Question

Polytetrafluoroethylene (Teflon) has a repeat unit with the formula \(\mathrm{F}_{2} \mathrm{C}-\mathrm{CF}_{2}\). A sample of the polymer consists of fractions with the following distribution of chains: $$ \begin{array}{ccc} & \text { Average Number } & \text { Amount (mol) } \\ \text { Fraction } & \text { of Repeat Units } & \text { of Polymer } \\ \hline 1 & 273 & 0.10 \\ 2 & 330 & 0.40 \\ 3 & 368 & 1.00 \\ 4 & 483 & 0.70 \\ 5 & 525 & 0.30 \\ 6 & 575 & 0.10 \end{array} $$ (a) Determine the molar mass of each fraction. (b) Determine the number-average molar mass of the sample. (c) Another type of average molar mass of a polymer sample is called the weight-average molar mass, \(\mathscr{H}_{\mathrm{w}}\) : \(\mathscr{M}_{\mathrm{w}}=\frac{\Sigma(\mathscr{M} \text { of fraction } \times \text { mass of fraction })}{\text { total mass of all fractions }}\) Calculate the weight-average molar mass of the sample of polytetrafluoroethylene.

Short answer

The number-average molar mass is 25763.38 g/mol and the weight-average molar mass is 47183.95 g/mol.

Step-by-step solution

- Molar Mass of a Repeat Unit

First, determine the molar mass of the repeat unit \(\mathrm{F}_2\mathrm{C}-\mathrm{CF}_2\). Each fluorine (F) has a molar mass of approximately 19 g/mol, and each carbon (C) has a molar mass of approximately 12 g/mol. Therefore, the molar mass of one repeat unit is: \[ 2(19)+2(12) = 62 \mathrm{g/mol} \]

- Calculate Molar Mass of Each Fraction

Next, calculate the molar mass of each fraction by multiplying the molar mass of the repeat unit by the average number of repeat units in that fraction. \[\text{Molar Mass of Fraction 1} = 62 \(\mathrm{g/mol}\) \times 273 = 16926 \(\mathrm{g/mol}\) \text{Molar Mass of Fraction 2} = 62 \(\mathrm{g/mol}\) \times 330 = 20460 \(\mathrm{g/mol}\) \text{Molar Mass of Fraction 3} = 62 \(\mathrm{g/mol}\) \times 368 = 22816 \(\mathrm{g/mol}\) \text{Molar Mass of Fraction 4} = 62 \(\mathrm{g/mol}\) \times 483 = 29946 \(\mathrm{g/mol}\) \text{Molar Mass of Fraction 5} = 62 \(\mathrm{g/mol}\) \times 525 = 32550 \(\mathrm{g/mol}\) \text{Molar Mass of Fraction 6} = 62 \(\mathrm{g/mol}\) \times 575 = 35650 \(\mathrm{g/mol}\) \]

- Determine Number-Average Molar Mass

The number-average molar mass (\(\mathscr{M}_n\)) is calculated as follows: \[ \mathscr{M}_n = \frac{\Sigma(\mathscr{M} \text{ of fraction} \times \text{amount (mol)})}{\text{total amount (mol)}} \] Using the values calculated in Step 2: \[ \mathscr{M}_n = \frac{16926 \times 0.10 + 20460 \times 0.40 + 22816 \times 1.00 + 29946 \times 0.70 + 32550 \times 0.30 + 35650 \times 0.10}{0.10 + 0.40 + 1.00 + 0.70 + 0.30 + 0.10} \] Simplify the expression: \[ \mathscr{M}_n = \frac{(1692.6 + 8184 + 22816 + 20962.2 + 9765 + 3565)}{2.60} = \frac{66984.8}{2.60} = 25763.38 \text{g/mol} \]

- Determine Weight-Average Molar Mass

The weight-average molar mass (\(\mathscr{M}_w\)) is calculated as follows: \[ \mathscr{M}_w = \frac{\Sigma(\mathscr{M} \text{ of fraction} \times \text{amount (mol)} \times \mathscr{M} \text{ of fraction})}{\Sigma(\mathscr{M} \text{ of fraction} \times \text{amount (mol)})} \] Using the values calculated in Step 2: \[ \mathscr{M}_w = \frac{(16926 \times 1692.6) + (20460 \times 8184) + (22816 \times 22816) + (29946 \times 20962.2) + (32550 \times 9765) + (35650 \times 3565)}{(16926 \times 0.10) + (20460 \times 0.40) + (22816 \times 1.00) + (29946 \times 0.70) + (32550 \times 0.30) + (35650 \times 0.10)} \] Simplify and solve: \[ \mathscr{M}_w = \frac{28701468 + 167309440 + 520474656 + 628797732 + 317332500 + 127119000}{1692.6 + 8184 + 22816 + 20962.2 + 9765 + 3565} \ = 3098256796.6 / 65640.80 \ = 47183.95 \text{g/mol} \]

Key concepts

molar mass calculation

In chemistry, the molar mass is the mass of one mole of a given substance. Specifically, for a polymer like Polytetrafluoroethylene (Teflon), its repeat unit is \(\text{F}_2\text{C}-\text{CF}_2\). To find the molar mass of one repeat unit, you sum up the atomic masses of all atoms within the unit.
Each fluorine (F) atom has a molar mass of approximately 19 g/mol, and each carbon (C) atom has a molar mass of approximately 12 g/mol. For the repeat unit of Teflon, which has 2 fluorine atoms and 2 carbon atoms, the molar mass calculation will be:

⇳ Molar mass of repeat unit = \[2 \times (F) + 2 \times (C)\right\]\ ⇳ Molar mass of repeat unit = \[2 \times 19 + 2 \times 12 = 62 \text{ g/mol} \]
Now, using the average number of repeat units for different fractions in a sample, we can calculate the molar mass of each fraction. For instance, for Fraction 1 with an average of 273 repeat units, the molar mass would be:
⇳ Molar mass of Fraction 1 = 62 \times 273 = 16926 \text{ g/mol}
By following the same steps, you'll calculate the molar mass for other fractions based on their average number of repeat units.

polymer chemistry

Polymer chemistry involves the study of large molecules made up of repeating units, known as monomers. In the context of Teflon, the main repeat unit is \(\text{F}_2\text{C}-\text{CF}_2)\), which is derived from the monomer tetrafluoroethylene. Each polymer chain consists of many of these repeat units.
Polymers like Teflon exhibit unique properties due to their large size and repetitive structure. They are known for their high chemical resistance and extremely low friction, making them suitable for applications like non-stick coatings and electrical insulation.
The distribution of chain lengths in a polymer sample can vary. In practice, polymers are synthesized such that the chains can have a range of lengths due to the nature of the polymerization process. This is why calculations such as number-average molar mass (\text{M}_n) and weight-average molar mass (\text{M}_w) are essential.
Understanding the distribution of the chain lengths helps in determining the physical properties of the polymer, such as tensile strength, elasticity, and melting point.

number-average molar mass

The number-average molar mass (\text{M}_n) is a way to average the molar masses of polymer molecules based on their number. It's calculated by taking each fraction's molar mass, multiplying it by the number of moles of that fraction, summing these values, and then dividing by the total number of moles in the sample.
In mathematical terms, \[\mathscr{M}_n = \frac{\text{Sum of (Molar mass of fraction} \times \text{Amount (mol)})} {\text{Total amount (mol)}} \right\]\For instance, to find the number-average molar mass for Teflon with the provided data:
⇳ Number-average molar mass = \[\frac{(1692.6 + 8184 + 22816 + 20962.2 + 9765 + 3565)}{2.60} \right\]
⇳ Number-average molar mass = 25763.38 \text{ g/mol}
This calculation helps in understanding the average size of the polymer molecules, which impacts various physical properties.

weight-average molar mass

The weight-average molar mass (\text{M}_w) provides a weighted average based on the mass of polymer molecules. It accounts for the fact that larger molecules contribute more to the polymer’s total mass. The formula for calculating \text{M}_w is:
\text{M}_w = \[\frac{\text{Sum of (Molar mass of fraction} \times \text{Mass of fraction)}}{\text{Total mass of all fractions}}\right\]
In the context of our Teflon example, it’s calculated by:
⇳ Weight-average molar mass = \[\frac{28701468 + 167309440 + 520474656 + 628797732 + 317332500 + 127119000}{1692.6 + 8184 + 22816 + 20962.2 + 9765 + 3565}\right\]
⇳ Weight-average molar mass = 47183.95 \text{ g/mol}
Weight-average molar mass is important because it emphasizes the impact larger molecules have on the overall properties, especially mechanical strength and viscosity, of the polymer material.
By comparing \text{M}_n and \text{M}_w, you can get insights into the distribution of molecular weights within the polymer sample. If \text{M}_w is significantly higher than \text{M}_n, it indicates the presence of many high-molar-mass molecules.