Question

For a condensation polymerization of a hydroxyacid with a residue weight of 200 , it is found that \(99 \%\) of the acid groups are used up. Calculate a) the number average molecular weight and b) the weight average molecular weights.

Short answer

The number average molecular weight (Mn) is approximately 19800, and the weight average molecular weight (Mw) is 20000.

Step-by-step solution

Calculate Degree of Polymerization (Xn)

First, we need to find the degree of polymerization (Xn). We know that 99% of the hydroxyacid groups are used up, so we have: Xn = 0.99 / (1 - 0.99) Calculating this, we get: Xn ≈ 99

Calculate Number Average Molecular Weight (Mn)

Now that we have the degree of polymerization (Xn), we can calculate the number average molecular weight (Mn). We can use the following formula: Mn = residue weight * Xn We know that the residue weight of the hydroxyacid is 200, so we have: Mn = 200 * 99 Mn ≈ 19800

Calculate Weight Average Molecular Weight (Mw)

Finally, we need to calculate the weight average molecular weight (Mw). We can use the following formula: Mw = residue weight * (Xn + 1) We know that the residue weight of the hydroxyacid is 200, and we have calculated the degree of polymerization (Xn) to be 99, so we have: Mw = 200 * (99 + 1) Mw = 200 * 100 Mw = 20000 So the number average molecular weight (Mn) is approximately 19800, and the weight average molecular weight (Mw) is 20000.

Key concepts

Degree of Polymerization

Understanding the degree of polymerization (DP) is fundamental when exploring the properties of polymers, particularly those formed through condensation polymerization. In essence, the DP represents the average number of repeating units in a single polymer chain. It's a pivotal concept because it directly impacts a polymer's physical characteristics such as strength, melting point, and solubility.

For instance, in the given exercise, the calculation of the degree of polymerization starts with the fact that 99% of the hydroxyacid's acid groups reacted. By applying the ratio \( X_n = \frac{0.99}{1 - 0.99} \), we discover that the DP (\( X_n \)) is approximately 99. This tells us that on average, each polymer chain in the sample is composed of about 99 repeating units. In a general setting, a higher degree of polymerization would typically denote a longer polymer chain, subsequently influencing its physical and mechanical properties.

Number Average Molecular Weight

Moving forward to molecular weight, particularly the number average molecular weight (\( M_n \)), this term describes an average that considers the mass of all polymer molecules in a sample divided by the total number of polymer molecules. The \( M_n \) offers a numerical representation of the molecular size distribution within a polymer sample. This weight-average is pivotal in predicting the behavior of polymers under various conditions because it reflects the majority of polymer species present.

In our exercise, to find the \( M_n \), we multiplied the residue weight, which is 200, by the degree of polymerization (\( M_n = 200 * 99 \)). Consequently, the calculated \( M_n \) of approximately 19800 represents an average molecular weight of the polymer chains. Higher numbers for \( M_n \) generally suggest a polymer with greater molecular weight and, likely, different physical properties than a similar polymer with a lower \( M_n \).

Weight Average Molecular Weight

The weight average molecular weight (\( M_w \)) is another type of average molecular weight used to characterize polymers. Unlike the \( M_n \), which considers all molecules equally, \( M_w \) provides a different perspective by placing more emphasis on the weight of the larger molecules. This leads to a weighted average in which heavier molecules skew the average. As a matter of fact, \( M_w \) is often higher than \( M_n \) because it reflects the presence of these larger molecules.

In the context of our hydroxyacid polymerization example, we calculated \( M_w \) by the expression \( M_w = 200 * (X_n + 1) \). After substituting in the known values, we determined the \( M_w \) to be 20000. This weight average molecular weight helps in inferring the distribution of molecular sizes and is particularly useful in understanding the polymer's potential performance in various applications, as larger molecules often confer different properties like toughness and viscosity to a polymer.