To determine which of the given ions does not liberate hydrogen gas when reacting with dilute acids, we need to consider the properties of these ions in relation to the reactivity series. The reactivity series of metals is a list that ranks metals by their reactivity, particularly with acids.
Reactivity with Dilute Acids: Metals or metal ions higher in the reactivity series than hydrogen have the ability to displace hydrogen ions from acids, thus liberating hydrogen gas. Conversely, metals lower in the reactivity series than hydrogen will not liberate hydrogen gas when reacting with dilute acids.
The question provides us with the ions ${Ti^{2+}}$, ${V^{2+}}$, ${Cr^{2+}}$, and ${Mn^{2+}}$. Let's analyze each:
Conclusion: The ion ${Mn^{2+}}$ does not liberate hydrogen when reacted with dilute acids, as manganese in the +2 oxidation state does not have sufficient reactivity compared to hydrogen. Hence, the correct answer is ${Mn^{2+}}$.