To determine which reaction is faster, let's examine the nature of the reactants and their states.
### Reaction Option a:
[tex]\[ \text{Ag}^+(aq) + \text{Cl}^-(aq) \rightarrow \text{AgCl}(s) \][/tex]
- This reaction takes place in an aqueous solution and involves ionic species, [tex]\(\text{Ag}^+\)[/tex] and [tex]\(\text{Cl}^-\)[/tex].
- In aqueous solutions, ionic reactants are dissociated and highly mobile. The ions are surrounded by water molecules, but strong electrostatic attractions between [tex]\(\text{Ag}^+\)[/tex] and [tex]\(\text{Cl}^-\)[/tex] cause them to react quickly to form the precipitate [tex]\(\text{AgCl}\)[/tex].
- Such ionic reactions in solution typically occur very rapidly because of these strong attractions and the high frequency of effective collisions.
### Reaction Option b:
[tex]\[ \text{H}_2(g) + \text{Cl}_2(g) \rightarrow 2 \text{HCl}(g) \][/tex]
- This reaction involves gaseous reactants, [tex]\(\text{H}_2\)[/tex] and [tex]\(\text{Cl}_2\)[/tex].
- Gas phase reactions can be slower compared to reactions in an aqueous solution because the reactant molecules must collide with sufficient energy and proper orientation to react.
- The frequency of effective collisions in gases is generally lower compared to the ionic interactions in solutions, leading to a slower reaction rate.
### Comparison and Conclusion:
- The reaction involving ions in an aqueous solution ([tex]\( \text{Ag}^+(aq) + \text{Cl}^-(aq) \rightarrow \text{AgCl}(s) \)[/tex]) is expected to be faster than the gas phase reaction ([tex]\(\text{H}_2(g) + \text{Cl}_2(g) \rightarrow 2 \text{HCl}(g)\)[/tex]).
- This is due to the rapid nature of ionic reactions in solution versus the slower, collision-dependent nature of gas phase reactions.
Therefore, the reaction we would expect to be faster is:
[tex]\[ \text{Ag}^+(aq) + \text{Cl}^-(aq) \rightarrow \text{AgCl}(s) \][/tex]
which corresponds to option (a).
