Problem 85 - Chemistry: The Central Science

Question

The As - As bond length in elemental arsenic is \(2.48 \AA\). The \(\mathrm{Cl}-\mathrm{Cl}\) bond length in \(\mathrm{Cl}_{2}\) is \(1.99 \AA\). (a) Based on these data, what is the predicted As - Cl bond length in arsenic trichloride, \(\mathrm{AsCl}_{3}\), in which each of the three \(\mathrm{Cl}\) atoms is bonded to the As atom? (b) What bond length is predicted for \(\mathrm{AsCl}_{3}\), using the atomic radii in Figure 7.7?

Step-by-Step Solution

Answer

The predicted As-Cl bond length in AsCl3 is approximately \(2.235 \AA\) based on the average of As-As and Cl-Cl bond lengths. To compare this with the bond length using atomic radii from Figure 7.7, add the atomic radii of As (r_As) and Cl (r_Cl) to get the As-Cl bond length as r_As + r_Cl. Compare the values to see if they are consistent or differ considerably.

Step 1: (a) Find As-Cl bond length using As-As and Cl-Cl bond lengths

First, we will find the average of the bond lengths of As-As and Cl-Cl to estimate the bond length of As-Cl in AsCl3. As-As bond length: \(2.48 \AA\) Cl-Cl bond length: \(1.99 \AA\) By averaging these values, we get: As-Cl bond length: \(\frac{2.48 + 1.99}{2} = 2.235 \AA\) The predicted As-Cl bond length in AsCl3 is approximately \(2.235 \AA\).

Step 2: (b) Find As-Cl bond length using atomic radii

Next, we will use the atomic radii values provided in Figure 7.7 to calculate the As-Cl bond length in AsCl3. Assuming that Figure 7.7 provides atomic radii for As and Cl (let's designate those values as r_As and r_Cl), we can add them together to find the As-Cl bond length: As-Cl bond length = r_As + r_Cl Check the values of r_As and r_Cl in Figure 7.7 and substitute them into the equation to find the As-Cl bond length. Finally, compare the predicted bond length from part (a) with the calculated bond length from part (b) using the atomic radii data. Note if their values are consistent or if they differ considerably.

Key Concepts

As-As bond length

The As-As bond length refers to the distance between two arsenic atoms in a molecule. In elemental arsenic, this bond length is approximately 2.48 angstroms ( ), which dictates how arsenic atoms interact at the molecular level. Bond lengths like this are essential because they provide insights into the structural stability and chemical characteristics of the element or compound.

When we measure the bond between two identical atoms, such as arsenic in this case, we determine a key physical characteristic that is used as a reference point. This process can then help predict the bond length of arsenic when it combines with other elements, for instance, chlorine in arsenic trichloride (AsCl3).

In elemental arsenic, the long As-As bond length showcases weaker interactions compared to bonds with smaller lengths, which typically indicate stronger interactions.

Cl-Cl bond length

The bond length for a chlorine molecule, Cl-Cl, is about 1.99 angstroms ( ). This measurement helps understand the physical and chemical properties of chlorine gas. Chlorine, being a diatomic molecule, naturally bonds with itself, and this bond length is a critical reference when evaluating its bonding with other elements.

Why is Cl-Cl bond length shorter compared to As-As? Various factors, including the atom size (atomic radii) and the nature of the bond. Chlorine is smaller and more electronegative than arsenic, resulting in a shorter bond. This shorter bond implies a stronger attraction between the two chlorine atoms than that between arsenic atoms in As-As bonds.

In practice, knowing the Cl-Cl bond length is useful when predicting how chlorine will combine with other substances, ensuring accurate estimations of compound structures.

Atomic radii

Atomic radii are crucial in understanding bond lengths in molecules, as they define the size of an atom and affect its bondability with other atoms. The atomic radius is generally half the distance between the nuclei of two atoms of the same element bonded together. In molecules where different elements are involved, such as arsenic combined with chlorine, atomic radii help in predicting bond lengths.

To calculate the As-Cl bond length using atomic radii:

• Consider the atomic radius of arsenic ( ) and chlorine ( ).
• Add these radii: As-Cl bond length = atomic radius of As + atomic radius of Cl.

Using atomic radii offers an alternative method for predicting bond lengths, complementing values derived from averaging known bond lengths like As-As and Cl-Cl.

This dual approach ensures reliable predictions, as it draws on the inherent size and bonding capacity of individual atoms.