Which is the higher density of polyethylene, linear or branched? Select the choice with the best accompanying explanation.

Linear polyethylene is the higher density form, as the chains are able to pack better and partially crystallize, resulting in better packing and higher density.

Branched is the higher density form, as the additional chains increase the mass, and therefore the density of the material.

They have the same density, as they have the same chemical formula. Density is independent of molecular geometry.
(b) High-density polyethylene (HDPE) is opaque (white) while low-density polyethylene (LDPE) is transparent to visible light. Select the best explanation for this observation.

The regularity of the packing of chains in HDPE creates a band-structure which results in the absorption of most wavelengths of light, similar to what happens with low-band gap semiconductors. The lack of band structure within LDPE means that it must be transparent.

The polymer chains are more closely packed in HDPE, so much so that visible light is unable to pass between them. The spacing in LDPE is sufficient that light does not interact with the chains at the molecular level.

The HDPE has both ordered and disordered regions, some are partially crystalline while others are amorphous. Light scatters at the interfaces between these regions since they have different refractive indices. Meanwhile, LDPE is entirely amorphous, and is transparent because the energy of visible light is not high enough to interact with the strong covalent bonds in the material.

Optical properties are unrelated to density, and as such this question cannot be answered with the known information. The observation is for different reasons.
(c) Shown below are several sketches of the temperature dependence of the molar volume of HDPE and LDPE, assuming both to be of comparable molecular weight. Select the graph which is most likely accurate.