Question

The \(\mathrm{P}-\mathrm{H}\) bond distance in \(\mathrm{PH}_{3}\) is \(1.42 \mathrm{~A}\) and the \(\mathrm{N}-\mathrm{H}\) bond distance in \(\mathrm{NH}_{3}\) is \(1.01 \mathrm{~A}\). Use the bond angle of \(93^{\circ}\) for \(\mathrm{PH}_{3}\) and \(107.3^{\circ}\) for \(\mathrm{NH}_{3}\) and calculate the distance between the hydrogens for each molecule. Would you expect the repulsion between the hydrogen nuclei in \(\mathrm{PH}_{3}\), to be more, or less, than in \(\mathrm{NH}_{3}\) ?

Short answer

The \(\mathrm{H} - \mathrm{H}\) distance in \(\mathrm{PH}_3\) is greater than in \(\mathrm{NH}_3\), so repulsion is less in \(\mathrm{PH}_3\).

Step-by-step solution

Understand the Molecular Geometry

First, understand that both \(\mathrm{PH}_3\) and \(\mathrm{NH}_3\) molecules have a trigonal pyramidal geometry. In these molecules, three hydrogen atoms are bonded to a central atom (either phosphorus or nitrogen), and the molecule's shape affects the distance between the hydrogen atoms.

Identify the Relevant Distances and Bond Angles

For \(\mathrm{PH}_3\), the bond distance \(\mathrm{P} - \mathrm{H}\) is given as \(1.42\mathrm{\, \text{Å} }\), and the bond angle is \(93^\circ\). For \(\mathrm{NH}_3\), the \(\mathrm{N} - \mathrm{H}\) bond distance is \(1.01\mathrm{\, \text{Å} }\), with a bond angle of \(107.3^\circ\). These parameters are crucial for calculating the distance between the hydrogen atoms.

Calculate the Hydrogen-Hydrogen Distance

Using the formula for calculating the distance between the two hydrogens in a pyramid, which is \(d = 2 \cdot r \cdot \sin\left(\frac{\theta}{2}\right)\), where \(r\) is the bond length and \(\theta\) is the bond angle:- For \(\mathrm{PH}_3\):\[d_{\mathrm{P}-\mathrm{H}} = 2 \times 1.42 \times \sin\left(\frac{93}{2}\right)\]Calculating this gives approximately \(2.65\, \text{Å} \).- For \(\mathrm{NH}_3\):\[d_{\mathrm{N}-\mathrm{H}} = 2 \times 1.01 \times \sin\left(\frac{107.3}{2}\right)\]Calculating this gives approximately \(1.63\, \text{Å} \).

Comparison of Repulsions

The larger the distance between hydrogen atoms, the less the inter-nuclear repulsion. Since the \( \mathrm{H} - \mathrm{H} \) distance in \(\mathrm{PH}_3\) is greater than in \(\mathrm{NH}_3\), the repulsion between the hydrogen nuclei in \(\mathrm{PH}_3\) is expected to be less than in \(\mathrm{NH}_3\).

Key concepts

Trigonometry in Chemistry

In chemistry, trigonometry plays a crucial role in understanding molecular geometry and bond angles. When studying molecules such as \(\mathrm{PH}_3\) and \(\mathrm{NH}_3\), trigonometrical calculations help determine the distances between atoms, based on bond angles and lengths. These calculations are essential to predict molecular properties and behaviors.
One common application is calculating the distance between hydrogen atoms in trigonal pyramidal structures. By using the formula for the distance between two hydrogen atoms, \(d = 2 \cdot r \cdot \sin\left(\frac{\theta}{2}\right)\), where \(r\) is the bond length and \(\theta\) is the bond angle, chemists can understand how geometry affects molecular interactions. For instance:

• \(\mathrm{PH}_3\) has a bond length of \(1.42\, \mathrm{\text{Å}}\) and an angle of \(93^{\circ}\).
• \(\mathrm{NH}_3\) has a bond length of \(1.01\, \mathrm{\text{Å}}\) and an angle of \(107.3^{\circ}\).

Calculating these distances reveals that the geometry and structure influence how the molecules interact physically and chemically.

Trigonal Pyramidal Structure

The trigonal pyramidal structure is a molecular shape that occurs when there is a central atom bonded to three other atoms, with a lone pair on the central atom. This results in a three-sided pyramid shape with the central atom at the apex. Molecules like \(\mathrm{PH}_3\) and \(\mathrm{NH}_3\) exhibit this geometry, which significantly affects their chemical properties.
The presence of a lone pair on the central atom pushes the bonded atoms closer together, altering the bond angles and the molecule's overall shape. This geometry leads to different effects on bond angles and the repulsion forces between atoms:

• \(\mathrm{NH}_3\), with its \(107.3^{\circ}\) angle, is pushed together more tightly compared to \(\mathrm{PH}_3\).
• The smaller \(93^{\circ}\) angle in \(\mathrm{PH}_3\) results in atoms being slightly further apart due to the larger central atom.

Understanding this geometry is essential for predicting the molecule's behavior, such as its reactivity and interactions with other molecules.

Intermolecular Forces

Intermolecular forces (IMFs) are the forces that mediate interaction between molecules, influencing properties like boiling and melting points, solubility, and molecular stability. These forces include hydrogen bonding, dipole-dipole interactions, and London dispersion forces.
In trigonal pyramidal molecules such as \(\mathrm{PH}_3\) and \(\mathrm{NH}_3\), the geometry impacts the type and strength of IMFs:

• \(\mathrm{NH}_3\) experiences hydrogen bonding due to its highly polar \(\mathrm{N}-\mathrm{H}\) bonds, giving it stronger intermolecular forces.
• In \(\mathrm{PH}_3\), with less polar bonds, weaker London dispersion forces typically predominate.

The distinction in intermolecular forces affects physical properties. For example, \(\mathrm{NH}_3\) can dissolve in water due to hydrogen bonds, whereas \(\mathrm{PH}_3\) does not interact similarly. Understanding how different molecular geometries affect intermolecular forces is key to grasping why molecules behave the way they do in various environments.