To determine which reaction is associated with the lattice energy of \(Li_2S\), we need to understand what lattice energy refers to. The lattice energy of a crystalline solid is the energy released when ions combine to form that crystalline lattice from gaseous ions.
Given this context, we need to identify the reaction that describes the formation of \(Li_2S\) from its constituent ions in the gaseous state. Let's analyze each option:
1. \(Li_2S (s) \rightarrow 2 Li^+ (g) + S^{2-} (g)\)
- This is the reverse of the process we need. Here, the solid \(Li_2S\) is decomposed into its gaseous ions, which would actually absorb energy rather than release it.
2. \(Li_2S (s) \rightarrow 2 Li^+ (aq) + S^{2-} (aq)\)
- This describes the dissolution of \(Li_2S\) into aqueous ions and not the formation of a solid lattice from gaseous ions.
3. \(2 Li^+ (g) + S^{2-} (g) \rightarrow Li_2S (s)\)
- This reaction correctly describes the formation of a solid \(Li_2S\) lattice from its gaseous ions, which is the direct definition of lattice energy. Therefore, this reflects the exothermic process that lattice energy is associated with.
4. \(2 Li (s) + \frac{1}{2} S_2 (g) \rightarrow Li_2S (s)\)
- This is a formation reaction from the elements in their standard states, not related directly to lattice energy.
5. \(2 Li^+ (aq) + S^{2-} (aq) \rightarrow Li_2S (s)\)
- This describes the formation of \(Li_2S\) from aqueous ions, not gaseous ions, and hence doesn't directly describe the lattice energy process.
From the above analysis, the correct reaction associated with the lattice energy of \(Li_2S\) is:
[tex]\[2 Li^+ (g) + S^{2-} (g) \rightarrow Li_2S (s)\][/tex]
Thus, the correct reaction is the third one. Therefore, the correct answer is 3.
