Consider that the coordinating atoms of the ligands in cis-[Co(NH3)4Cl2]Cl and mer-[Co(NH3)3Cl3] oct

Q: Why don't we consider the Cl outside the square bracket in the first complex?

The $Cl$ outside the bracket is a counter ion. It exists in the outer coordination sphere to balance the charge and does not form the vertices of the octahedral geometry around the central Cobalt atom. Only the ligands inside the square brackets dictate the shape.

Q: What exactly does 'mer' mean in coordination chemistry?

'Mer' stands for meridional. It means three identical ligands are arranged along a meridian of the octahedron. They form a T-shape and occupy three positions that are coplanar with the central metal atom, unlike the 'fac' (facial) isomer where they form a single triangular face.

Q: How do I know which faces share an edge without drawing it?

It's a fundamental property of the octahedral solid! In any regular octahedron, every single edge acts as a hinge connecting exactly two triangular faces. So, if you identify one edge connecting two $Cl$ atoms, you automatically know it belongs to exactly two faces.

Q: Could a face have three Cl atoms in the mer-isomer?

No. In the mer-isomer, the three $Cl$ atoms are arranged in a T-shape (e.g., positions 1, 2, and 3 around the equator). They do not form a mutually adjacent triangle. A face with three $Cl$ atoms would require the 'fac' (facial) isomer arrangement.

Q: Is it possible to have a face with two N atoms and one Cl atom?

Yes, absolutely! Both complexes have faces with two N atoms and one Cl atom. However, the question specifically restricts us to counting ONLY the faces that have exactly one N atom and two Cl atoms. Always read the constraints carefully!

Question

Consider that the coordinating atoms of the ligands in $cis	ext{-}[Co(NH_3)_4Cl_2]Cl$ and $mer	ext{-}[Co(NH_3)_3Cl_3]$ octahedral complexes are at the vertices of an octahedron. The sum of total number of the triangular faces in both the complexes having one N atom and two Cl atoms at their corners is ____.

Sigma Percentile · Accepted Answer

Welcome to a beautiful journey into the 3D world of coordination chemistry! This problem is a fantastic test of your spatial visualization skills. We are asked to count specific triangular faces on two different octahedral complexes. Let's break down the geometry and decode the structures step by step.

### The Anatomy of an Octahedron

Before we dive into the specific molecules, let's make sure we perfectly understand the stage on which this play is happening: the **octahedron**. An octahedron is a highly symmetric 3D shape featuring a central atom surrounded by six ligands.

Geometrically, it consists of **6 vertices**, **12 edges**, and **8 triangular faces**. Every single face is a triangle formed by three mutually adjacent vertices. A crucial property to remember is that **every edge in an octahedron is shared by exactly two triangular faces**.

The question asks us to find the total number of triangular faces that have exactly **one nitrogen atom** and **two chlorine atoms** at their corners. In simpler terms, we are looking for faces that contain an edge connecting two $Cl$ atoms, with the third vertex of the triangle being an $NH_3$ molecule.

### Decoding the Cis Isomer

Our first protagonist is the $cis	ext{-}[Co(NH_3)_4Cl_2]^+$ complex. The prefix **'cis'** is the key here. It tells us that the two identical chlorine ligands are placed adjacent to each other, separated by a $90^\circ$ bond angle.

Because these two $Cl$ atoms are adjacent, they form exactly **one edge** of the octahedron. Now, recall our geometric rule: a single edge is shared by exactly two faces. Therefore, there are exactly two triangular faces that contain this specific $Cl-Cl$ edge.

What about the third vertex of these two faces? Since all the remaining four positions in the complex are occupied by $NH_3$ ligands, the third vertex for both of these faces must be a nitrogen atom.

Thus, the cis isomer contributes exactly **$2$ faces** that meet our criteria.

### Decoding the Mer Isomer

Now, let's turn our attention to the $mer	ext{-}[Co(NH_3)_3Cl_3]$ complex. The prefix **'mer'** stands for meridional. This indicates that the three identical chlorine ligands are arranged along a meridian of the octahedron, forming a T-shape. They occupy three positions that are coplanar with the central cobalt atom.

If you visualize this T-shape arrangement, you will notice that it creates **two separate pairs** of adjacent chlorine atoms. For instance, if the chlorines are at positions 1, 2, and 3 around the equator, the adjacent pairs are (1,2) and (2,3). This means we have **two distinct $Cl-Cl$ edges**.

Applying our geometric rule again: each of these two edges is shared by two faces. 
- The first $Cl-Cl$ edge is shared by 2 faces.
- The second $Cl-Cl$ edge is shared by 2 faces.

This gives us a total of $2 	imes 2 = 4$ faces containing two chlorine atoms. Since the remaining three vertices of the octahedron are occupied by $NH_3$ ligands, the third vertex for all four of these faces is guaranteed to be a nitrogen atom.

Thus, the mer isomer contributes exactly **$4$ faces** that meet our criteria.

### The Final Tally

Bringing it all together is the easiest part. We simply sum the valid faces from both complexes:

$$	ext{Total Faces} = (	ext{Faces in } cis) + (	ext{Faces in } mer)$$
$$	ext{Total Faces} = 2 + 4 = \mathbf{6}$$

There are exactly 6 such triangular faces across both complexes. This problem beautifully demonstrates how chemical nomenclature directly translates into precise 3D geometric properties!

Enter Numerical Value:

Visualized Solution

$O c t ah e d r a l C o m pl e x es$

$T a r g e t : F a ces w i t h 2 C l an d 1 N$

$c i s - [C o (N H_{3})_{4} C l_{2}]^{+}$

$F a ces in c i s I so m er$

$m er - [C o (N H_{3})_{3} C l_{3}]$

$F a ces in m er I so m er$

$T o t a l T r ian g u l a r F a ces$

The Sigma Insight: Nomenclature, Isomerism, Importance and Werner's Theory

The Anatomy of an Octahedron

Decoding the Cis Isomer

Decoding the Mer Isomer

The Final Tally

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Enter Numerical Value:

Visualized Solution

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Octahedral Complexes

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Target: Faces with 2 Cl and 1 N

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } cis-[Co(NH3​)4​Cl2​]+

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Faces in cis Isomer

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } mer-[Co(NH3​)3​Cl3​]

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Faces in mer Isomer

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Total Triangular Faces