Question

Given that the \(\mathrm{H}\) -to \(-\mathrm{H}\) distance in \(\mathrm{NH}_{3}\) is \(0.1624 \mathrm{~nm}\) and N-H distance is \(0.101 \mathrm{~nm}\), calculate the bond angle \(\mathrm{H}-\mathrm{N}-\mathrm{H}\).

Short answer

The bond angle H-N-H in ammonia (NH₃) is approximately \(107.8^{\circ}\).

Step-by-step solution

Understand the Law of Cosines

The Law of Cosines is a formula used in geometry to relate the lengths of the sides of a triangle to the cosine of one of its angles. It can be represented as: \(c^2 = a^2 + b^2 - 2ab \cdot cos(\phi)\) Where a, b, and c are the side lengths of the triangle and φ is the angle opposite to the side c. In our case, let the sides a and b be the N-H distances, and the side c be the H-H distance. The angle φ we need to find is the H-N-H bond angle.

Plug in the given values

Now, we need to plug in the given values into the Law of Cosines formula: \((0.1624)^2 = (0.101)^2 + (0.101)^2 - 2(0.101)(0.101) \cdot cos(\phi)\)

Solve for Cosine of the Bond Angle

First, simplify the equation and solve for the cosine of the bond angle: \(0.02634 = 0.0202 + 0.0202 - 0.020402 \cdot cos(\phi)\) Now, subtract 0.0202 + 0.0202 from both sides of the equation and then divide by -0.020402: \(cos(\phi) \approx -0.2877\)

Calculate the Bond Angle (H-N-H)

Finally, to find the bond angle, we need to find the inverse cosine (arccos) of -0.2877: \(\phi = arccos(-0.2877)\) \(\phi \approx 107.8^{\circ}\) Therefore, the bond angle H-N-H in ammonia (NH₃) is approximately \(107.8^{\circ}\).

Key concepts

Law of Cosines

The Law of Cosines is a geometric principle that connects the sides and angles of a triangle. It's particularly useful in non-right triangles. Imagine trying to determine one angle inside a triangle when all three sides are known. This is where the Law of Cosines becomes handy. The formula is written as: \[ c^2 = a^2 + b^2 - 2ab \cdot \cos(\phi) \] Here, \(a\) and \(b\) are two side lengths, \(c\) is the side opposite the angle \(\phi\), and \(\phi\) is the angle you want to find. By rearranging this formula, you can determine the cosine of the angle if you have the lengths of each side. This becomes especially practical in chemistry when calculating bond angles in molecules with complex geometrical configurations, like ammonia.

Bond Angles

A bond angle is the angle formed between two atoms that are bonded to a common third atom. It helps in determining the spatial arrangement of atoms in a molecule, which is crucial for understanding the molecule's shape and reactivity.

• In molecular geometry, bond angles provide insights into how atoms are oriented in three-dimensional space.
• Typically, these angles are influenced by factors like electron pair repulsion and bond lengths.

For a molecule like ammonia, knowing the bond angle helps predict how the nitrogen and hydrogen atoms are positioned relative to each other.

Ammonia Structure

Ammonia is a simple yet intriguing molecule with the chemical formula \(NH_{3}\). Its structure consists of three hydrogen atoms bonded to a central nitrogen atom. One key aspect is its trigonal pyramidal shape. This shape is due to the presence of a lone pair of electrons on the nitrogen atom, which repels the hydrogen atoms slightly inward, compressing the bond angle compared to that of a perfect tetrahedron.

• The nitrogen atom in ammonia forms covalent bonds with each hydrogen atom.
• This position allows ammonia to act as a base, readily accepting protons due to the lone pair on nitrogen.

This structure explains a lot about ammonia's chemical properties and how it interacts with other molecules.

H-N-H Angle

The H-N-H angle is an important feature in the structure of ammonia. This specific angle gives insight into the spatial arrangement and bonding characteristics of the molecule. When calculating this angle using the Law of Cosines, the side lengths in the triangle are the distances between the nitrogen and hydrogen atoms.To find the bond angle \( \phi \), you first determine the cosine value given by \[ \cos(\phi) = \frac{a^2 + b^2 - c^2}{2ab} \] With the calculated cosine value, you use the inverse cosine function to find \( \phi \), yielding an angle of approximately \( 107.8^{\circ} \). This angle confirms the slightly compressed bond structure due to the lone pair on nitrogen, which influences the spatial distribution of atoms around the nitrogen.