The dark crystals of potassium permanganate are iso-structural with:

\(KClO_4\)

The correct answer is option 2. \(KClO_4\).

Isostructural compounds are those that share the same or very similar crystal structures. This means that the arrangement of ions or molecules within the crystal lattice is almost identical, even if the specific atoms or ions involved are different. For two compounds to be isostructural, they must have the same type of lattice (such as cubic, tetragonal, etc.) and the same arrangement of their constituent ions or molecules.

Structure of Potassium Permanganate \((KMnO_4)\):

Crystal Structure: \(KMnO_4\) crystallizes in an orthorhombic system.

Anion Structure: The \(MnO_4^-\) anion has a tetrahedral geometry, where one manganese atom is surrounded by four oxygen atoms.

Lattice Arrangement: The potassium ions \((K^+)\) are situated between these tetrahedral \(MnO_4^-\) ions, forming the crystal lattice.

Detailed Analysis of Each Option:

1. \(MnO_2\) (Manganese Dioxide):

Crystal Structure: \(MnO_2\) has a rutile-type structure, which is tetragonal. In this structure, manganese atoms are surrounded by six oxygen atoms in an octahedral geometry.

Comparison with \(KMnO_4\): The structure of \(MnO_2\) is very different from that of \(KMnO_4\), both in terms of the geometry of the anion and the overall crystal lattice. Therefore, \(MnO_2\) is not isostructural with \(KMnO_4\).

2. \(KClO_4\) (Potassium Perchlorate):

Crystal Structure: \(KClO_4\) crystallizes in the orthorhombic system, similar to \(KMnO_4\).

Anion Structure: The \(ClO_4^-\) anion in \(KClO_4\) has a tetrahedral structure, just like the \(MnO_4^-\) anion in \(KMnO_4\).

Lattice Arrangement: The potassium ions \((K^+)\) are similarly situated between the tetrahedral \(ClO_4^-\) ions in the crystal lattice.

Comparison with \(KMnO_4\): Both compounds have the same type of crystal lattice and similar arrangements of ions. Thus, **\(KClO_4\) is isostructural with \(KMnO_4\)**.

3. \(MnCl_2\) (Manganese(II) Chloride):

Crystal Structure: \(MnCl_2\) crystallizes in a different structure, typically a layered structure where each manganese ion is surrounded by six chloride ions in an octahedral arrangement.

Comparison with \(KMnO_4\): The structure and arrangement in \(MnCl_2\) are quite different from that of \(KMnO_4\), making it non-isostructural with \(KMnO_4\).

4. \(KIO_3\) (Potassium Iodate):

Crystal Structure: \(KIO_3\) has a trigonal structure, and the \(IO_3^-\) anion has a trigonal pyramidal geometry.

Comparison with \(KMnO_4\): The geometry of the anion and the crystal lattice in \(KIO_3\) are different from those in \(KMnO_4\). Therefore, \(KIO_3\) is not isostructural with \(KMnO_4\).

Conclusion:

Among the options, \(KClO_4\) (Potassium Perchlorate) is the compound that is isostructural with \(KMnO_4\). Both compounds have an orthorhombic crystal structure with tetrahedral anions (\(ClO_4^-\) in \(KClO_4\) and \(MnO_4^-\) in \(KMnO_4\)) and similarly arranged potassium ions in the lattice. Therefore, the correct answer is option 2: \(KClO_4\).