Osmometry has been widely used in the polymer industry because it is a sensitive technique for determining the huge molar masses of polymer molecules. Now imagine that you are a polymer chemist; you have devised a new way to make polyethylene and wish to know the molar mass of your new material. The osmotic pressure due to 2.20 g of polyethylene (PE) dissolved in enough benzene to produce 100.0 mL of solution was 1.10 * 10^–2 atm at 25°C. Calculate the molar mass of the polymer, which is a nonelectrolyte. The answer will be an average molar mass as not all the polymer molecules are the same length.

ANTICIPATE Because so many atoms are linked together in each polymer molecule, you should expect a high molar mass.

PLAN Use the procedure for osmometry in Toolbox 5E.1. Because polyethylene is a nonelectrolyte, i = 1. Use R in the units that match the data—in this case, liters and atmospheres.

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Toolbox 5F.1 HOW TO USE COLLIGATIVE PROPERTIES TO DETERMINE MOLAR MASS CONCEPTUAL BASIS The lowering of freezing point and the generation of osmotic pressure both depend on the total concentration of solute parti- cles. Therefore, by using these colligative properties to deter- mine the amount of solute present and knowing its mass, its molar mass can be inferred. PROCEDURE 1. Cryoscopy Step 1 Convert the observed freezing-point depression, AT₁, into solute molality b (Topic 5E) by writing Eq. 1b in the form ATF ik, b Take the freezing-point constant from Table 5F.1. Step 2 Calculate the amount of solute, nsolute (in moles), in the sample by multiplying the molality by the mass of solvent, msolvent (in kilograms): nsolute = bx msolvent Step 3 Determine the molar mass of the solute by dividing the given mass of solute, meolute (in grams), by the calculated amount in moles (step 2). msolute Msolute 1solute This procedure is illustrated in Example 5F.1. 2. Osmometry Step 1 Convert the observed osmotic pressure into solute molar concentration (c) by writing Eq. 3 in the form II iRT In some cases, it may be necessary to calculate the osmotic pressure from the height, h, of the solution (in an apparatus like that in Fig. 5F.3) by using II = gdh, where d is the density of the solution and g is the acceleration of free fall (see inside back cover). Step 2 Because molar concentration is defined as c = 1solute/V, where nolute is the amount of solute (in moles) and V is the volume of solution (in liters), calculate the amount of solute from 1solute = CV Step 3 Determine the molar mass of the solute by dividing the given mass of solute, msolute (in grams), by its amount (step 2), as in the procedure for cryoscopy. This procedure is illustrated in Example 5F.2.