Problem 116 - Chemistry The Molecular Science

Question

The bond length can be considered as approximately the sum of the atomic radii of the two bonded atoms. The atomic radii for single-bonded carbon and oxygen are \(77 \mathrm{pm}\) and \(74 \mathrm{pm}\), respectively. For double-bonded \(\mathrm{C}\) and \(\mathrm{O}\) the values are \(67 \mathrm{pm}\) and \(60 \mathrm{pm}\), respectively. Use these data to estimate the carbon-to-oxygen bond length in (a) methanol, \(\mathrm{CH}_{3} \mathrm{OH} ;\) (b) dimethyl ether, \(\mathrm{CH}_{3} \mathrm{OCH}_{3}\); (c) formaldehyde, \(\mathrm{H}_{2} \mathrm{CO}\). Explain any difference among the bond lengths (see Table 6.1 as a reference).

Step-by-Step Solution

Answer

In methanol and dimethyl ether, the bond length is 151 pm. In formaldehyde, it's 127 pm. Single bonds are longer than double bonds.

Step 1: Identify Bond Types

First, determine the types of bonds present in each molecule. Methanol (\(\mathrm{CH}_3 \mathrm{OH}\)) and dimethyl ether (\(\mathrm{CH}_3 \mathrm{OCH}_3\)) have a single bond between carbon and oxygen, while formaldehyde (\(\mathrm{H}_2 \mathrm{CO}\)) has a double bond between carbon and oxygen.

Step 2: Atomic Radii for Single Bond

For single-bonded carbon and oxygen, use atomic radii of 77 pm for carbon and 74 pm for oxygen.

Step 3: Calculate Bond Length for Single Bond

The bond length for a single \(\mathrm{C-O}\) bond in methanol and dimethyl ether is the sum of the atomic radii: \(77 \, \text{pm} + 74 \, \text{pm} = 151 \, \text{pm}\).

Step 4: Atomic Radii for Double Bond

For double-bonded carbon and oxygen, use atomic radii of 67 pm for carbon and 60 pm for oxygen.

Step 5: Calculate Bond Length for Double Bond

The bond length for a double \(\mathrm{C=O}\) bond in formaldehyde is the sum of the atomic radii: \(67 \, \text{pm} + 60 \, \text{pm} = 127 \, \text{pm}\).

Step 6: Explain Differences

The bond lengths differ based on the bond type: a single bond is longer (151 pm) because the shared electron cloud is more extended, while a double bond (127 pm) is shorter due to additional shared electrons, pulling the atoms closer.

Key Concepts

Atomic Radii

The concept of atomic radii is crucial in understanding bond lengths. Atomic radii refer to the size of an atom, typically measured from the center of the nucleus to the edge of the electron cloud surrounding it.
The atomic radius is a decisive factor for determining the bond length between atoms.

When atoms form bonds, their radii partially dictate how close together the nuclei of the bonded atoms can come.

• For single bonds, carbon has an atomic radius of 77 pm, while oxygen measures 74 pm.
• For double bonds, these radii decrease to 67 pm for carbon and 60 pm for oxygen.

The reason for the change in radius with different bond types relates to electron sharing. Double bonds involve more extensive sharing, pulling the atoms closer together and thus reducing the atomic radius.
The bond length, then, is often considered the sum of these atomic radii, making the calculation straightforward for simple molecules.

C-O Single Bond

Single bonds are the most basic form of chemical linkage between two atoms, involving the sharing of a pair of electrons. In the case of a carbon-oxygen single bond, as seen in molecules like methanol (\(\mathrm{CH}_3\mathrm{OH}\) and dimethyl ether, \(\mathrm{CH}_3\mathrm{OCH}_3\), the bond length can be easily estimated.

To calculate the bond length:

• Use the atomic radii of the single-bonded atoms: 77 pm (carbon) and 74 pm (oxygen).
• Add these radii together: 77 pm + 74 pm = 151 pm.

This sum gives us the estimated length of the C-O single bond, approximately 151 pm.

The longer distance is due to the fact that a single bond allows for a relatively loose connection, as the electron cloud overlap is limited to just one pair of shared electrons, providing less strong pull on the atoms compared to a double bond.

C=O Double Bond

Double bonds involve the sharing of two pairs of electrons between atoms, which enhances the connectivity and brings the atoms closer together. This is why a C=O double bond, such as that in formaldehyde (\(\mathrm{H}_2\mathrm{CO}\), is shorter than a C-O single bond.

Here's how the bond length is determined:

• The atomic radii for double-bonded carbon is 67 pm, and for oxygen, it's 60 pm.
• Add these values: 67 pm + 60 pm = 127 pm.

The resulting bond length for a C=O double bond is about 127 pm.

This shortened bond length results from the increased electron sharing, which reinforces the interaction between the atoms' nuclei and pulls them significantly closer together, compared to a single bond.
Double bonds introduce rigidity and strength into molecular structures, influencing chemical reactivity and stability.