Question

Calculate the molar mass of a particular polyethylene sample, \(\left.+\mathrm{CH}_{2}-\mathrm{CH}_{2}\right)_{n},\) where \(n=4600 .\)

Short answer

The molar mass of the polyethylene sample is 129,076 g/mol.

Step-by-step solution

Identify the Repeating Unit

Polyethylene is represented by the formula \((\text{–CH}_2–CH}_2)_n\). Each of these units, known as the repeating unit, consists of two carbon atoms and four hydrogen atoms.

Calculate the Molar Mass of the Repeating Unit

To find the molar mass of the repeating unit \(-\text{CH}_2–CH}_2\), we sum up the molar masses of all atoms in it. The molar mass of a carbon atom \(\text{C}\) is approximately \(12.01\ \text{g/mol}\), and for a hydrogen atom \(\text{H}\) it is about \(1.01\ \text{g/mol}\). Therefore, the molar mass of the repeating unit is:\[2 \times 12.01\ \text{g/mol (C)} + 4 \times 1.01\ \text{g/mol (H)} = 28.06\ \text{g/mol}\]

Calculate the Total Molar Mass

Given that the sample has \(n = 4600\) repeating units, multiply the molar mass of a single unit by 4600 to find the total molar mass of the polyethylene. \[\text{Total}\ \text{Molar}\ \text{Mass} = 4600 \times 28.06\ \text{g/mol} = 129076\ \text{g/mol}\]

Conclusion

The molar mass of the entire polyethylene sample is thus \(129076\ \text{g/mol}\).

Key concepts

Polyethylene

Polyethylene is a widely used plastic, and it's essential to our everyday life. You’ll find it in items such as plastic bags, bottles, and even highway barriers. This polymer is formed by linking thousands of ethylene molecules (\(\text{CH}_2=\text{CH}_2\)) together. Polyethylene is known for its toughness, chemical resistance, and flexibility. These properties make it an excellent choice for various applications.

• **Versatility**: Used for both consumer goods and industrial applications.
• **Durability**: Known for its long-lasting performance.
• **Chemical Structure**: Embraces a simple, repeating structure.

Understanding polyethylene on a molecular level can help us see why it's such a versatile material. Learning to calculate the molar mass of polyethylene is a small yet crucial skill in chemistry.

Repeating Unit

The concept of a repeating unit in polymers like polyethylene is foundational to understanding their structure and properties. In the case of polyethylene, a repeating unit is \(\text{CH}_2-\text{CH}_2\), a simple pair of carbon atoms each bonded to two hydrogen atoms. This basic unit repeats along the length of the polymer chain.

• **Simple Structure**: Each repeating unit contains 2 carbons (C) and 4 hydrogens (H).
• **Building Blocks**: These units are the building blocks of the polyethylene chain.
• **Molar Mass Calculation**: Understanding the repeating unit helps in calculating the total mass.

Repetition of these small units results in the formation of long chains that provide polyethylene its distinct mechanical and chemical properties.

Step-by-Step Solution

Breaking down the problem into manageable steps simplifies complex calculations. We've detailed a step-by-step guide on how to calculate the molar mass of polyethylene by focusing on each individual element within the repeating unit. 1. **Identify the Repeating Unit**: Recognizing that polyethylene consists of repeating [-CH₂–CH₂-] units. 2. **Calculate the Repeating Unit's Molar Mass**: - Each carbon weighs approximately 12.01 g/mol. - Each hydrogen weighs about 1.01 g/mol. - The total for one [-CH₂–CH₂-] unit is therefore 28.06 g/mol. 3. **Multiply by the Number of Units**: Multiply this molar mass by the number of repeating units (n=4600). 4. **Total Molar Mass**: Calculate to find the total mass as 129076 g/mol. With this step-by-step framework, students can tackle similar problems by breaking them into distinct parts, enhancing understanding and efficiency.

Chemical Formula

The chemical formula \((\text{CH}_2–\text{CH}_2)_n\) is vital when dissecting the composition of polyethylene. Knowing this formula is the key to understanding not just the composition but also how to calculate significant properties such as molar mass.

• **Symbolic Representation**: The formula represents the entire polyethylene molecule when n is specified.
• **Chain Formation**: Each unit, \(-\text{CH}_2-\text{CH}_2-\), indicates continuation along the chain.
• **Calculation Basis**: Forms the foundation for performing calculations involving mass and chemical properties.

By understanding the chemical formula, we can appreciate the simplicity and elegance of polymer chemistry, linking straightforward hydrocarbon units to produce remarkable materials.