To determine the correct formula and geometry of the first complex `Ni(NH₃)₄(NO₃)₂·2H₂O` with a magnetic moment of zero, we can follow these steps:
### Step 1: Understand the Magnetic Moment
The magnetic moment (μ) of a complex is related to the presence of unpaired electrons. A magnetic moment of zero indicates that there are no unpaired electrons in the complex.
**Hint:** Recall that the formula for magnetic moment is μ = √(n(n+2)), where n is the number of unpaired electrons. If μ = 0, then n = 0.
### Step 2: Identify the Coordination Number and Geometry
The empirical formula indicates that nickel (Ni) is coordinated with four ammonia (NH₃) ligands and two nitrate (NO₃) ligands. This suggests a coordination number of 6, which is typical for octahedral complexes.
**Hint:** Coordination number can help determine the geometry of the complex. Common geometries include tetrahedral (4) and octahedral (6).
### Step 3: Determine the Electronic Configuration of Ni
Nickel has an atomic number of 28. The electronic configuration of Ni is [Ar] 3d⁸ 4s². In a coordination complex, the 4s electrons are typically lost first, so we focus on the 3d electrons.
**Hint:** Remember that the oxidation state of Ni in the complex will affect its electronic configuration.
### Step 4: Determine the Oxidation State of Nickel
In the complex `Ni(NH₃)₄(NO₃)₂·2H₂O`, NH₃ is a neutral ligand, and NO₃ is a negatively charged ligand. Each NO₃ has a charge of -1, so the total charge contributed by the two NO₃ ligands is -2. To balance this and keep the complex neutral, Ni must be in the +2 oxidation state.
**Hint:** The overall charge of the complex must be neutral. Calculate the charge contributed by the ligands to find the oxidation state of the metal.
### Step 5: Write the Electronic Configuration for Ni²⁺
For Ni²⁺, we remove two electrons from the 4s and 3d orbitals, resulting in the configuration [Ar] 3d⁸.
**Hint:** When determining the electronic configuration for a cation, remember to remove electrons from the outermost shell first.
### Step 6: Determine the Geometry
With a d⁸ configuration and a coordination number of 6, the most stable geometry for Ni²⁺ is square planar. In a square planar geometry, the 3d orbitals split into two sets, allowing for pairing of electrons.
**Hint:** Different geometries can lead to different arrangements of electrons. Square planar complexes often lead to paired electrons in d⁸ configurations.
### Step 7: Confirm the Absence of Unpaired Electrons
In a square planar arrangement, the 3d electrons will fill as follows:
- 3dxy, 3dxz, and 3dyz will have paired electrons, and the 3dz² and 3dx²-y² will also be filled, resulting in no unpaired electrons.
Thus, the magnetic moment is zero.
**Hint:** Visualize the d-orbital splitting in different geometries to understand how electrons are arranged.
### Conclusion
The correct formula and geometry of the first complex `Ni(NH₃)₄(NO₃)₂·2H₂O` is:
- **Formula:** [Ni(NH₃)₄(NO₃)₂] (as a neutral complex)
- **Geometry:** Square planar
