Ligand Substitution in Cr(III) Complexes

The question asks about the rate of hydration of a complex ion with the general formula $\text{[Cr}^\text{III}\text{(H}_2\text{O)}_5\text{X]}^\text{n+}$. Hydration is a type of ligand substitution reaction where the ligand 'X' is replaced by a water molecule. $\text{Cr}^\text{III}$ complexes are known to be kinetically inert, meaning their ligand substitution reactions are generally slow.

Acidic Medium Effect on Reaction Rate

The problem states that the reaction takes place in neutral aqueous medium and is compared to what happens in acidic medium at room temperature. Acidic medium contains a significant concentration of $\text{H}^+$ ions. These $\text{H}^+$ ions can interact with the ligands present in the complex or in the solution.

The rate of a reaction can be affected by the conditions, such as the presence of acid or base, especially if the reactants or intermediates can react with acid or base. In this case, $\text{H}^+$ ions from the acidic medium can potentially interact with the ligand 'X' or other parts of the complex.

Ligand Analysis and Protonation

Let's consider the nature of the ligand 'X' in each option:

• $\text{N}_3^-$ (Azide): This is an anionic ligand. It is the conjugate base of hydrazoic acid ($\text{HN}_3$), which is a weak acid (pKa $\approx$ 4.7). This means $\text{N}_3^-$ is a weak base and can be protonated by $\text{H}^+$ ions in an acidic medium ($\text{N}_3^- + \text{H}^+ \rightleftharpoons \text{HN}_3$).
• $\text{F}^-$ (Fluoride): This is an anionic ligand. It is the conjugate base of hydrofluoric acid ($\text{HF}$), which is a weak acid (pKa $\approx$ 3.17). This means $\text{F}^-$ is a weak base and can be protonated by $\text{H}^+$ ions in an acidic medium ($\text{F}^- + \text{H}^+ \rightleftharpoons \text{HF}$).
• $\text{CN}^-$ (Cyanide): This is an anionic ligand. It is the conjugate base of hydrogen cyanide ($\text{HCN}$), which is a weak acid (pKa $\approx$ 9.21). This means $\text{CN}^-$ is a weak base and can be protonated by $\text{H}^+$ ions in an acidic medium ($\text{CN}^- + \text{H}^+ \rightleftharpoons \text{HCN}$).
• $\text{NH}_3$ (Ammonia): This is a neutral ligand. While free ammonia is a base (can be protonated to $\text{NH}_4^+$, pKa of $\text{NH}_4^+$ $\approx$ 9.25), when coordinated to a metal ion, its basicity is reduced. Crucially, it is a neutral molecule.

Effect of Acidic Medium on Ligand X

The rate of ligand substitution in $\text{Cr}^\text{III}$ complexes is sensitive to the nature of the leaving group (X) and the incoming group ($\text{H}_2\text{O}$). If the leaving group 'X' is an anionic ligand that can be protonated in acidic medium, this protonation can potentially affect the bond strength between $\text{Cr}^\text{III}$ and 'X', or change the reaction pathway. For example, protonation of a coordinated anionic ligand might make it a better leaving group, thus increasing the reaction rate. This means the rate would be *affected* by the acidic medium.

However, if 'X' is a neutral ligand like $\text{NH}_3$, its interaction with $\text{H}^+$ ions in acidic medium is different. Coordinated $\text{NH}_3$ is less likely to undergo protonation in a way that significantly alters the rate-determining step of the hydration reaction compared to the effects seen with protonation of anionic ligands. The departure of a neutral ligand like $\text{NH}_3$ is less likely to be facilitated or inhibited by the presence of $\text{H}^+$ ions compared to an anionic ligand whose charge and bonding character can be altered by protonation.

Conclusion for Unaffected Rate

The question states that the rate of hydration remains unaffected in acidic medium. This suggests that the ligand 'X' is not significantly influenced by the $\text{H}^+$ ions present in the acidic medium in a way that changes the kinetics of the substitution reaction. Among the given options, the neutral ligand $\text{NH}_3}$ is the least likely to have its leaving group properties or the reaction mechanism significantly altered by the presence of acid, compared to the anionic ligands which are basic and susceptible to protonation. Therefore, when X is $\text{NH}_3}$, the rate of hydration is likely to remain unaffected by the change from neutral to acidic medium.

Thus, the ligand 'X' for which the rate of hydration remains unaffected in acidic medium is $\text{NH}_3}$.