In x-ray studies of crystalline peptides, Linus Pauling and Robert Corey found that the C—N bond in the peptide link is intermediate in length (1.32 Å) between a typical C—N single bond (1.49 Å) and a C=N double bond (1.27 Å). They also found that the peptide bond is planar (all four atoms attached to the C—N group are located in the same plane) and that the two α-carbon atoms attached to the C—N are always trans to each other (on opposite sides of the peptide bond). Required:
a. What does the length of the C—N bond in the peptide linkage indicate about its strength and its bond order (i. e., whether it is single, double, or triple)?
b. What do the observations of Pauling and Corey tell us about the ease of rotation about the C—N peptide bond?

The C-N bond is a double bond.

There is no free rotation about the C-N bond

Explanation:

Linus Pauling and Robert Corey carried out a painstaking study of the bond lengths of peptide bonds as well as the stereo chemistry of atoms and groups around the peptide bonds in the crystal structures of molecules containing one or a few peptide bonds.

Their findings indicate that the bond length of the C-N bond in crystalline peptides is intermediate in length (1.32 Å) between a typical C—N single bond (1.49 Å) and a C=N double bond (1.27 Å). This is because the peptide bond has some double bond character as a result of resonance involving the C=O and -NH moieties of the amide.

As a result of resonance as shown above, the peptide bonds are found to be planar hence free rotation about the C-N bond is hindered. The peptide bond is mostly found in the trans configuration because it is more energetically favourable than the cis configuration.

Summarily, owing to the existence of a partial double bond character between the α carbon and the amine nitrogen in the peptide, there is no free rotation around the C-N bond.