Determine the number of 3d-electrons in each ion.
Step 1: Electron configuration of neutral atoms.
- Chromium (Cr), Z=24: \[ \text{Cr}: [Ar] \, 3d^5 \, 4s^1 \]
- Copper (Cu), Z=29: \[ \text{Cu}: [Ar] \, 3d^{10} \, 4s^1 \]
- Titanium (Ti), Z=22: \[ \text{Ti}: [Ar] \, 3d^2 \, 4s^2 \]
- Manganese (Mn), Z=25: \[ \text{Mn}: [Ar] \, 3d^5 \, 4s^2 \]
Step 2: Electron configuration of the ions.
- For Cr$^{2+}$: Remove 2 electrons from 4s, then 3d. Configuration: \[ \text{Cr}^{2+}: [Ar] \, 3d^4 \] Number of 3d-electrons = 4.
- For Cu$^{2+}$: Remove 2 electrons from 4s, then 3d. Configuration: \[ \text{Cu}^{2+}: [Ar] \, 3d^9 \] Number of 3d-electrons = 9.
- For Ti$^{3+}$: Remove 3 electrons from 4s, then 3d. Configuration: \[ \text{Ti}^{3+}: [Ar] \, 3d^1 \] Number of 3d-electrons = 1.
- For Mn$^{2+}$: Remove 2 electrons from 4s, then 3d. Configuration: \[ \text{Mn}^{2+}: [Ar] \, 3d^5 \] Number of 3d-electrons = 5.
Step 3: Arrange ions in increasing order of 3d-electrons.
- Ti$^{3+}$: 1 3d-electron.
- Cr$^{2+}$: 4 3d-electrons.
- Mn$^{2+}$: 5 3d-electrons.
- Cu$^{2+}$: 9 3d-electrons.
The increasing order is: Ti$^{3+}$ (1)<Cr$^{2+}$ (4)<Mn$^{2+}$ (5)<Cu$^{2+}$ (9).
Final Answer:
\[ \boxed{\text{The correct order is Ti^{3+}, Cr^{2+}, Mn^{2+}, Cu^{2+}.}} \]