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Chapter 14: Problem 7

Is it possible to have a poly(methyl methacrylate) homopolymer with the following molecular weight data and a degree of polymerization of \(527 ?\) Why or why not? \begin{tabular}{lcc} \hline \multicolumn{1}{c}{ Molecular Weight Range \((\mathrm{g} /\) mol \()\)} & \(\boldsymbol{w}_{i}\) & \(\boldsymbol{x}_{\boldsymbol{i}}\) \\ \hline \(8,000-20,000\) & \(0.02\) & \(0.05\) \\ \(20,000-32,000\) & \(0.08\) & \(0.15\) \\ \(32,000-44,000\) & \(0.17\) & \(0.21\) \\ \(44,000-56,000\) & \(0.29\) & \(0.28\) \\ \(56,000-68,000\) & \(0.23\) & \(0.18\) \\ \(68,000-80,000\) & \(0.16\) & \(0.10\) \\ \(80,000-92,000\) & \(0.05\) & \(0.03\) \\ \hline \end{tabular}

Short Answer

Expert verified
Answer: No, it is not possible to have a poly(methyl methacrylate) homopolymer with the given molecular weight data and a degree of polymerization of 527.

Step by step solution

01

Calculate the middle value of the Molecular Weight Range

We will find the middle value of each molecular weight range by taking the average of the upper and lower limits of the molecular weight range. For example, the middle value for the first range (\(8,000-20,000\) g/mol) will be calculated as: Middle Value = \(\frac{8000 + 20000}{2} = 14,000\) g/mol We will repeat this for all molecular weight ranges.
02

Calculate the Weight-average Molecular Weight (\(\overline{M}_{w}\))

We will use the formula for the weight-average molecular weight, \(\overline{M}_{w}\) and the given data: \(\overline{M}_{w} = \sum_{i} w_{i}M_{i}\), where \(w_{i}\) represents the weight fraction and \(M_{i}\) is the middle value of the molecular weight range. \(\begin{aligned} \overline{M}_{w} = & (0.02)(14,000)+(0.08)(26,000)+(0.17)(38,000)+(0.29)(50,000) \\ & +(0.23)(62,000)+(0.16)(74,000)+(0.05)(86,000) \\ & = 7,240 \end{aligned}\)
03

Calculate the Number-average Molecular Weight (\(\overline{M}_{n}\))

We will use the formula for the number-average molecular weight, \(\overline{M}_{n}\), and the given data: \(\overline{M}_{n} = \frac{\sum_{i} w_{i}M_{i}}{\sum_{i} x_{i}}\), where \(x_{i}\) represents the mole fraction. \(\begin{aligned} \overline{M}_{n} = & \frac{(0.02)(14,000)+(0.08)(26,000)+(0.17)(38,000)+(0.29)(50,000)+(0.23)(62,000)+(0.16)(74,000)+(0.05)(86,000)}{(0.05)+(0.15)+(0.21)+(0.28)+(0.18)+(0.10)+(0.03)} \\ & = \frac{7,240}{1} \\ & = 7,240 \end{aligned}\)
04

Determine the degree of polymerization

We will use the given degree of polymerization, \(DP = 527\), and the molar mass of the repeating unit, Methyl Methacrylate, which is approximately 100 g/mol. Then we can calculate the expected number-average molecular weight based on the given degree of polymerization: Expected \(\overline{M}_{n}\) = \(DP \times M_{repeating\:unit}\) = \(527 \times 100\) = \(52,700\)
05

Comparing the calculated and expected molecular weights

The calculated number-average molecular weight (\(\overline{M}_{n} = 7,240\) g/mol) does not match the expected number-average molecular weight based on the degree of polymerization (\(52,700\) g/mol). Therefore, it is not possible to have a poly(methyl methacrylate) homopolymer with the given molecular weight data and a degree of polymerization of 527.

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Most popular questions from this chapter

On the basis of the structures presented in this chapter, sketch repeat unit structures for the following polymers: (a) polychlorotrifluoroethylene, and (b) poly(vinyl alcohol).

The number-average molecular weight of a poly(styrene-butadiene) alternating copolymer is \(1,350,000 \mathrm{~g} / \mathrm{mol}\); determine the average number of styrene and butadiene repeat units per molecule.

Make comparisons of thermoplastic and thermosetting polymers (a) on the basis of mechanical characteristics upon heating and (b) according to possible molecular structures.

For each of the following pairs of polymers, do the following: (1) state whether it is possible to determine whether one polymer is more likely to crystallize than the other; (2) if it is possible, note which is the more likely and then cite reason(s) for your choice; and (3) if it is not possible to decide, then state why. (a) Linear and syndiotactic poly(vinyl chloride); linear and isotactic polystyrene (b) Network phenol-formaldehyde; linear and heavily crosslinked \(c i s\)-isoprene (c) Linear polyethylene; lightly branched isotactic polypropylene (d) Alternating poly(styrene-ethylene) copolymer; random poly(vinyl chloridetetrafluoroethylene) copolymer

For a linear freely rotating polymer molecule, the total extended chain length \(L\) depends on the bond length between chain atoms \(d\), the total number of bonds in the molecule \(N\), and the angle between adjacent backbone chain atoms \(\theta\), as follows: $$ L=N d \sin \left(\frac{\theta}{2}\right) $$ Furthermore, the average end-to-end distance \(r\) for a randomly winding polymer molecule in Figure \(14.6\) is equal to $$ r=d \sqrt{N} $$ A linear polytetrafluoroethylene has a numberaverage molecular weight of \(500,000 \mathrm{~g} / \mathrm{mol}\); compute average values of \(L\) and \(r\) for this material.
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