Molar Mass of Polyvinyl Alcohol by Viscosity

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2 and as E α < E β, the number of head-to-head linkages increases with polymerization temperature. Each of these head-to-head linkages forms a 1,2-glycol which can be quantitatively cleaved with Periodate (IO - 4 ). Therefore, a determination of molar mass before and after treatment with Periodate should provide an estimate of the fraction of head-to-head linkages in the molecule. A convenient method for the determination of polymers is via viscosity measurements. The apparatus to be used in these determinations is the Ostwald viscosimeter. The size of the capillary is marked '100'. mark a sample mark b The viscosity η of the solution is given by η = B ρ t (4) where B is an apparatus constant, ρ is the liquid density at the measurement temperature, and t is the time for the meniscus to go from the top fiducial mark (a) to the bottom mark (b). This will take about 50 s - 60 s for pure water. It has been shown that the ratio of the volume v of suspended particles in a total fluid volume V is proportional to the specific viscosity η sp ; defined as: η sp = η 1 v η 0 V (5)

3 where η is the viscosity of the solution, and η 0 the viscosity of the solvent. Note that η is directly proportional to t! The ratio of η sp to the solute concentration c (in grams per 100 ml of solution!!) in the limit of infinite dilution is called the intrinsic viscosity [η]: η sp [η] = lim c 0 c = lim 1 c ln η c 0 η 0. (6) Thus, 1/c. ln(η/η 0 ) and η sp /c can be plotted vs. concentration and extrapolated to c = 0 to give [η]. The problem now is to relate the molar mass to the viscosity. Since these molecules are long chains with many single bonds about which rotation can take place, parts of the molecule more than a few atoms apart are essentially randomly oriented and can therefore be treated statistically. As such it can be shown that the mean distance between the chain ends, which is also the effective mean diameter d of the molecule, is proportional to the square root of the chain length (or the molar mass M): d M 1 2. (7) The volume of a molecule can be approximated as (v M ) = d 3 and equation (7) then becomes: v M M 3 2. (8) Thus, the volume of all the molecules (v) is: v cv M M3 2 = cvm 1 2. (9) From equations (9), (5) and (6) follows KM 1 2 = η sp c = [η] (10) where K is a constant.

4 These calculations ignore the significant effect of the solvent by assuming that it is a poor solvent with no solvating effect which would make the molecular size increase; i.e., the statistical treatment assumes random orientation but in fact an atom in the molecule cannot be in the same space at the same time as a solvent molecule. Thus equation (10) above must be modified and is found for PVOH in aqueous solution at 25 C to be: [η] = (M v ) a where a = 0.76, and M v has units of g/mol. (11) where a = 0.76, and M v has units of g/mol. Equation (11) was derived by Flory and Leutner for monodisperse samples (i. e., all molecules have the same weight) but also holds for polydisperse ones (i. e. the sample is made up of molecules with a distribution of molar masses). In the latter case, the molar mass is M v, the viscosity average molar mass given by equation (12): ( M v ) a = M 1+a P(M)dM 0. (12) MP(M)dM 0 where P(M) is the molar mass distribution function. Other experimental methods (i. e. measurements of osmotic pressure) give number average molar masses M n : M n = MP(M)dM 0. (13) P(M)dM 0 For monodisperse polymers, M v = M n, but for polydisperse polymers the relationship depends on "a" and the distribution function P(M). If the chain lengths are random, this function is: P(M) = 1 M n e M /M n. (14) Along with the aforementioned value of "a" for PVOH of 0.76, it can be shown that M v and M n are related as follows: M v M n = (15)

5 As previously mentioned, PVOH can be specifically cleaved at "head-to-head" sites, thereby changing the number average molar mass M n of the uncleaved sample to M n ' of the cleaved sample. The fraction of the total monomer units bonded head-to-head is equal to the increase in number of molecules during cleavage divided by the total number of units present, as number is inversely proportional to molar mass in each case. = 1 / M n 1 / M n 1 / M0 (16) where M 0 is the monomer molar mass (= 44 g/mol for PVOH). Therefore (with all molar masses in g/mol), 1 = 44 M n 1 M n (17) and from (15) and (17) 1 = 83 M v 1 M v (18)

7 of cleaved PVOH, and then prepare and determine t for 1/4 and 1/8 dilutions. Finally, carefully flush the apparatus with water and dry with two small aliquots of acetone.

8 Data Analysis 1) Given that the coefficient of viscosity of water η 0 at 30 C is Pa s, use the data from your water measurement to determine the apparatus constant B from equation (4). Calculate the error. 2) Calculate c (in grams per 100 ml) of each solution. 3) Calculate the viscosity η of each solution. Determine the error from the standard deviation. 4) Calculate η sp. for both the cleaved and uncleaved polymer. Determine the error from the standard deviation. 5) a) Tabulate the values for η sp /c and c including the error for η sp /c. b) Make plots for both the cleaved and uncleaved polymer of η sp /c versus c. Don't forget the error bars. c) Obtain [η] from the intercepts at c = 0 and graphically determine the error for [η]. d) Determine the correlation coefficients for these linear fits. 7) Use equations (11) and (15) to find M v and M n for the original and degraded polymers. Estimate the error for both quantities. Discuss whether your result is reasonable. 8) Use the results from analysis step (7) to determine and calculate the error. 9) Tabulate your results of (M v, M n and ). 10) How is affected by k α and k β (equation 3)?

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