It is given that for conformational isomers, the net dipole moment
`mu_(obs)=sum mu_(i)x_(i)`
Where `mu_(obs)` = obsereved dipole moment of the compound
`mu_(i)` = dipole moment of the stable conformational isomers
`X_(1)` = mole fraction of stable conformers
of the compound `Z-CH_(2)-CH_(2)-Z` draw the Newman projection formula of all stable conformational isomers, if `mu_(obs) = 1D, " and " X_(anti) = 0.82, " find " mu_(gauche)`. Now draw the Newman projection formula of the most stable conformation of meso Y-CHD-CHD-Y.
(a) If Y is `CH_(3) ("rotation about "C_(2)-C_(3) " bond")`
(b) If Y is OH `("rotation about "C_(1)-C_(2) " bond")`

Answer

To solve the problem step by step, we will first analyze the given information and then calculate the required values.

### Step 1: Identify the conformational isomers
The compound given is Z-CH₂-CH₂-Z. This compound can exist in two stable conformations:
1. **Gauche Conformation**
2. **Anti Conformation**

### Step 2: Determine the mole fractions
We are given:
- The mole fraction of the anti conformer, \( X_{anti} = 0.82 \)
- Therefore, the mole fraction of the gauche conformer can be calculated as:
\[ X_{gauche} = 1 - X_{anti} = 1 - 0.82 = 0.18 \]

### Step 3: Use the dipole moment equation
The observed dipole moment \( \mu_{obs} \) is given as 1 D. According to the formula:
\[ \mu_{obs} = \mu_{gauche} \cdot X_{gauche} + \mu_{anti} \cdot X_{anti} \]
Since the anti conformer has a dipole moment of 0 (as the dipoles cancel out), we can simplify the equation:
\[ 1 = \mu_{gauche} \cdot 0.18 + 0 \cdot 0.82 \]
This simplifies to:
\[ 1 = \mu_{gauche} \cdot 0.18 \]

### Step 4: Solve for \( \mu_{gauche} \)
Now, we can solve for \( \mu_{gauche} \):
\[ \mu_{gauche} = \frac{1}{0.18} \approx 5.56 \, D \]

### Step 5: Draw the Newman projection for the most stable conformation of meso Y-CHD-CHD-Y
Now we need to draw the Newman projection for the most stable conformation of meso Y-CHD-CHD-Y for two cases:

#### (a) If Y is CH₃ (rotation about C₂-C₃ bond)
1. **Draw the front carbon (C₂)** with the CH₃ group at the top.
2. **Draw the back carbon (C₃)** with the two CHD groups staggered.
3. This arrangement minimizes steric hindrance and is the most stable conformation.

#### (b) If Y is OH (rotation about C₁-C₂ bond)
1. **Draw the front carbon (C₁)** with the OH group at the top.
2. **Draw the back carbon (C₂)** with the two CHD groups staggered.
3. This arrangement also minimizes steric hindrance and is the most stable conformation.

### Summary of Results
- The dipole moment of the gauche conformer \( \mu_{gauche} \) is approximately 5.56 D.
- The Newman projections for both cases (Y as CH₃ and Y as OH) show the most stable staggered conformations.