1. Understanding the Complex

The complex given is cis[Mn(en)₂Cl₂]. Let's break down its components:

• Central Metal Ion: Manganese (Mn).
• Ligands:
  • en$: Ethylenediamine ($NH_2CH_2CH_2NH_2$), which is a bidentate ligand. There are two $en ligands, providing a total of 4 nitrogen donor atoms.
  • Cl: Chloro, a monodentate ligand. There are two Cl ligands.
• Coordination Number: The total number of donor atoms attached to the central metal ion is $2 \times 2$ (from en) $+ 2 \times 1$ (from Cl) $= 6$. A coordination number of 6 corresponds to an octahedral geometry.
• Isomer: The prefix cis indicates that the two Cl ligands are adjacent to each other, with a Cl-Mn-Cl bond angle of 90°.

2. Visualizing the Structure

To count the bond angles, we can visualize the octahedral complex by placing the Mn atom at the origin of a 3D Cartesian coordinate system. The six ligand positions will be along the positive and negative x, y, and z axes.

Let the six positions be:

• P1: (+x, 0, 0)
• P2: (-x, 0, 0)
• P3: (0, +y, 0)
• P4: (0, -y, 0)
• P5: (0, 0, +z)
• P6: (0, 0, -z)

In this arrangement:

• Any two positions on the same axis are trans to each other (180° angle), e.g., P1 and P2.
• Any two positions on different axes are cis to each other (90° angle), e.g., P1 and P3.

3. Placing the Ligands

• Since the complex is cis, we place the two Cl ligands at any two adjacent (cis) positions. Let's place Cl₁ at P1 and Cl₂ at P3.
  • Cl₁ is at (+x, 0, 0).
  • Cl₂ is at (0, +y, 0).
  • The angle Cl₁-Mn-Cl₂ is 90°.
• The remaining four positions (P2, P4, P5, P6) are occupied by the four nitrogen atoms from the two en ligands. Let's call them N₁, N₂, N₃, N₄.

4. Counting the cis N-Mn-Cl Bond Angles

The question asks for the total number of cis N-Mn-Cl bond angles. A cis bond angle in an octahedral complex is 90°. We need to find the number of pairs of (N, Cl) ligands that are at a 90° angle to each other.

We can do this by considering each Cl ligand separately.

A) Angles involving Cl₁ (at position P1):

• A ligand at P1 (+x axis) is cis (90°) to ligands at the y and z axes (P3, P4, P5, P6).
• A ligand at P1 is trans (180°) to the ligand at P2 (-x axis).
• The four cis positions relative to P1 are occupied by Cl₂ (at P3) and three N atoms (at P4, P5, P6).
• The trans position is occupied by one N atom (at P2).
• Therefore, Cl₁ forms a 90° angle with three N atoms.
• This gives us 3 cis N-Mn-Cl bond angles.

B) Angles involving Cl₂ (at position P3):

• A ligand at P3 (+y axis) is cis (90°) to ligands at the x and z axes (P1, P2, P5, P6).
• A ligand at P3 is trans (180°) to the ligand at P4 (-y axis).
• The four cis positions relative to P3 are occupied by Cl₁ (at P1) and three N atoms (at P2, P5, P6).
• The trans position is occupied by one N atom (at P4).
• Therefore, Cl₂ forms a 90° angle with three N atoms.
• This gives us 3 cis N-Mn-Cl bond angles.

5. Total Count

The total number of cis N-Mn-Cl bond angles is the sum of the counts from both Cl ligands.

Total angles = (Angles with Cl₁) + (Angles with Cl₂) = 3 + 3 = 6.

This counting is consistent with the fact that the en ligands must chelate by connecting two cis positions. For instance, the N atoms at P2 and P5 can belong to one en ligand, and those at P4 and P6 can belong to the other, as the angles P2-Mn-P5 and P4-Mn-P6 are both 90°.

Thus, there are a total of 6 cis N-Mn-Cl bond angles in the molecule.