For the compound `A-CH_(2)-CH_(27)-A` draw the newman projection formula of all the stable conformational it `mu_(obs) = 2D " and " X_(anti) = 0.75 " then find " mu_(gauche.)("If "A= NO_(2))`

Answer

To solve the problem step by step, we will first address the drawing of the Newman projection for the compound \( A-CH_2-CH_2-A \) and then calculate \( \mu_{\text{gauche}} \) based on the given data.

### Step 1: Understand the Compound Structure
The compound can be represented as \( A-CH_2-CH_2-A \), where \( A \) is a substituent (in this case, \( NO_2 \)). The two \( CH_2 \) groups are connected to the substituents \( A \).

### Step 2: Draw the Newman Projection
To draw the Newman projection, we will look down the bond connecting the two \( CH_2 \) groups. The three main conformations to consider are:

1. **Eclipsed Conformation**: In this conformation, the substituents on the front and back carbon atoms are aligned with each other.
2. **Gauche Conformation**: In this conformation, the substituents are staggered but not opposite each other (i.e., they are at a dihedral angle of 60 degrees).
3. **Anti Conformation**: In this conformation, the substituents are staggered and positioned opposite each other (i.e., at a dihedral angle of 180 degrees).

#### Eclipsed Conformation
- Draw the \( NO_2 \) groups on the front and back carbon atoms aligned with each other.
- The remaining positions will have hydrogen atoms.

#### Gauche Conformation
- Draw the \( NO_2 \) groups staggered at a 60-degree angle relative to each other.
- The remaining positions will have hydrogen atoms.

#### Anti Conformation
- Draw the \( NO_2 \) groups staggered and directly opposite each other.
- The remaining positions will have hydrogen atoms.

### Step 3: Label the Conformations
- Label the first conformation as **Eclipsed**.
- Label the second conformation as **Gauche**.
- Label the third conformation as **Anti**.

### Step 4: Calculate \( \mu_{\text{gauche}} \)
Given:
- \( \mu_{\text{obs}} = 2D \)
- \( X_{\text{anti}} = 0.75 \)

Using the formula:
\[ \mu_{\text{obs}} = \mu_{\text{gauche}} \cdot X_{\text{gauche}} + \mu_{\text{anti}} \cdot X_{\text{anti}} \]

Since \( X_{\text{gauche}} + X_{\text{anti}} = 1 \), we can find \( X_{\text{gauche}} \):
\[ X_{\text{gauche}} = 1 - X_{\text{anti}} = 1 - 0.75 = 0.25 \]

Substituting the known values into the equation:
\[ 2 = \mu_{\text{gauche}} \cdot 0.25 + 0 \cdot 0.75 \]
\[ 2 = \mu_{\text{gauche}} \cdot 0.25 \]

Now, solving for \( \mu_{\text{gauche}} \):
\[ \mu_{\text{gauche}} = \frac{2}{0.25} = 8D \]

### Final Answer
The value of \( \mu_{\text{gauche}} \) when \( A = NO_2 \) is \( 8D \).