Chapter 12, Problem 12.131P

(a)

Interpretation Introduction

Interpretation:

The molar mass of each fraction is to be calculated.

Concept introduction:

Molar mass is defined as the summation of the masses of the atoms that are present in one mole of a particular molecule.

A polymer is a macromolecule that is formed from the covalently bonded structure of small molecules known as monomers. Polymers are of two types as follows:

1. Natural polymers

2. Synthetic polymers

The molecular mass of polymer depends upon the mass of monomer and number of monomer. The formula to calculate the molecular mass of polymer is as follows:

  $M_{\text{polymer}}=\left(M_{\text{monomer}}\right)\left(n\right)$        (1)

Here,

$M_{\text{polymer}}$ is the molar mass of the polymer.

$M_{\text{monomer}}$ is the molar mass of monomer.

$n$ is the degree of polymerization.

Answer to Problem 12.131P

The molar mass of fraction 1, 2, 3, 4, 5, 6 are $2.73\times {10}^4\text{g/mol}$, $3.30\times {10}^4\text{g/mol}$, $3.68\times {10}^4\text{g/mol}$, $4.83\times {10}^4\text{g/mol}$, $5.25\times {10}^4\text{g/mol}$ and $5.75\times {10}^4\text{g/mol}$ respectively.

Explanation of Solution

Substitute 273 for $n$ and $100.02\text{ g/mol}$ for $M_{\text{monomer}}$ in equation (1) to calculate the molar mass of fraction 1.

  $\begin{array}{c}{M}_{\text{polymer}}=\left(100.02\text{ g/mol}\right)\left(273\right)\\ =2.7305\times {10}^4\text{g/mol}\\ \approx 2.73\times {10}^4\text{g/mol}\end{array}$

Substitute 330 for $n$ and $100.02\text{ g/mol}$ for $M_{\text{monomer}}$ in equation (1) to calculate the molar mass of fraction 2.

  $\begin{array}{c}{M}_{\text{polymer}}=\left(100.02\text{ g/mol}\right)\left(330\right)\\ =3.30066\times {10}^4\text{g/mol}\\ \approx 3.30\times {10}^4\text{g/mol}\end{array}$

Substitute 368 for $n$ and $100.02\text{ g/mol}$ for $M_{\text{monomer}}$ in the equation (1) to calculate the molar mass of fraction 3.

  $\begin{array}{c}{M}_{\text{polymer}}=\left(100.02\text{ g/mol}\right)\left(368\right)\\ =3.6807\times {10}^4\text{g/mol}\\ \approx 3.68\times {10}^4\text{g/mol}\end{array}$

Substitute 483 for $n$ and $100.02\text{ g/mol}$ for $M_{\text{monomer}}$ in equation (1) to calculate the molar mass of fraction 4.

  $\begin{array}{c}{M}_{\text{polymer}}=\left(100.02\text{ g/mol}\right)\left(483\right)\\ =4.83097\times {10}^4\text{g/mol}\\ \approx 4.83\times {10}^4\text{g/mol}\end{array}$

Substitute 525 for $n$ and $100.02\text{ g/mol}$ for $M_{\text{monomer}}$ in equation (1) to calculate the molar mass of fraction 5.

  $\begin{array}{c}{M}_{\text{polymer}}=\left(100.02\text{ g/mol}\right)\left(525\right)\\ =5.25105\times {10}^4\text{g/mol}\\ \approx 5.25\times {10}^4\text{g/mol}\end{array}$

Substitute 575 for $n$ and $100.02\text{ g/mol}$ for $M_{\text{monomer}}$ in equation (1) to calculate the molar mass of fraction 6.

  $\begin{array}{c}{M}_{\text{polymer}}=\left(100.02\text{ g/mol}\right)\left(575\right)\\ =5.75115\times {10}^4\text{g/mol}\\ \approx 5.75\times {10}^4\text{g/mol}\text{.}\end{array}$

Conclusion

The molecular mass of polymer depends upon the mass of monomer and number of monomer.

(b)

Interpretation Introduction

Interpretation:

The number-average molar mass of the sample is to be calculated.

Concept introduction:

The number-average molar mass is a method to determine the molar mass of the polymer. It is calculated by the average of the molecular masses of the individual monomer unit that combine to form a polymer. The formula to calculate the number-average molar mass of the sample is as follows:

  $\text{Number-average molar mass}=\frac{\text{total sum of all chains}}{\text{number of moles of chains}}$

Answer to Problem 12.131P

The number-average molar mass is $4.2\times {10}^4\text{g/mol}$.

Explanation of Solution

The formula to calculate the number-average molar mass of the sample is as follows:

  $\text{Number-average molar mass}=\frac{{\displaystyle \sum \left[\begin{array}{l}\left(M_{\text{monomer}}\right)\\ \left(\begin{array}{l}\left(\begin{array}{l}averagenumberof\\ repeatingunit\end{array}\right)\\ \left(molofpolymer\right)\end{array}\right)\end{array}\right]}}{\text{number of moles of chains}}$        (2)

Substitute the values in the equation (2) to calculate the number-average molar mass of the sample.

  $\begin{array}{c}\text{Number-average molar mass}=\frac{\left[\left(100.02\text{ g/mol}\right)\left(\begin{array}{l}\left(273\right)\left(0.10\right)+\left(330\right)\left(0.40\right)+\\ \left(368\right)\left(1.00\right)+\left(483\right)\left(0.70\right)\\ +\left(525\right)\left(0.30\right)+\left(575\right)\left(0.10\right)\end{array}\right)\right]}{\left(0.10+0.40+1.00+0.70+0.30+0.10\right)}\\ =4.1562\times {10}^4\text{g/mol}\\ \approx 4.2\times {10}^4\text{g/mol}\text{.}\end{array}$

Conclusion

The number-average molar mass is one of the method to determine the molar mass of the polymer. The number of monomer varies and therefore it is difficult to determine the molar mass of polymer.

(c)

Interpretation Introduction

Interpretation:

The weight-average molar mass of the sample of polytetra fluoroethylene is to be calculated.

Concept introduction:

The weight-average molar mass is a method to determine the molar mass of the polymer. The formula to calculate the weight-average molar mass of the sample is as follows:

  $\text{Weight-average molar mass}=\frac{{\displaystyle \sum \left(\text{Mass of fraction}\right)\left(\text{mass of fraction}\right)}}{\text{total mass of all fractions}}$

Answer to Problem 12.131P

The weight-average molar mass of the sample is $4.3\times {10}^4\text{g/mol}$.

Explanation of Solution

The formula to calculate weight-average molar mass of the sample is as follows:

  $\text{weight-average}\text{molar mass}=\frac{{\displaystyle \sum \left[\left(\begin{array}{l}{\left(\text{molar mass}of\text{fraction}\right)}^2\\ \left(molofpolymer\right)\end{array}\right)\right]}}{{\displaystyle \sum \left[\left(\begin{array}{l}\left(\text{molar mass}of\text{fraction}\right)\\ \left(molofpolymer\right)\end{array}\right)\right]}}$        (3)

Substitute the values in the equation (3) to calculate the weight-average molar mass of the sample.

  $\begin{array}{c}{M}_{\text{w}}=\frac{\left(\begin{array}{l}{\left(2.73\times {10}^4\text{g/mol}\right)}^2\left(\text{0}\text{.10}\text{mol}\right)+{\left(3.30\times {10}^4\text{g/mol}\right)}^2\left(0.40\text{mol}\right)+\\ {\left(3.68\times {10}^4\text{g/mol}\right)}^2\left(1.00\text{mol}\right)+{\left(4.83\times {10}^4\text{g/mol}\right)}^2\left(0.70\text{mol}\right)+\\ {\left(5.25\times {10}^4\text{g/mol}\right)}^2\left(0.30\text{mol}\right)+{\left(5.75\times {10}^4\text{g/mol}\right)}^2\left(0.10\text{mol}\right)\end{array}\right)}{\left(\begin{array}{l}\left(2.73\times {10}^4\text{g/mol}\right)\left(\text{0}\text{.10}\text{mol}\right)+\left(3.30\times {10}^4\text{g/mol}\right)\left(0.40\text{mol}\right)+\\ \left(3.68\times {10}^4\text{g/mol}\right)\left(1.00\text{mol}\right)+\left(4.83\times {10}^4\text{g/mol}\right)\left(0.70\text{mol}\right)+\\ \left(5.25\times {10}^4\text{g/mol}\right)\left(0.30\text{mol}\right)+\left(5.75\times {10}^4\text{g/mol}\right)\left(0.10\text{mol}\right)\end{array}\right)}\\ =4.3085\times {10}^4\text{g/mol}\\ \approx 4.3\times {10}^4\text{g/mol}\text{.}\end{array}$

Conclusion

The weight-average molar mass is one of the method to determine the molar mass of the polymer. The number of monomer varies and therefore it is difficult to determine the molar mass of a polymer