1 14-1 CHAPTER 14 POLYMER STRUCTURES PROBLEM SOLUTIONS Hydrocarbon Molecules Polymer Molecules The Chemistry of Polymer Molecules 14.1 The repeat unit structures called for are sketched below. (a Polychlorotrifluoroethylene (b Poly(vinyl alcohol
2 14-2 Molecular Weight 14.2 Repeat unit weights for several polymers are asked for in this problem. (a For polytetrafluoroethylene, each repeat unit consists of two carbons and four fluorines (Table If A C and A F represent the atomic weights of carbon and fluorine, respectively, then m 2(A C + 4(A F (2(12.01 g/mol + (4(19.00 g/mol g/mol (b For poly(methyl methacrylate, from Table 14.3, each repeat unit has five carbons, eight hydrogens, and two oxygens. Thus, m 5(A C + 8(A H + 2(A O (5(12.01 g/mol + (8(1.008 g/mol + (2(16.00 g/mol g/mol (c For nylon 6,6, from Table 14.3, each repeat unit has twelve carbons, twenty-two hydrogens, two nitrogens, and two oxygens. Thus, m 12(A C + 22(A H + 2(A N + 2(A O (12(12.01 g/mol + (22(1.008 g/mol + (2(14.01 g/mol + (2(16.00 g/mol g/mol (d For poly(ethylene terephthalate, from Table 14.3, each repeat unit has ten carbons, eight hydrogens, and four oxygens. Thus, m 10(A C + 8(A H + 4(A O (10(12.01 g/mol + (8(1.008 g/mol + (4(16.00 g/mol g/mol
3 We are asked to compute the degree of polymerization for polystyrene, given that the numberaverage molecular weight is 500,000 g/mol. The repeat unit molecular weight of polystyrene is just m 8(A C + 8(A H (8(12.01 g/mol + (8(1.008 g/mol g/mol Now it is possible to compute the degree of polymerization using Equation 14.6 as DP M n m 500,000 g/mol g/mol 4800
4 (a The repeat unit molecular weight of polypropylene is called for in this portion of the problem. For polypropylene, from Table 14.3, each repeat unit has three carbons and six hydrogens. Thus, m 3(A C + 6(A H (3(12.01 g/mol + (6(1.008 g/mol g/mol (b We are now asked to compute the number-average molecular weight. Since the degree of polymerization is 15,000, using Equation 14.6 M n (DPm (15,000(42.08 g/mol 631,000 g/mol
5 (a From the tabulated data, we are asked to compute M n, the number-average molecular weight. This is carried out below. Molecular wt Range Mean M i x i x i M i 10,000-20,000 15, ,000-30,000 25, ,000-40,000 35, ,000-50,000 45, ,250 50,000-60,000 55, ,100 60,000-70,000 65, ,000-80,000 75, ,000-90,000 85, M n x i M i 49, 800 g/mol (b From the tabulated data, we are asked to compute M w, the weight-average molecular weight. Molecular wt. Range Mean M i w i w i M i 10,000-20,000 15, ,000-30,000 25, ,000-40,000 35, ,000-50,000 45, ,350 50,000-60,000 55, ,200 60,000-70,000 65, ,700 70,000-80,000 75, ,000-90,000 85, M w w i M i 55,200 g/mol (c Now we are asked to compute the degree of polymerization, which is possible using Equation For polytetrafluoroethylene, the repeat unit molecular weight is just
6 14-6 m 2(A C + 4(A F (2(12.01 g/mol + (4(19.00 g/mol g/mol And DP M n m 49,800 g/mol g/mol 498
7 (a From the tabulated data, we are asked to compute M n, the number-average molecular weight. This is carried out below. Molecular wt. Range Mean M i x i x i M i 8,000-20,000 14, ,000-32,000 26, ,000-44,000 38, ,000-56,000 50, ,000 56,000-68,000 62, ,160 68,000-80,000 74, ,000-92,000 86, M n x i M i 47, 720 g/mol (b From the tabulated data, we are asked to compute M w, the weight-average molecular weight. This determination is performed as follows: Molecular wt. Range Mean M i w i w i M i 8,000-20,000 14, ,000-32,000 26, ,000-44,000 38, ,000-56,000 50, ,500 56,000-68,000 62, ,260 68,000-80,000 74, ,840 80,000-92,000 86, M w w i M i 53,720 g/mol (c We are now asked if the degree of polymerization is 477, which of the polymers in Table 14.3 is this material? It is necessary to compute m in Equation 14.6 as m M n DP 47,720 g/mol g/mol
8 14-8 The repeat unit molecular weights of the polymers listed in Table 14.3 are as follows: Polyethylene g/mol Poly(vinyl chloride g/mol Polytetrafluoroethylene g/mol Polypropylene g/mol Polystyrene g/mol Poly(methyl methacrylate g/mol Phenol-formaldehyde g/mol Nylon 6, g/mol PET g/mol Polycarbonate g/mol Therefore, polytetrafluoroethylene is the material since its repeat unit molecular weight is closest to that calculated above.
9 This problem asks if it is possible to have a poly(vinyl chloride homopolymer with the given molecular weight data and a degree of polymerization of The appropriate data are given below along with a computation of the number-average molecular weight. Molecular wt. Range Mean M i x i x i M i 8,000-20,000 14, ,000-32,000 26, ,000-44,000 38, ,000-56,000 50, ,000 56,000-68,000 62, ,160 68,000-80,000 74, ,000-92,000 86, M w x i M i 47,720 g/mol For PVC, from Table 14.3, each repeat unit has two carbons, three hydrogens, and one chlorine. Thus, m 2(A C + 3(A H + (A Cl (2(12.01 g/mol + (3(1.008 g/mol + (35.45 g/mol g/mol Now, we will compute the degree of polymerization using Equation 14.6 as DP M n m 47,720 g/mol g/mol 764 Thus, such a homopolymer is not possible since the calculated degree of polymerization is 764 not 1120.
10 (a For chlorinated polyethylene, we are asked to determine the weight percent of chlorine added for 8% Cl substitution of all original hydrogen atoms. Consider 50 carbon atoms; there are 100 possible side-bonding sites. Ninety-two are occupied by hydrogen and eight are occupied by Cl. Thus, the mass of these 50 carbon atoms, m C, is just m C 50(A C (50(12.01 g/mol g Likewise, for hydrogen and chlorine, m H 92(A H (92(1.008 g/mol g m Cl 8(A Cl (8(35.45 g/mol g Thus, the concentration of chlorine, C Cl, is determined using a modified form of Equation 4.3 as C Cl m Cl m C + m H + m Cl x g g g g x wt% (b Chlorinated polyethylene differs from poly(vinyl chloride, in that, for PVC, (1 25% of the sidebonding sites are substituted with Cl, and (2 the substitution is probably much less random.
11 14-11 Molecular Shape 14.9 This problem first of all asks for us to calculate, using Equation 14.11, the average total chain length, L, for a linear polyethylene polymer having a number-average molecular weight of 300,000 g/mol. It is necessary to calculate the number-average degree of polymerization, DP, using Equation For polyethylene, from Table 14.3, each repeat unit has two carbons and four hydrogens. Thus, m 2(A C + 4(A H (2(12.01 g/mol + (4(1.008 g/mol g/mol and DP M n m 300,000 g/mol g/mol 10,695 which is the number of repeat units along an average chain. Since there are two carbon atoms per repeat unit, there are two C C chain bonds per repeat unit, which means that the total number of chain bonds in the molecule, N, is just (2(10,695 21,390 bonds. Furthermore, assume that for single carbon-carbon bonds, d nm and θ 109 (Section 14.4; therefore, from Equation θ L Nd sin 2 (21,390(0.154 nm sin nm 2 It is now possible to calculate the average chain end-to-end distance, r, using Equation as r d N (0.154 nm 21, nm
12 (a This portion of the problem asks for us to calculate the number-average molecular weight for a linear polytetrafluoroethylene for which L in Equation is 2000 nm. It is first necessary to compute the value of N using this equation, where, for the C C chain bond, d nm, and θ 109. Thus N L d sin θ nm (0.154 nm sin ,900 Since there are two C C bonds per PTFE repeat unit, there is an average of N/2 or 15,900/ repeat units per chain, which is also the degree of polymerization, DP. In order to compute the value of M n using Equation 14.6, we must first determine m for PTFE. Each PTFE repeat unit consists of two carbon and four fluorine atoms, thus m 2(A C + 4(A F (2(12.01 g/mol + (4(19.00 g/mol g/mol Therefore M n (DPm (7950( g/mol 795,000 g/mol (b Next, we are to determine the number-average molecular weight for r 15 nm. Solving for N from Equation leads to N r2 d 2 (15 nm 2 (0.154 nm which is the total number of bonds per average molecule. Since there are two C C bonds per repeat unit, then DP N/2 9490/ Now, from Equation 14.6 M n (DPm (4745( g/mol 474,600 g/mol
13 14-13 Molecular Configurations We are asked to sketch portions of a linear polypropylene molecule for different configurations (using two-dimensional schematic sketches. (a Syndiotactic polypropylene (b Atactic polypropylene (c Isotactic polypropylene
14 This problem asks for us to sketch cis and trans structures for butadiene and chloroprene. (a The structure for cis polybutadiene (Table 14.5 is The structure of trans butadiene is (b The structure of cis chloroprene (Table 14.5 is The structure of trans chloroprene is
15 14-15 Thermoplastic and Thermosetting Polymers This question asks for comparisons of thermoplastic and thermosetting polymers. (a Thermoplastic polymers soften when heated and harden when cooled, whereas thermosetting polymers, harden upon heating, while further heating will not lead to softening. (b Thermoplastic polymers have linear and branched structures, while for thermosetting polymers, the structures will normally be network or crosslinked.
16 (a It is not possible to grind up and reuse phenol-formaldehyde because it is a network thermoset polymer and, therefore, is not amenable to remolding. (b Yes, it is possible to grind up and reuse polypropylene since it is a thermoplastic polymer, will soften when reheated, and, thus, may be remolded.
17 14-17 Copolymers This problem asks for sketches of the repeat unit structures for several alternating copolymers. (a For poly(ethylene-propylene (b For poly(butadiene-styrene (c For poly(isobutylene-isoprene
18 For a poly(acrylonitrile-butadiene alternating copolymer with a number-average molecular weight of 1,000,000 g/mol, we are asked to determine the average number of acrylonitrile and butadiene repeat units per molecule. Since it is an alternating copolymer, the number of both types of repeat units will be the same. Therefore, consider them as a single repeat unit, and determine the number-average degree of polymerization. For the acrylonitrile repeat unit, there are three carbon atoms, three hydrogen atoms, and one nitrogen atom, while the butadiene repeat consists of four carbon atoms and six hydrogen atoms. Therefore, the acrylonitrile-butadiene combined repeat unit weight is just m 7(A C + 9(A H + 1(A N (7(12.01 g/mol + (9(1.008 g/mol + (14.01 g/mol g/mol From Equation 14.6, the degree of polymerization is just DP M n m 1,000,000 g/mol g/mol 9333 Thus, there is an average of 9333 of both repeat unit types per molecule.
19 This problem asks for us to calculate the number-average molecular weight of a random poly(isobutylene-isoprene copolymer. For the isobutylene repeat unit there are four carbon and eight hydrogen atoms. Thus, its repeat unit molecular weight is m Ib 4(A C + 8(A H (4(12.01 g/mol + (8(1.008 g/mol g/mol The isoprene repeat unit is composed of five carbon and eight hydrogen atoms. Thus, its repeat unit molecular weight is m Ip 5(A C + 8(A H (5(12.01 g/mol + (8(1.008 g/mol g/mol From Equation 14.7, the average repeat unit molecular weight is just m f Ib m Ib + f Ip m Ip (0.25(56.10 g/mol + (0.75(68.11 g/mol g/mol Since DP 1500 (as stated in the problem, M n may be computed using Equation 14.6 as M n m (DP (65.11 g/mol( ,700 g/mol
20 For an alternating copolymer that has a number-average molecular weight of 100,000 g/mol and a degree of polymerization of 2210, we are to determine one of the repeat unit types if the other type is ethylene. It is first necessary to calculate m using Equation 14.6 as m M n DP 100,000 g/mol g/mol Since this is an alternating copolymer we know that chain fraction of each repeat unit type is 0.5; that is f e f x 0.5, f e and f x being, respectively, the chain fractions of the ethylene and unknown repeat units. Also, the repeat unit molecular weight for ethylene is m s 2(A C + 4(A H 2(12.01 g/mol + 4(1.008 g/mol g/mol Now, using Equation 14.7, it is possible to calculate the repeat unit weight of the unknown repeat unit type, m x. Thus m x m f e m e f x g/mol - (0.5(28.05 g/mol g/mol Finally, it is necessary to calculate the repeat unit molecular weights for each of the possible other repeat unit types. These are calculated below: m styrene 8(A C + 8(A H 8(12.01 g/mol + 8(1.008 g/mol g/mol m propylene 3(A C + 6(A H 3(12.01 g/mol + 6(1.008 g/mol g/mol m TFE 2(A C + 4(A F 2(12.01 g/mol + 4(19.00 g/mol g/mol m VC 2(A C + 3(A H + (A Cl 2(12.01 g/mol + 3(1.008 g/mol g/mol g/mol Therefore, vinyl chloride is the other repeat unit type since its m value is almost the same as the calculated m x.
21 (a This portion of the problem asks us to determine the ratio of butadiene to acrylonitrile repeat units in a copolymer having a weight-average molecular weight of 250,000 g/mol and a degree of polymerization of It first becomes necessary to calculate the average repeat unit molecular weight of the copolymer, m, using Equation 14.6 as m M n DP 250,000 g/mol g/mol If we designate f b as the chain fraction of butadiene repeat units, since the copolymer consists of only two repeat unit types, the chain fraction of acrylontrile repeat units f a is just 1 f b. Now, Equation 14.7 for this copolymer may be written in the form m f b m b + f a m a f b m b + (1 f b m a in which m b and m a are the repeat unit molecular weights for butadiene and acrylontrile, respectively. These values are calculated as follows: m b 4(A C + 6(A H 4(12.01 g/mol + 6(1.008 g/mol g/mol m a 3(A C + 3(A H + (A N 3(12.01 g/mol + 3(1.008 g/mol + (14.01 g/mol g/mol. Solving for f b in the above expression yields f b m m a m b m a g/mol g/mol g/mol g/mol 0.80 Furthermore, f a 1 f b ; or the ratio is just f b f a (b Of the possible copolymers, the only one for which there is a restriction on the ratio of repeat unit types is alternating; the ratio must be 1:1. Therefore, on the basis of the result in part (a, the possibilities for this copolymer are random, graft, and block.
22 For a copolymer consisting of 35 wt% ethylene and 65 wt% propylene, we are asked to determine the fraction of both repeat unit types. In 100 g of this material, there are 35 g of ethylene and 65 g of propylene. The ethylene (C 2 H 4 molecular weight is m(ethylene 2(A C + 4(A H (2(12.01 g/mol + (4(1.008 g/mol g/mol The propylene (C 3 H 6 molecular weight is m(propylene 3(A C + 6(A H (3(12.01 g/mol + (6(1.008 g/mol g/mol Therefore, in 100 g of this material, there are 35 g 1.25 mol of ethylene g /mol and 65 g 1.54 mol of propylene g /mol Thus, the fraction of the ethylene repeat unit, f(ethylene, is just f (ethylene 1.25 mol 1.25 mol mol 0.45 Likewise, f (propylene 1.54 mol 1.25 mol mol 0.55
23 For a random poly(styrene-butadiene copolymer in which M n 350,000 g/mol and DP 5000, we are asked to compute the fractions of styrene and butadiene repeat units. From Table 14.5, the styrene repeat unit has eight carbon and eight hydrogen atoms. Thus, m st (8(12.01 g/mol + (8(1.008 g/mol g/mol Also, from Table 14.5, the butadiene repeat unit has four carbon and six hydrogen atoms, and m bu (4(12.01 g/mol + (6(1.008 g/mol g/mol From Equation 14.7 m f st m st + f bu m bu Now, let x f st, such that m x + (54.09(1 x since f st + f bu 1. Also, from Equation 14.6 Or DP M n m ,000 g /mol [ x (1 x] g /mol Solving for x leads to x f st f(styrene Also, f(butadiene 1 x
24 14-24 Polymer Crystallinity The tendency of a polymer to crystallize decreases with increasing molecular weight because as the chains become longer it is more difficult for all regions along adjacent chains to align so as to produce the ordered atomic array.
25 For each of four pairs of polymers, we are asked to (1 state whether it is possible to decide which is more likely to crystallize; (2 if so, which is more likely and why; and (3 it is not possible to decide then why. (a No, it is not possible to decide for these two polymers. On the basis of tacticity, the isotactic PP is more likely to crystallize than the atactic PVC. On the other hand, with regard to side-group bulkiness, the PVC is more likely to crystallize. (b Yes, it is possible to decide for these two copolymers. The linear and syndiotactic polypropylene is more likely to crystallize than crosslinked cis-isoprene since linear polymers are more likely to crystallize than crosslinked ones. (c Yes, it is possible to decide for these two polymers. The linear and isotactic polystyrene is more likely to crystallize than network phenol-formaldehyde; network polymers rarely crystallize, whereas isotactic ones crystallize relatively easily. (d Yes, it is possible to decide for these two copolymers. The block poly(acrylonitrile-isoprene copolymer is more likely to crystallize than the graft poly(chloroprene-isobutylene copolymer. Block copolymers crystallize more easily than graft ones.
26 For this problem we are given the density of nylon 6,6 (1.213 g/cm 3, an expression for the volume of its unit cell, and the lattice parameters, and are asked to determine the number of repeat units per unit cell. This computation necessitates the use of Equation 3.5, in which we solve for n. Before this can be carried out we must first calculate V C, the unit cell volume, and A the repeat unit molecular weight. For V C V C abc 1 cos 2 α cos 2 β cos 2 γ + 2 cosα cosβ cos γ (0.497(0.547( (0.664(0.232( nm x cm 3 The repeat unit for nylon 6,6 is shown in Table 14.3, from which the value of A may be determined as follows: A 12(A C + 22(A H + 2(A O + 2(A N 12(12.01 g/mol + 22(1.008 g/mol + 2(16.00 g/mol + 2(14.01 g/mol g/mol Finally, solving for n from Equation 3.5 leads to n ρv C N A A (1.213 g/cm3 (3.098 x cm 3 /unit cell(6.023 x repeat units/mol g/mol 1 repeat unit/unit cell
27 (a We are asked to compute the densities of totally crystalline and totally amorphous poly(ethylene terephthalate (ρ c and ρ a from Equation From Equation 14.8 let C % crystallinity, such that 100 C ρ c (ρ s ρ a ρ s (ρ c ρ a Rearrangement of this expression leads to ρ c (C ρ s ρ s + ρ c ρ a Cρ s ρ a 0 in which ρ c and ρ a are the variables for which solutions are to be found. Since two values of ρ s and C are specified in the problem statement, two equations may be constructed as follows: ρ c (C 1 ρ s1 ρ s1 + ρ c ρ a C 1 ρ s1 ρ a 0 ρ c (C 2 ρ s2 ρ s2 + ρ c ρ a C 2 ρ s2 ρ a 0 In which ρ s g/cm 3, ρ s g/cm 3, C , and C Solving the above two equations for ρ a and ρ c leads to ρ a ρ s1 ρ s2 (C 1 C 2 C 1 ρ s1 C 2 ρ s2 (1.408 g/cm 3 (1.343 g/cm 3 ( (0.743(1.408 g/cm 3 (0.312(1.343 g/cm g/cm3 And ρ c ρ s1 ρ s2 (C 2 C 1 ρ s2 (C 2 1 ρ s1 (C 1 1 (1.408 g/cm 3 (1.343 g/cm 3 ( (1.343 g/cm 3 ( (1.408 g/cm 3 ( g/cm3 (b Now we are to determine the % crystallinity for ρ s g/cm 3. Again, using Equation 14.8
28 14-28 % crystallinity ρ c (ρ s ρ a ρ s (ρ c ρ a 100 (1.450 g/cm3 (1.382 g/cm g/cm 3 (1.382 g/cm 3 (1.450 g/cm g/cm %
29 (a We are asked to compute the densities of totally crystalline and totally amorphous polypropylene (ρ c and ρ a from Equation From Equation 14.8 let C % crystallinity, such that 100 C ρ c (ρ s ρ a ρ s (ρ c ρ a Rearrangement of this expression leads to ρ c (C ρ s ρ s + ρ c ρ a C ρ s ρ a 0 in which ρ c and ρ a are the variables for which solutions are to be found. Since two values of ρ s and C are specified in the problem, two equations may be constructed as follows: ρ c (C 1 ρ s1 ρ s1 + ρ c ρ a C 1 ρ s1 ρ a 0 ρ c (C 2 ρ s2 ρ s2 + ρ c ρ a C 2 ρ s2 ρ a 0 In which ρ s g/cm 3, ρ s g/cm 3, C , and C Solving the above two equations for ρ a and ρ c leads to ρ a ρ s1 ρ s2 (C 1 C 2 C 1 ρ s1 C 2 ρ s2 (0.904 g /cm3 (0.895 g /cm 3 ( (0.628(0.904 g /cm 3 (0.544(0.895 g /cm g/cm3 And ρ c ρ s1 ρ s2 (C 2 C 1 ρ s2 (C 2 1 ρ s1 (C 1 1 (0.904 g /cm 3 (0.895 g /cm 3 ( (0.895 g /cm 3 ( (0.904 g /cm g/cm3 ( (b Now we are asked to determine the density of a specimen having 74.6% crystallinity. Solving for ρ s from Equation 14.8 and substitution for ρ a and ρ c which were computed in part (a yields
30 14-30 ρ s ρ c ρ a C (ρ c ρ a ρ c (0.946 g /cm 3 (0.841 g /cm 3 (0.746(0.946 g /cm g /cm g /cm g/cm 3
31 14-31 Diffusion in Polymeric Materials This is a permeability problem in which we are asked to compute the diffusion flux of oxygen through a 15-mm thick sheet of low density polyethylene. In order to solve this problem it is necessary to employ Equation The permeability coefficient of O 2 through LDPE is given in Table 14.6 as 2.2 x (cm 3 STP- cm/cm 2 -s-pa. Thus, from Equation 14.9 J P M P x P M P 2 P 1 x and taking P kpa (150,000 Pa and P kpa (2,000,000 Pa we get 2.2 x (cm 3 STP(cm 2,000,000 Pa - 150, 000 Pa cm 2 -s-pa 1.5 cm 2.7 x 10-6 (cm 3 STP cm 2 -s
32 This problem asks us to compute the permeability coefficient for carbon dioxide through high density polyethylene at 325 K given a steady-state permeability situation. It is necessary for us to Equation 14.9 in order to solve this problem. Rearranging this expression and solving for the permeability coefficient gives P M J x P J x P 2 P 1 Taking P kpa (2,500,000 Pa and P kpa (4,000,000 Pa, the permeability coefficient of CO 2 through HDPE is equal to 2.2 x 10-8 (cm 3 STP cm 2 (5 cm -s P M (4,000,000 Pa - 2,500,000 Pa 0.73 x (cm 3 STP(cm cm 2 -s-pa
33 This problem asks that we compute the diffusion flux at 350 K for water in polystyrene. It is first necessary to compute the value of the permeability coefficient at 350 K. The temperature dependence of P M is given in the problem statement, as follows: P M P M0 exp Q p RT And, incorporating values provided for the constants P M0 and Q p, we get P M 9.0 x 10-5 (cm 3 STP(cm cm 2 -s-pa exp 42,300 J/mol (8.314 J/mol- K(350 K 4.4 x (cm 3 STP(cm cm 2 -s-pa And, using Equation 14.9, the diffusion flux is equal to J P M P x P M P 2 P 1 x 4.4 x (cm 3 STP(cm cm 2 -s-pa 20,000 Pa -1,000 Pa 3.0 cm 2.8 x 10-7 (cm 3 STP cm 2 -s
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14:635:407:02 Homework III Solutions 4.1 Calculate the fraction of atom sites that are vacant for lead at its melting temperature of 327 C (600 K). Assume an energy for vacancy formation of 0.55 ev/atom.
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