

In order to obtain the number-average molar mass of a particular sample, osmotic coefficient (II/c) data, measured at various low concentrations, must be extrapolated to the zero concentration limit. In addition to the ideal gas contribution [Eq. (1.73)] that arises from individual polymers, the osmotic pressure also has a contribution from polymer-polymer interactions. The contribution to osmotic pressure from interaction... 

This procedure is analogous to determination of the number-average molar mass and weight-average second virial coefficient from the measurements of osmotic pressure II at different concentrations [Eq. (1.76) and Fig. 1.22]. 

Lysozyme is an enzyme that cleaves bacterial cell walls. A sample of lysozyme extracted from egg white has a molar mass of 13,930 g. A quantity of 0.100 g of this enzyme is dissolved in 150 g of water at 25°C. Calculate the vapor-pressure lowering, the depression in freezing point, the elevation in boiling point, and the osmotic pressure of this solution. (The vapor pressure of water at 25°C is 23.76 mmHg.)... 

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