For each of the following pairs of polymers, do the following: (1) state whether or not it is possible to decide whether one polymer has a higher tensile modulus than the other; (2) if this is possible, note which has the higher tensile modulus and then cite the reason(s) for your choice; and (3) if it is not possible to decide, then state why. (a) Branched and atactic poly(vinyl chloride) with a weight-average molecular weight of \(100,000 \mathrm{g} / \mathrm{mol} ;\) linear and isotactic poly \((\) vinyl chloride) having a weight-average molecular weight of \(75,000 \mathrm{g} / \mathrm{mol}\) (b) Random styrene-butadiene copolymer with \(5 \%\) of possible sites crosslinked; block styrene-butadiene copolymer with \(10 \%\) of possible sites crosslinked (c) Branched polyethylene with a numberaverage molecular weight of \(100,000 \mathrm{g} / \mathrm{mol}\) atactic polypropylene with a number-average molecular weight of \(150,000 \mathrm{g} / \mathrm{mol}\)

Poly(vinyl chloride), commonly known as PVC, is a widely used synthetic polymer prized for its versatile properties. The physical characteristics of PVC can vary significantly depending on its molecular structure. When we speak about atactic PVC, we refer to a polymer where the pendant groups are arranged randomly along the polymer chain. On the other hand, isotactic polymers have pendant groups arranged in the same orientation at each repeating unit.

Linear, isotactic PVC has longer chains that pack closely together due to their regular structure, leading to stronger intermolecular forces such as van der Waals forces and potential hydrogen bonding sites. This close packing results in a higher tensile modulus, which is a measure of the stiffness of the polymer, meaning it resists deformation under tensile stress.

Conversely, branched and atactic PVC consists of chains that are more randomly arranged, prohibiting tight packing and reducing intermolecular interactions. Therefore, the tensile modulus is lower as the material is less rigid and more likely to deform under stress. The ordered structure of isotactic PVC leads to improved strength and stiffness, facilitating its use in applications that require these particular mechanical properties.

Styrene-butadiene rubber (SBR) is another important class of polymers that exhibits diverse properties based on structural variations. Crosslinking is a process where polymer chains are linked together through covalent bonds, and this can significantly influence the mechanical strength of the material.

In the context of SBR copolymers, the difference between random and block configurations, as well as the degree of crosslinking, plays a key role in their physical properties. The term block copolymer refers to the arrangement of monomers in long sequences or 'blocks', which can aggregate into domains, enhancing certain properties such as tensile strength. Crosslinking within these block sequences often results in a network structure that further restricts chain movement, thereby increasing tensile modulus.

A random copolymer, on the other hand, has styrene and butadiene monomers distributed in a non-sequential manner, which leads to a more disordered material. When there is a 5% crosslinking, it indicates fewer covalent bonds between chains, leading to more flexibility and lesser stiffness compared to a block copolymer with a higher level of crosslinking. Thus, with a 10% crosslinking, block SBRs demonstrate a higher tensile modulus and are suitable for applications where enhanced durability and strength are crucial.

The molecular weight of a polymer is a critical factor that affects its mechanical, thermal, and chemical properties. It essentially reflects the size of the polymer chains, which directly impacts the material’s tensile properties. Higher molecular weight polymers tend to have longer chains, which can become entangled, leading to increased strength and toughness.

As the molecular weight increases, so does the viscosity of the polymer in its melted state, and the polymer exhibits greater resistance to flow. In the solid state, these long chains result in higher tensile strength and modulus because the entanglements act like physical crosslinks, which help the material resist deformation under tensile loads. Moreover, high molecular weight polymers often display better impact resistance and stress-crack resistance.

However, it is important to note that the increase in tensile properties with molecular weight does level off at a certain point. This is due to the fact that beyond a certain molecular weight, the additional chain entanglements do not yield substantial increases in strength or stiffness. This concept of molecular weight is fundamental to understanding the exercise solution provided, where the mechanical properties of different polymers cannot be determined solely by molecular weight, but also require consideration of the structural configuration and chain orientation.