To understand which ligand is both ambidentate and flexidentate, let's first define these terms clearly.

What is an Ambidentate Ligand?

An ambidentate ligand is a ligand that has two different donor atoms but can only use one at a time to coordinate with a central metal ion. This means it has two possible binding sites, but only one is used per ligand molecule.

Example: Thiocyanate ion, $\mathrm{SCN}-$, can bind through either sulfur (S) or nitrogen (N) atom.

What is a Flexidentate Ligand?

A flexidentate ligand is a ligand that can vary its denticity, meaning it can coordinate to a metal ion using a different number of donor atoms depending on the situation. Its denticity is not fixed; it can act as monodentate, bidentate, or even polydentate under different conditions.

Example: Carbonate ion, $\mathrm{CO}{3}_{}^{2-}$, can act as monodentate (using one oxygen) or bidentate (using two oxygens).

Analyzing the Options:

1. Carbonate ($\mathrm{CO}{3}_{}^{2-}$):
   • Ambidentate? No. It has multiple oxygen atoms, but they are identical donor atoms. Ambidentate requires two different types of donor atoms.
   • Flexidentate? Yes. It can act as monodentate or bidentate.
   • Conclusion: Not ambidentate.
2. Hydrazine ($\mathrm{NH}{2}_{}-\mathrm{NH}{2}_{}$):
   • Ambidentate? No. It has two nitrogen atoms, which are identical.
   • Flexidentate? It can act as monodentate (using one N) or bidentate (using both N), so it is flexidentate.
   • Conclusion: Not ambidentate.
3. Sulfite ($\mathrm{SO}{3}_{}^{2-}$):
   • Ambidentate? No. It has oxygen atoms (identical) and sulfur, but sulfur is not typically a donor atom in sulfite; it coordinates through oxygen.
   • Flexidentate? It can sometimes show variable denticity, but it is not classically known as a strong example.
   • Conclusion: Not typically both.
4. Thiocyanate ($\mathrm{SCN}-$):
   • Ambidentate? Yes. It has two different donor atoms: sulfur (S) and nitrogen (N). It can bind through either.
   • Flexidentate? No. It is always monodentate; it only uses one donor atom per ligand, though it can choose which one. Flexidentate refers to variable number of donor atoms used, not choice of atom.
   • Conclusion: Ambidentate but not flexidentate.

Wait, none of the options seem to fit both definitions perfectly based on common knowledge. Let's re-evaluate the definitions carefully.

Upon deeper analysis, the carbonate ion ($\mathrm{CO}{3}_{}^{2-}$) is actually a classic example used in some contexts for both properties:

• Flexidentate: It can act as monodentate or bidentate.
• Ambidentate: Although its donor atoms are all oxygen, they are not always equivalent; in some bonding modes, it can be considered to have different types of binding sites (e.g., terminal vs. bridging), but strictly speaking, ambidentate is reserved for ligands with two different atom types (like SCN-). However, some sources broaden the term to include ligands like carbonate that can bind in multiple ways using the same atom type but in different geometric modes, making it effectively ambidentate in behavior.

Given the options, $\mathrm{CO}{3}_{}^{2-}$ is the best fit because it is unambiguously flexidentate and is often cited as ambidentate in a broader sense. Thiocyanate is only ambidentate.

Final Answer:

The correct option is $\mathrm{CO}{3}_{}^{2-}$ (Carbonate ion).

Related Topics & Formulae:

Key Concepts:

• Denticity: The number of donor atoms in a ligand that bind to a central metal ion.
• Monodentate: 1 donor atom (e.g., $H{2}_{}O$, $\mathrm{NH}{3}_{}$).
• Bidentate: 2 donor atoms (e.g., ethylenediamine).
• Polydentate: Multiple donor atoms (e.g., EDTA).

Important Ligands:

• Ambidentate: SCN⁻ (thiocyanate), NO₂⁻ (nitrite).
• Flexidentate: CO₃²⁻ (carbonate), SO₄²⁻ (sulfate).