Animated Solution for Chemistry - Hydrocarbons: In the following skew conformation of ethane, ${\text{H}}^{\prime }-\text{C}-\text{C}-{\text{H}}^{\prime \prime }$ dihedral angle is

When we look at a Newman projection, we are staring directly down the carbon-carbon bond axis. The front carbon is represented by a central dot, and the back carbon is represented by a large circle. In this specific problem, we are given a skew conformation of ethane, which is an intermediate state between the fully eclipsed and fully staggered forms.

Our primary objective is to find the dihedral angle between two specific hydrogen atoms: ${\text{H}}^{\prime }$ and ${\text{H}}^{\prime \prime }$. By carefully observing the diagram, we can identify that ${\text{H}}^{\prime }$ is attached to the back carbon, while ${\text{H}}^{\prime \prime }$ is attached to the front carbon. The dihedral angle is simply the angle between these two bonds when projected onto the 2D plane of the paper.

To find the total dihedral angle $\theta$, we can break the angular distance into two manageable segments using the top front hydrogen (let's call it ${\text{H}}_{\text{top}}$) as our reference point.

First, the problem explicitly gives us the angle between the back ${\text{H}}^{\prime }$ and the front ${\text{H}}_{\text{top}}$. This offset is exactly $29^{\circ }$.

Second, we need the angle between ${\text{H}}_{\text{top}}$ and ${\text{H}}^{\prime \prime }$. Since both of these hydrogen atoms are attached to the same front carbon, their bonds are separated by perfect tetrahedral geometry projected onto a 2D plane. This means the angle between any two adjacent bonds on the same carbon in a Newman projection is always exactly $120^{\circ }$.

Now, we simply add these two segments together to find the total angular distance from ${\text{H}}^{\prime }$ to ${\text{H}}^{\prime \prime }$:

Substituting our known values:

This elegant geometric breakdown leads us directly to the correct answer. Understanding how to navigate the fixed $120^{\circ }$ intervals of a Newman projection is a powerful tool for solving complex conformational analysis problems in organic chemistry.