Using the definitions for total chain molecule length \(L\) (Equation 14.11) and average chain end-to-end distance \(r\) (Equation 14.12), for a linear polyethylene determine the following: (a) the number-average molecular weight for \(L=2500 \mathrm{~nm}\) (b) the number-average molecular weight for \(r=20 \mathrm{~nm}\)

When we analyze polymers like polyethylene, an important characteristic is the chain length. A polyethylene molecule can be envisioned as a long string of carbon atoms, each with two hydrogen atoms attached, resembling a chain of many links. The overall length of this polymer chain, denoted by 'L', is a measure of the number of repeating units (each repeating unit is a 'link') from one end to the other.

For students dealing with calculations regarding molecular weight, the total chain length is essential to finding how many monomer units are present in the polymer. By knowing the average length of each monomer unit, you can calculate the total number of units by dividing the total chain length by the length of a single unit. This gives us 'N', the number of repeat units. Multiplying 'N' by the molecular weight of a single repeat unit then provides the molecular weight of the entire chain.

In polymer science, besides the total chain length, another intriguing concept is the end-to-end distance, represented by 'r'. While the chain length measures the full extension of the polymer chain, the end-to-end distance measures the straight-line distance from the starting point of the polymer chain to its endpoint. This distance tends to be less than the chain length due to the coiling and folding of the chain.

This end-to-end distance can reveal information about the chain's flexibility and is calculated using the root-mean-square end-to-end distance of a random coil. For simple estimations, the end-to-end distance can help in assessing the packing and strength properties of the polymer. In the calculation of molecular weight, 'r' is squared and divided by the square of the average bond length to find 'N', which is then used in further calculations.

The repeat unit molecular weight is a fundamental value in understanding polymers. Each polymer is made up of repeating units, and polyethylene, for instance, has a repeat unit of CH2. The 'molecular weight' of the repeat unit is determined by adding up the atomic masses of each atom within the unit

For CH2, it's the atomic mass of carbon (approximately 12 g/mol) plus twice the atomic mass of hydrogen (approximately 1 g/mol each), summing up to 14 g/mol for polyethylene. This information is crucial when we tie it back to chain length and end-to-end distances to find out the molecular weight of the entire polymer chain. Knowing this value is pivotal in understanding polymers' physical properties, such as tensile strength, durability, and melting point, which differ widely based on molecular weight.