To determine the correct order of first ionization energies for the given elements (Titanium, Manganese, Zinc, and Nickel), we will analyze their electronic configurations and the stability associated with them.
### Step-by-Step Solution:
1. **Identify the Elements and Their Atomic Numbers:**
- Titanium (Ti) - Atomic number 22
- Manganese (Mn) - Atomic number 25
- Zinc (Zn) - Atomic number 30
- Nickel (Ni) - Atomic number 28
2. **Write the Electronic Configurations:**
- Titanium (Ti): \( [Ar] \, 4s^2 \, 3d^2 \)
- Manganese (Mn): \( [Ar] \, 4s^2 \, 3d^5 \)
- Zinc (Zn): \( [Ar] \, 4s^2 \, 3d^{10} \)
- Nickel (Ni): \( [Ar] \, 4s^2 \, 3d^8 \)
3. **Analyze the Stability of the Configurations:**
- **Zinc (Zn)** has a fully filled \(3d^{10}\) subshell, making it very stable. Removing an electron from a stable configuration requires a high amount of energy.
- **Nickel (Ni)** has a \(3d^8\) configuration. Although it has a filled \(4s^2\), it is less stable than Zn but more stable than Ti and Mn.
- **Manganese (Mn)** has a half-filled \(3d^5\) configuration, which is relatively stable, thus requiring a significant amount of energy to remove an electron.
- **Titanium (Ti)** has \(3d^2\) configuration, which is neither half-filled nor fully filled, making it the least stable and requiring the least ionization energy.
4. **Determine the Order of Ionization Energies:**
- From the analysis, we can conclude:
- Highest Ionization Energy: Zinc (Zn)
- Second Highest: Nickel (Ni)
- Third: Manganese (Mn)
- Lowest: Titanium (Ti)
5. **Final Order of Ionization Energies:**
- The correct order of first ionization energies is:
\[
\text{Zn} > \text{Ni} > \text{Mn} > \text{Ti}
\]
