# Computation of bond angles and other angles in tetrahedral

• MHB
• WMDhamnekar
In summary, the bond angle of $109.5^\circ$ in tetrahedral molecular shapes is computed using the dot product method, where the angle is calculated from the definition of the dot product between two bonds. The points (+1,+1,+1), (+1,-1,-1), (-1,+1,-1), (-1,-1,+1) form a regular tetrahedron with its center at the origin, and the angle $\phi$ can be calculated as $\phi=\arccos\left(-\frac 13\right)\approx 109.5^\circ$. To compute other angles in a tetrahedral structure, a similar method can be used.
WMDhamnekar
MHB
Hello,
I didn't understand the geometry of molecules in which central atom has no lone pairs of electrons. for example, in $CH_4, NH_4^+$ molecular shape is tetrahedral and bond angle is $109.5^\circ$. How is that bond angle computed? $CH_4$ stands for liquid methane and $NH_4^+$ is a polyatomic cation. Now my other question involve mathematics as well.

If i want to compute other angles of this tetrahedral, how can i compute it?

Dhamnekar Winod said:
Hello,
I didn't understand the geometry of molecules in which central atom has no lone pairs of electrons. for example, in $CH_4, NH_4^+$ molecular shape is tetrahedral and bond angle is $109.5^\circ$. How is that bond angle computed? $CH_4$ stands for liquid methane and $NH_4^+$ is a polyatomic cation.Now my other question involve mathematics as well.

If i want to compute other angles of this tetrahedral, how can i compute it?

Hi,

I got the answer to the question how is $109.5^\circ$ angle between all the bonds in tetrahedral structure computed.

Dhamnekar Winod said:
Hi,

I got the answer to the question how is $109.5^\circ$ angle between all the bonds in tetrahedral structure computed.
Here is one way to do it.

First consider that the points (+1,+1,+1), (+1,-1,-1), (-1,+1,-1), (-1,-1,+1) span a regular tetrahedron with its center at the origin.
\begin{tikzpicture}
%preamble \usepackage{tikz-3dplot}
\tdplotsetmaincoords{80}{110}
\begin{scope}[scale=3,tdplot_main_coords]
\coordinate[label=below:O] (O) at (0,0,0);
\coordinate[label=A] (A) at (+1,+1,+1);
\coordinate[label=left:B] (B) at (+1,-1,-1);
\coordinate[label=right:C] (C) at (-1,+1,-1);
\coordinate[label=D] (D) at (-1,-1,+1);

\draw[-latex] (O) -- (1,0,0) node[ left ] {x};
\draw[-latex] (O) -- (0,1,0) node[ right ] {y};
\draw[-latex] (O) -- (0,0,1) node[ above ] {z};

\draw[help lines] (-1,1,1) -- (-1,-1,1) -- (1,-1,1);
\draw[help lines] (1,1,-1) -- (-1,1,-1) -- (-1,1,1) -- (1,1,1);
\draw[help lines] (0,-1,1) -- (0,1,1) -- (0,1,-1);
\draw[help lines] (1,1,1) -- (1,-1,1) -- (1,-1,-1) -- (1,1,-1) -- cycle;
\draw[help lines] (1,-1,0) -- (1,1,0) -- (-1,1,0) (1,0,-1) -- (1,0,1) -- (-1,0,1);
\draw[dotted] (C) -- (D);
\draw[dashed] (O) -- (A);
\draw[dashed] (O) -- (B);
\draw[dashed] (O) -- (C);
\draw[dashed] (O) -- (D);
\draw[thick] (B) -- (C) -- (A) -- (D) -- (B);
\draw[ultra thick] (A) -- (B);
\end{scope}
\end{tikzpicture}

The angle between 2 bonds is $\phi=\angle AOB$.
We can calculate the angle $\phi$ from the definition of the dot product:
$\overrightarrow{OA} \cdot \overrightarrow{OB} = OA\cdot OB \cdot \cos\phi \\ \cos\phi = \frac{\overrightarrow{OA} \cdot \overrightarrow{OB}}{OA\cdot OB} = \frac{(+1,+1,+1)\cdot(+1,-1,-1)}{\|(+1,+1,+1)\|\cdot \|(+1,-1,-1)\|} = \frac{-1}{\sqrt 3\cdot \sqrt 3} = -\frac 13 \\ \phi=\arccos\left(-\frac 13\right)\approx 109.5^\circ$

Last edited:

## What is a tetrahedral molecule?

A tetrahedral molecule is a molecule in which the central atom is surrounded by four other atoms or groups of atoms, forming a tetrahedron shape. This arrangement results in bond angles of approximately 109.5 degrees.

## How are bond angles calculated in a tetrahedral molecule?

Bond angles in a tetrahedral molecule are calculated using the valence shell electron pair repulsion (VSEPR) theory. This theory states that the electron pairs in the valence shell of an atom will arrange themselves in a way that minimizes repulsion, resulting in specific bond angles.

## What is the relationship between bond angles and molecular geometry?

Bond angles play a crucial role in determining the molecular geometry of a molecule. The specific bond angles in a molecule will determine its overall shape, which can affect its chemical and physical properties.

## What factors can influence bond angles in a tetrahedral molecule?

The main factors that can influence bond angles in a tetrahedral molecule are the number of lone pairs on the central atom and the electronegativity of the surrounding atoms. Lone pairs can create greater repulsion, resulting in slightly smaller bond angles, while electronegative atoms can pull bonding electrons closer, resulting in slightly larger bond angles.

## How can the bond angles in a tetrahedral molecule be measured experimentally?

The bond angles in a tetrahedral molecule can be measured experimentally using techniques such as X-ray crystallography or nuclear magnetic resonance (NMR) spectroscopy. These methods allow for the visualization and measurement of the molecular structure, including bond angles.

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