To find the sum of X, Y, and Z where:
- X is the O-N-O bond angle in \( NO_3^{-} \)
- Y is the O-N-O bond angle in \( NO_2^{+} \)
- Z is the F-Xe-F adjacent bond angle in \( XeF_{4} \)
Let's calculate each one step by step.
### Step 1: Calculate X (O-N-O bond angle in \( NO_3^{-} \))
1. **Determine the hybridization of the nitrogen in \( NO_3^{-} \)**:
- Nitrogen has 5 valence electrons.
- There are 3 oxygen atoms (monovalent).
- The charge is -1, so we add 1.
- The formula for hybridization is:
\[
\text{Hybridization} = \frac{\text{Valence electrons of central atom} + \text{Monovalent atoms} + \text{Charge}}{2}
\]
- Plugging in the values:
\[
\text{Hybridization} = \frac{5 + 3 + 1}{2} = \frac{9}{2} = 4.5 \quad \text{(which indicates sp}^2\text{ hybridization)}
\]
2. **Determine the molecular geometry**:
- With sp² hybridization and no lone pairs, the geometry is trigonal planar.
3. **Determine the bond angle**:
- In a trigonal planar arrangement, the bond angle is 120 degrees.
- Therefore, \( X = 120^\circ \).
### Step 2: Calculate Y (O-N-O bond angle in \( NO_2^{+} \))
1. **Determine the hybridization of nitrogen in \( NO_2^{+} \)**:
- Nitrogen has 5 valence electrons.
- There are 2 oxygen atoms (monovalent).
- The charge is +1, so we subtract 1.
- Using the hybridization formula:
\[
\text{Hybridization} = \frac{5 + 2 - 1}{2} = \frac{6}{2} = 3 \quad \text{(which indicates sp hybridization)}
\]
2. **Determine the molecular geometry**:
- With sp hybridization and no lone pairs, the geometry is linear.
3. **Determine the bond angle**:
- In a linear arrangement, the bond angle is 180 degrees.
- Therefore, \( Y = 180^\circ \).
### Step 3: Calculate Z (F-Xe-F adjacent bond angle in \( XeF_{4} \))
1. **Determine the hybridization of xenon in \( XeF_{4} \)**:
- Xenon has 8 valence electrons.
- There are 4 fluorine atoms (monovalent).
- Using the hybridization formula:
\[
\text{Hybridization} = \frac{8 + 4}{2} = 6 \quad \text{(which indicates sp}^3\text{d}^2\text{ hybridization)}
\]
2. **Determine the molecular geometry**:
- With sp³d² hybridization, the geometry is octahedral.
- However, the shape is square planar due to the presence of 2 lone pairs.
3. **Determine the bond angle**:
- In a square planar arrangement, the adjacent bond angles are 90 degrees.
- Therefore, \( Z = 90^\circ \).
### Final Calculation: Sum of X, Y, and Z
Now we can find the sum:
\[
X + Y + Z = 120^\circ + 180^\circ + 90^\circ = 390^\circ
\]
Thus, the final answer is:
\[
\text{Sum of } X, Y, \text{ and } Z = 390^\circ
\]
