# Viscosity of Solutions of Macromolecules

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1 Vscosty of Solutons of Macromolecules When a lqud flows, whether through a tube or as the result of pourng from a vessel, layers of lqud slde over each other. The force f requred s drectly proportonal to the area A and velocty < of the layers, and nversely proportonal to the dstance d between them, as shown n fgure-1. The proportonalty constant s the coeffcent of vscosty or more commonly, the vscosty. Unts of Vscosy f Aν d The tradtonal unt of vscosty s the pose, orgnally a cgs unt that has the symbol P. The dmensons based on Equaton-2 are the g/cm sec. ( 1) 2 fd gcm cm gcm P aν sec cm cm sec 1 ( 2) 2 sec f d Fgure.1 In SI unts the pose has no specal symbol and has unts of kg/m s. Thus, 1 P.1 kg m -1 s -1. Vscostes of gases and lquds are often reported n µp and cp, respectvely. For example, at 25< C the vscosty of N 2 s 178 µp and the vscosty of water s.8937 cp. Measurement The rate of flow R (cm 3 /sec) of a lqud through a cylndrcal tube of radus r and length l under a pressure head P s gven by the Pouslle equaton. R V 4 Π Pr t 8l ( 3) Measurement of P, r, t, V, and l permts the calculaton of the vscosty: 4 Π Pr t ( 4) 8Vl It s easer to measure the vscosty of a lqud by comparng t wth another lqud of known vscosty ( presumably already measured wth Equaton-4). Snce P Dgh 1 2 ρ1t1 (5) ρ t 2 2

2 The Oswald vscometer (Fgure-2) s a smple devce for comparng the flow tmes of two lquds of known densty. If the vscosty of one lqud s known, the other can be calculated. After the reservor s flled wth a lqud, t s pulled by sucton above the upper mark. The tme requred for the lqud to fall from mark 1 to mark 2 s recorded. Then the tme requred for the same volume of a lqud of known vscosty to flow under dentcal condtons s recorded, and the vscosty s calculated wth Equaton-5. Fgure.2 Vscosty of Solutons of Hgh Polymers Whle the vscosty of water and molasses arses from hydrogen bondng between flowng lqud layers, the vscosty of solutons of macromolecules arses from the entanglement of large molecules flowng along wth the solvent. One would expect ths effect to depend on the sze, shape, and molecular weght of the macromolecules. Indeed, the problem attracted the attenton of Ensten, who showed that for large sphercal molecules, n sp 25. ( V / V 25 ). φ ( 6) where N s the fractonal volume occuped by the macromolecules and sp, the specfc vscosty, s defned as sp ( 7)

3 where and are the vscostes of the soluton and solvent, respectvely. Note that the specfc vscosty s dmensonless. Intrnsc Vscosty The propertes of ndvdual macromolecules separated from ther nteracton wth each other are found by extrapolatng to nfnte dluton. [ ] lm sp lm 1 n ln ( 8 ) c c c c n The quantty [], the ntrnsc vscosty s mportant because t s most drectly related to propertes such as molecular weght and shape. Its dmenson s that of recprocal concentraton, usually expressed as 1/(g/1 ml). In dlute soluton, both sp /c and (1/c)ln(/ ) are lnearly dependent upon concentraton c and can be extrapolated to nfnte dluton to determne []. [ n] KM a ( 9) Molecular Weght It s found emprcally that the ntrnsc vscosty of a soluton depends on the molecular weght of the solute: where a falls between ½ and 2 and s a parameter that s related to the shape of the molecule. When the solute molecule s nearly sphercal, a s about.5, but when the solute molecule s long, extended. or rod shaped, ts value may reach 1.7 to 1.8. Both K and a must be determned from vscosty measurements wth solutons of macromolecules of known molecular weght, determned by other methods, such as osmotc pressure measurements. Some typcal values for K and a are lsted n Table-1. Because chan termnaton n polymerzaton reactons occurs randomly, polymers have dstrbuton of molecular masses about an average value. Expermental measurement of vscosty or collgatve propertes lke osmotc pressure leads to M n, the number average molecular weght. If the polymer sample s consdered to be made up of a large number of fractons consstng of n 1 moles of mass M 1, n 2 moles of mass M 2, and so on.., The number average molecular weght s defned as: M n n + n Collgatve propertes such as osmotc pressure depend on the number of partcles n soluton but not on the ndvdual mass. Lght scatterng of polymer solutons, however, depends on the mass and number of partcles and leads to the mass average molecular weght M w (Mathews, 1984) where M w 2 n

4 Table.1 Parameters for Molecular Weght Determnaton (Unts of [] and K are 1/(gm/1ml)) []KM a Macromolecule Solvent K x 1 4 a Cellulose Acetate Acetone Methyl methacrylate Benzene Polystyrene Toluene Polyvnyl Alcohol Water If all the molecules have the same molecular mass, the polymer s sad to be monodspersed and M w /M n 1.. Synthetc polymers wth Mw/Mn as low as 1.4 have been prepared. Vnyl polymers have Mw/Mn ratos from 2 to about 1 and low densty polymers has Mw/Mn as hgh as 2 (Rudn, 1969). Apparatus Ths experment requres an Oswald vscometer wth a water flow tme of about 1 sec; a 25<C thermostat; 1 and 25 ml ppets; rubber bulb and tube, two or more 5 ml volumetrc flasks; cleanng soluton; polystyrene ( the peanut-sze peces used as protectve packng materal are satsfactory); and 3 ml toluene. Expermental Procedure Dssolve about 1g polystyrene n toluene n a 5 ml volumetrc flask. Whle the polymer s dssolvng clean the vscometer wth cleanng soluton. If t s not avalable, prepare the cleanng soluton by dssolvng 12g sodum dchromate (Na 2 Cr 2 O 7 ) n ml hot water. Cool. Cautously, slowly, and wth strrng, add 225 ml concentrated sulfurc acd. Store n a 25 ml glass-stoppered bottle. Use carefully on glassware that has been prevously cleaned wth detergent and rnsed wth water. Do not dspose the cleanng soluton n the snk. Calbraton - After cleanng the vscometer wth cleanng soluton, rnse t wth dstlled water; then add a 1 ml alquot of water and let t come to thermal equlbrum wth the thermostat. Both fducal marks should be below the water level of the thermostat. Wth a rubber bulb, pull the water n the vscometer above the upper fducal mark and measure the tme t requres to flow from the upper to the lower mark. Repeat to determne the uncertanty n measurng tme. The flow tme wth water, along wth ts densty at 25 C ( g/cm3) and vscosty (.8937 cp) are used wth Equaton-5 to determne the vscostes of toluene and the polystyrene solutons. Dran the water from the vscometer; rnse t frst wth a few mlllters of acetone, then wth toluene. Add a 1 ml alquot of toluene, and after the vscometer and toluene are at thermal equlbrum, measure the flow tme as before. Measurements-Take a 25 ml alquots of the polystyrene soluton and dlute to 5ml wth toluene n a volumetrc flask. Agan measure the flow tme. Dlute repeatedly untl the fnal soluton s 1/8 the orgnal soluton concentraton; measure the flow tme of each soluton.

5 Results and Calculatons The densty of toluene at 25 C s.866g/cm3. The densty of the soluton may be assumed to be that of toluene. Tabulate c,, sp, sp/c and (1/c)ln(/ ). On a sngle sheet of paper plot both sp/c and (1/c)ln(/ ) verses c. Extrapolate to c to obtan []. If your extrapolaton s lnear, you may run a least squares analyss of the data. Calculate the molecular weght of your polystyrene sample wth Equaton-9 and the parameters lsted n Table-1. The parameter K n Table-1 requres the concentraton c to be expressed n g/1 ml. References 1 Allcock, H.R. and F.W. Lampe, Contemporary Polymer Chemstry, Prentce Hall, NJ, Bradbury, J.H., Polymerzaton Knetcs and Voscometrc Characterzaton of Polystyrene, J.Chem.Edu, 4: , Carather, Jr.,C.E., Generaton of Poly (vnyl alcohol) and Arrangement of Structural Unts., J.Chem.Edu.,55: , Flory P.J., Prncples of Polymer Chemstry, Cornell Unversty Press,NY, Flory P.J, and F.S. Leutner, Occurance of Head-to-Head Arrangemants of Structural Unt n Polyvnyl Alcohol, J.Ploy.Sc.3:88, Goldberg, A.I, W.P.Hohensten and H Mark, Intrnsc Vscosty Molecular Weght Relatonshps for Polystyrene J.Poly.Sc.2:52, Johnson, J.F, R Martn and R.s. Porter, A Wssberger and B.W. Rostter, Technques of Chemstry, Vol 1, Physcal Methods of Chemstry Vol 1, chap. 2, Wley- Interscence,NY, Mathas, L.J., Evaluaton of Vty-Molecular Weght Relatonshps, J.Chem.Edu.,6, 422,1977.

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