Problem 45 - Essential Calculus Early Transcendentals

Question

A molecule of methane, \(\mathrm{CH}_{4},\) is structured with the four hydrogen atoms at the vertices of a regular tetrahedron and the carbon atom at the centroid. The bond angle is the angle formed by the \(\mathrm{H}-\mathrm{C}-\mathrm{H}\) combination; it is the angle between the lines that join the carbon atom to two of the hydrogen atoms. Show that the bond angle is about \(109.5^{\circ}\) Hint: Take the vertices of the tetrahedron to be the points \((1,0,0),(0,1,0),(0,0,1),\) and \((1,1,1),\) as shown in the figure. Then the centroid is \(\left(\frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right) . ]\)

Step-by-Step Solution

Answer

The bond angle is approximately \(109.5^{\circ}\).

Step 1: Identify the Positions of Atoms

The vertices of the tetrahedron are given as \((1,0,0), (0,1,0), (0,0,1), (1,1,1)\). The carbon atom (centroid) is at \(\left(\frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right)\). Choose two hydrogen atoms, say at \((1,0,0)\) and \((0,1,0)\), to form the angle with the carbon atom.

Step 2: Calculate Vector Directions

Construct vectors from the carbon atom to two of the hydrogen atoms: - Vector \(\vec{CA}\) from \(C\) to \((1,0,0)\) is \(\left(\frac{1}{2}, -\frac{1}{2}, -\frac{1}{2}\right)\).- Vector \(\vec{CB}\) from \(C\) to \((0,1,0)\) is \(\left(-\frac{1}{2}, \frac{1}{2}, -\frac{1}{2}\right)\).

Step 3: Use Dot Product to Find Cosine of Angle

The dot product of \(\vec{CA}\) and \(\vec{CB}\) is given by: \(\vec{CA} \cdot \vec{CB} = \left(\frac{1}{2}\right)^2 - \left(\frac{1}{2}\right)^2 - \left(\frac{1}{2}\right)^2 = -\frac{1}{4}\).Compute the magnitudes: \(|\vec{CA}| = |\vec{CB}| = \sqrt{\left(\frac{1}{2}\right)^2 + \left(\frac{1}{2}\right)^2 + \left(\frac{1}{2}\right)^2} = \frac{\sqrt{3}}{2}\).

Step 4: Compute the Cosine and the Angle

The cosine of the bond angle \(\theta\) is:\[ \cos \theta = \frac{\vec{CA} \cdot \vec{CB}}{|\vec{CA}| \cdot |\vec{CB}|} = \frac{-\frac{1}{4}}{\left(\frac{\sqrt{3}}{2}\right)^2} = -\frac{1}{3} \]Thus, \(\theta = \cos^{-1}(-\frac{1}{3})\) which is approximately \(109.5^{\circ}\).

Key Concepts

tetrahedron geometry

The concept of a tetrahedron is central to understanding the bond angles in molecules like methane. A tetrahedron is a geometric shape that consists of four triangular faces, six edges, and four vertices. In the case of a regular tetrahedron, all four faces are equilateral triangles, and all edges are of equal length. This symmetry plays an important role in determining the spatial configuration of atoms in certain molecules.

In a molecule like methane, the carbon atom is located at the centroid of this tetrahedral structure, while the hydrogen atoms are situated at the vertices. This arrangement maximizes distance between the hydrogen atoms, reducing repulsion and creating a particularly stable structure known as tetrahedral geometry.

The bond angle, which is approximately 109.5°, is the angle between any two lines drawn from the carbon atom to the vertices of the tetrahedron. This angle ensures that the structure is as symmetrical as possible, allowing each hydrogen atom to be equidistant from its neighbors, which leads to optimal molecular stability.

methane molecule

Methane ( CH 4 ) is a simple hydrocarbon molecule composed of one carbon atom bonded to four hydrogen atoms. It is one of the simplest organic molecules and is the primary component of natural gas. The fascinating aspect of methane is how its atoms are arranged in space, as this arrangement is the reason behind its chemical and physical properties.

The carbon atom in methane is at the geometric center or centroid of a tetrahedron, while the hydrogen atoms are at its vertices. This positions the carbon atom in such a way that the bonds with hydrogen are directed towards the corners of the tetrahedron, forming an angle of about 109.5° between each pair of bonds. Such a spatial arrangement leads to a very stable molecule with tetrahedral geometry.

Methane's tetrahedral shape is an example of a bond hybridization known as sp^3 hybridization, where one s-orbital and three p-orbitals in carbon combine to form four equivalent sp^3 hybrid orbitals. These orbitals are directed towards the vertices of a tetrahedron, allowing strong σ-bonds to form with hydrogen atoms.

dot product

The dot product, also known as the scalar product, is a fundamental mathematical operation used to find the angle between two vectors. It is key to determining the bond angle in the methane molecule.

For two vectors, \( \vec{A} \) and \( \vec{B} \), the dot product is defined as:\[\vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos(\theta)\]where \( \theta \) is the angle between the vectors, and \(|\vec{A}|\) and \(|\vec{B}|\) are the magnitudes of the vectors. By rearranging this formula, we can solve for \( \cos(\theta) \) and thus find the angle \( \theta \).

In the context of methane, vectors are constructed from the carbon atom to the hydrogen atoms. By calculating the dot product of two such vectors and dividing by the product of their magnitudes, we arrive at \( \cos \theta = -\frac{1}{3} \), leading to the bond angle of approximately 109.5°. The negative value indicates that the vectors form an angle greater than 90°, which is consistent with the tetrahedral angle.