Question

In a polymer sample \(30 \%\) of molecules have a molecular mass \(20,000,40 \%\) have 30,000 and the rest \(60,000 .\) What is the weight average molecular mass of the polymer? (a) 40,300 (b) 30,600 (c) 43,333 (d) 50,400

Short answer

The weight average molecular mass is 36,000, option not listed.

Step-by-step solution

Identify the Given Information

We are given a polymer sample with molecules of different molecular masses: \(30\%\) have a mass of 20,000, \(40\%\) have 30,000, and the remainder, which is \(30\%\), have 60,000. We need to calculate the weight average molecular mass.

Recall the Formula for Weight Average Molecular Mass

The weight average molecular mass \( \overline{M}_w \) is calculated by the formula: \[ \overline{M}_w = \frac{w_1 \times M_1 + w_2 \times M_2 + w_3 \times M_3}{w_1 + w_2 + w_3} \] where \( w_i \) is the weight fraction of molecules with molecular mass \( M_i \). Since we're given percentages, these percentages will be used as weights.

Convert Percentages to Weight Fractions

Convert the percentages to decimal weight fractions: - For the mass 20,000: \(w_1 = 0.30\),- For the mass 30,000: \(w_2 = 0.40\),- For the mass 60,000: \(w_3 = 0.30\).

Calculate the Weight Average Molecular Mass

Using the formula from Step 2, substitute the mass and weight fractions:\[ \overline{M}_w = \frac{(0.30 \times 20,000) + (0.40 \times 30,000) + (0.30 \times 60,000)}{0.30 + 0.40 + 0.30} \]Calculate each term:\(0.30 \times 20,000 = 6,000\)\(0.40 \times 30,000 = 12,000\)\(0.30 \times 60,000 = 18,000\)Add these to get the numerator: \(6,000 + 12,000 + 18,000 = 36,000\).Since the sum of the fractions is 1.0, the weight average molecular mass is 36,000.

Key concepts

Polymer Chemistry

Polymer chemistry is all about the magic of binding smaller units into long chains, known as polymers. These smaller units, called monomers, link together in varied numbers and sequences to form diverse structures, giving rise to myriad materials from plastics to DNA.

There are two crucial aspects in polymer chemistry: *the way monomers are organized* and *the size of the polymers*. How these are configured determines the polymer's properties, such as flexibility, strength, and melting point.

• **Types of Polymerization**: Most polymers are synthesized through processes such as addition polymerization or condensation polymerization. In addition polymerization, monomers add to each other without the loss of any atoms, whereas in condensation polymerization, monomers join together with the loss of a small molecule, typically water.
• **Structure and Properties**: The arrangement of monomers within the polymer can be linear, branched, or cross-linked, influencing the polymer's physical properties.

Understanding the basics of polymer chemistry forms the foundation for countless applications in materials science and industry.

Molecular Mass Calculations

**Molecular mass calculations** in polymer chemistry often revolve around measuring an average of masses due to the diversity in chain lengths and structures.

One important way to measure this is by calculating the weight-average molecular mass ( \( \overline{M}_w \)), which provides insight into the polymer’s molecular distribution.

• **Weight-Average Molecular Mass**: This calculates the mass where each mass element is weighted by the mass fraction, giving more weight to heavier polymer molecules.
  Formula: \( \overline{M}_w = \frac{w_1 \times M_1 + w_2 \times M_2 + w3 \times M_3}{w_1 + w_2 + w3} \) , where \(w_i\) represents the weight fraction, and \(M_i\) the molecular masses of fractions.
• **Importance**: This average is particularly vital for understanding the behavior of polymers in solution or melted form, influencing viscosity and mechanical properties.

Therefore, mastering how to convert percentages to weight fractions and use them in formulas is crucial for anyone delving into polymer science.

Polymer Science

Polymer science combines chemistry, physics, and engineering to study polymers' synthesis, characteristics, and applications. It aims to unravel how molecular structure affects a polymer’s properties, aiding in the design of new materials.

Understanding polymer science opens the door to innovations in technology and materials, from everyday plastic goods to cutting-edge biomedical devices.

• **Material Properties**: Variables such as the temperature and the polymer’s molecular weight can drastically change a polymer’s flexibility, toughness, and resilience. These factors are critical in applications ranging from packaging materials to tire manufacturing.
• **Applications**: Polymers are used in numerous fields, from creating lightweight automotive parts to developing drug delivery systems in medicine. Each application demands specific polymer characteristics, underscoring the importance of knowledgeable design and synthesis.

Exploring polymer science is like unfolding a world of potential, where small changes at the molecular level can lead to groundbreaking advancements.