In CH4 molecule, there are four C-H bonds. If two adjacent bonds are in î+ĵ+k̂ and î-ĵ-k̂ direct

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In CH4 molecule, there are four C-H bonds. If two adjacent...

In $\mathrm{CH}_{4}$ molecule, there are four $\mathrm{C}-\mathrm{H}$ bonds. If two adjacent bonds are in $\hat{i}+\hat{j}+\hat{k}$ and $\hat{i}-\hat{j}-\hat{k}$ direction, then find the angle between these bonds. (A) $\sin ^{-1}\left(\frac{-1}{3}\right)$ (B) $\cos ^{-1}\left(\frac{1}{3}\right)$ (C) $\sin ^{-1}\left(\frac{1}{3}\right)$ (D) $\cos ^{-1}\left(\frac{-1}{3}\right)$

In $\mathrm{CH}_{4}$ molecule, there are four $\mathrm{C}-\mathrm{H}$ bonds. If two adjacent bonds are in $\hat{i}+\hat{j}+\hat{k}$ and $\hat{i}-\hat{j}-\hat{k}$ direction, then find the angle between these bonds.
(A) $\sin ^{-1}\left(\frac{-1}{3}\right)$
(B) $\cos ^{-1}\left(\frac{1}{3}\right)$
(C) $\sin ^{-1}\left(\frac{1}{3}\right)$
(D) $\cos ^{-1}\left(\frac{-1}{3}\right)$

Key Concepts

Angle Calculation Using Dot Product

Calculating the angle between two vectors involves using the dot product formula: a · b = |a||b| cos(?). When the vectors represent bond directions, this formula allows determination of the angle between the bonds by isolating the cosine term. This approach is widely used in chemistry and physics to analyze bond angles and other angular relationships between directional quantities.

Vector Dot Product

The dot product is a mathematical operation that takes two vectors and returns a scalar. It is defined as the product of the magnitudes of the two vectors and the cosine of the angle between them. This concept is crucial in determining the angle between bonds when represented as vectors, as it directly relates the components of the vectors to the cosine of the angle.

Molecular Geometry

This concept involves understanding the three?dimensional arrangement of atoms within a molecule. In the context of CH4, the molecule exhibits a tetrahedral geometry due to the repulsion between electron pairs, resulting in bond angles of approximately 109.5° between the hydrogen atoms. This arrangement is central to predicting molecular properties such as bond angles and reactivity.

Tetrahedral Geometry

Tetrahedral geometry is a specific molecular shape where a central atom is equidistant from four surrounding atoms, forming a shape similar to a pyramid with a triangular base. This structure minimizes electron pair repulsions according to VSEPR theory, leading to bond angles of about 109.5°, which is fundamental in understanding the spatial distribution of bonds in molecules like methane.

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