Question

The repeat unit of the polystyrene of a coffee cup has the formula \(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{CHCH}_{2}\). If the molar mass of the polymer is \(3.5 \times 10^{5} \mathrm{~g} / \mathrm{mol},\) what is the degree of polymerization?

Short answer

The degree of polymerization is approximately 4369.

Step-by-step solution

Determine the Molar Mass of the Repeat Unit

Calculate the molar mass of the repeat unit \(C_6H_5CHCH_2\). Add the atomic masses of all the atoms in the formula: \(6 \times 12.01\) for carbon, \(8 \times 1.01\) for hydrogen.

Calculate the Molar Mass of the Repeat Unit

The molar mass of Carbon (C) is 12.01 g/mol and Hydrogen (H) is 1.01 g/mol. Therefore, the molar mass of the repeat unit is:\[6 \times 12.01 + 8 \times 1.01 = 72.06 + 8.08 = 80.14 \text{ g/mol}\]

Find the Degree of Polymerization

The degree of polymerization (n) is the ratio of the molar mass of the polymer to the molar mass of the repeat unit. Use the given molar mass of the polymer: \(3.5 \times 10^5 \text{ g/mol}\). Divide this by the molar mass of the repeat unit calculated in Step 2. \[n = \frac{3.5 \times 10^5}{80.14} = 4368.8 \]

Key concepts

Molar Mass Calculation

Understanding how to calculate the molar mass is crucial for many chemistry problems, such as determining the degree of polymerization. The molar mass is the mass of one mole of a given molecule or compound, usually expressed in g/mol.
To calculate the molar mass of a repeat unit, you need to sum the atomic masses of all the atoms in its chemical formula.

• Consider each element in the formula and multiply its atomic mass by the number of times it appears.
• For instance, in the repeat unit \(\text{C}_6\text{H}_5\text{CHCH}_2\), there are 6 carbon atoms, 8 hydrogen atoms, and another carbon and hydrogen in the extra group.
• The atomic mass of carbon (C) is approximately 12.01 g/mol, and hydrogen (H) is 1.01 g/mol.

Calculate the molar mass step-by-step:

• For carbon: \[6 \times 12.01 = 72.06 \] g/mol
• For hydrogen: \[8 \times 1.01 = 8.08 \] g/mol

Add these values together: \[72.06 + 8.08 = 80.14 \] g/mol.
This final value, 80.14 g/mol, is the molar mass of one repeat unit of polystyrene.

Repeat Unit

A repeat unit in polymer chemistry is the smallest segment of the polymer chain that repeats to form the entire polymer.
For polystyrene, the repeat unit is \(\text{C}_6\text{H}_5\text{CHCH}_2\). It contains:

• A benzene ring (\text{C}_6\text{H}_5) attached to a vinyl group (\text{CHCH}_2).

The repeat unit plays a vital role because it allows us to determine the properties and structure of the entire polymer.

• Knowing the repeat unit helps in calculating the polymer's molar mass when scaled to many repeat units.
• It simplifies understanding the polymer's physical and chemical characteristics.

When multiple repeat units join, they form long chains, known as macromolecules, which define the polymer's properties.

Polystyrene Chemistry

Polystyrene is a synthetic aromatic hydrocarbon polymer made from the monomer styrene. It's widely used due to its clarity, rigidity, and insulation properties.
Some key points about polystyrene in chemistry include:

• It is produced via the polymerization of styrene monomers (\text{CH}_2\text{=CH}\text{C}_6\text{H}_5).
• The repeat unit of polystyrene is \(\text{C}_6\text{H}_5\text{CHCH}_2\).

The degree of polymerization (n) is a measure of how many repeat units are linked together in a polymer chain.

• This is calculated by dividing the molar mass of the entire polymer by the molar mass of its repeat unit.

For example, for a polystyrene polymer with a molar mass of \3.5 \times 10^5\} g/mol:

• \mathit{n} = \frac{3.5 \times 10^5}{80.14} \
• \mathit{n} \approx 4368.8 \

Thus, the degree of polymerization is approximately 4369, indicating how many repeat units combine to form the full polymer.