Question

A sample of polymer contains \(0.50\) mole fraction with molecular weight 100,000 and \(0.50\) mole fraction with molecular weight 200,000 . Calculate (a) the number average molecular weight, \(\mathrm{M}_{\mathrm{n}}\) and (b) the weight average molecular weight, \(\mathrm{M}_{\mathrm{w}}\).

Short answer

The number average molecular weight, \(\mathrm{M}_{\mathrm{n}}\) is \(150,000\) and the weight average molecular weight, \(\mathrm{M}_{\mathrm{w}}\) is \(166,667\).

Step-by-step solution

Calculate the number average molecular weight

First, let's calculate the number average molecular weight, \(\mathrm{M}_{\mathrm{n}}\). We are given the mole fractions and the molecular weights of each component. Recall that mole fraction equals the number of mols of each component divided by the total number of mols: $$ X_i = \frac{N_i}{\sum N_{i}} $$ Let N be the total number of moles. We can write the number of moles of component A and B as: $$ N_A = X_A \times N = 0.50 \times N $$ $$ N_B = X_B \times N = 0.50 \times N $$ Now, we substitute the values in the \(\mathrm{M}_{\mathrm{n}}\) equation: $$ \mathrm{M}_{\mathrm{n}} = \frac{\sum N_{i}M_{i}}{\sum N_{i}} \approx \frac{(0.50 \times N)(100000)+(0.50 \times N)(200000)}{N} $$

Simplify the expression for the number average molecular weight

Combine the terms in the numerator and simplify the expression: $$ \mathrm{M}_{\mathrm{n}} = \frac{50,000N + 100,000N}{N} $$ $$ \mathrm{M}_{\mathrm{n}} = \frac{150,000N}{N} $$ Cancel N: $$ \mathrm{M}_{\mathrm{n}} = 150{{,}}000 $$

Calculate the weight average molecular weight

Next, let's use the formula for \(\mathrm{M}_{\mathrm{w}}\) to find the weight average molecular weight: $$ \mathrm{M}_{\mathrm{w}} = \frac{\sum N_{i}M_{i}^{2}}{\sum N_{i}M_{i}} \approx \frac{(0.50 \times N)(100{{,}}000)^{2} + (0.50 \times N)(200{{,}}000)^{2}}{(0.50 \times N)(100{{,}}000)+(0.50 \times N)(200{{,}}000)} $$

Simplify the expression for the weight average molecular weight

Combine the terms in the numerator and simplify the expression: $$ \mathrm{M}_{\mathrm{w}} = \frac{5 \times {10}^{9}N + 20 \times {10}^{9}N}{0.5 \times {10}^{5}N + {10^{5}}N} $$ $$ \mathrm{M}_{\mathrm{w}} = \frac{25 \times {10}^{9}N}{1.5 \times {10}^{5}N} $$ Cancel N and divide the numerator by the denominator: $$ \mathrm{M}_{\mathrm{w}} \approx 166{{,}}667 $$ Now we have the two required values: a) The number average molecular weight, \(\mathrm{M}_{\mathrm{n}} = 150{{,}}000\) b) The weight average molecular weight, \(\mathrm{M}_{\mathrm{w}} = 166{{,}}667\)

Key concepts

Mole Fraction

The mole fraction is a way of expressing the concentration of a component in a mixture. It is defined as the number of moles of that component divided by the total number of moles of all components in the mixture. Mathematically, it's denoted as:

\[ X_i = \frac{N_i}{\sum N_{i}} \]

In polymer science, determining the mole fraction of different polymer molecules is essential for understanding their behavior and properties in a mixture. For instance, when blending two polymers, the mole fraction can affect the material's mechanical and thermal characteristics. By calculating the mole fractions, as done in the exercise, it assists scientists and engineers in predicting the behavior of polymeric materials under various conditions.

Number Average Molecular Weight

The number average molecular weight (Mn) is the mean of the molecular weights of the individual polymers in a sample, weighted by the number of molecules (or moles) of each. To calculate Mn, we use the formula:

\[ \mathrm{M}_{\mathrm{n}} = \frac{\sum N_{i}M_{i}}{\sum N_{i}} \]

In the context of the provided exercise, Mn represents an average that favors the presence of smaller molecules, as they are more numerous even if their individual weights are lower. This parameter is particularly informative for understanding the behavior of a polymer in solution, where the number of molecules can directly influence the viscosity and other solution properties. The calculation of Mn in the step-by-step solution showcases how this value is a balance of the various molecular weights present in a mixture, based on their relative molar amounts.

Weight Average Molecular Weight

The weight average molecular weight (Mw) is another crucial parameter in polymer science used to characterize polymer samples. Unlike Mn, the Mw takes into account the mass of each polymer molecule. It gives a higher weighting to molecules with a greater mass. It's calculated using the formula:

\[ \mathrm{M}_{\mathrm{w}} = \frac{\sum N_{i}M_{i}^{2}}{\sum N_{i}M_{i}} \]

This calculation is important because, in polymers, the material properties can be more reflective of the heavier molecules due to their greater contribution to the total mass. Thus, Mw provides insight into the distribution of molecular weights within a polymer sample and often correlates with properties like tensile strength and impact resistance. As demonstrated in the example provided, calculating Mw is a bit more complex than Mn due to the squared masses, but it is a critical step in fully characterizing a polymer's molecular weight distribution.

Polymer Science

Polymer science is the field that studies polymers, which are long chains of molecules made up of repeating units called monomers. Polymers include natural materials like DNA and proteins, as well as synthetic materials such as plastics and rubbers. The physical and chemical properties of polymers depend heavily on their molecular weight distribution, which includes parameters like Mn and Mw.

Understanding the molecular weight distribution is critical because it influences how the polymer behaves during processing and in its final application. For example, a polymer with a broad molecular weight distribution may be more tough and processable, while a narrow distribution could lead to more consistent material characteristics. The combination of mole fraction, Mn, and Mw calculations offer a wealth of information for scientists and engineers looking to develop and apply polymers in various industries.