To choose the thermochemical equation that illustrates the formation enthalpy ([tex]\( \Delta H_f \)[/tex]) for [tex]\( \text{Li}_2\text{SO}_4 \)[/tex], we need to identify the equation that represents the formation of 1 mole of [tex]\( \text{Li}_2\text{SO}_4 \)[/tex] from its elements in their standard states.
Let's examine each option:
1. [tex]\( 2 \text{Li}^+(aq) + \text{SO}_4^{2-}(aq) \rightarrow \text{Li}_2\text{SO}_4(aq) \)[/tex]
This equation represents the formation of [tex]\( \text{Li}_2\text{SO}_4 \)[/tex] in the aqueous phase from ions, not from elements in their standard states.
2. [tex]\( 16 \text{Li}(s) + \text{S}_8(s,\text{rhombic}) + 16 \text{O}_2(g) \rightarrow 8 \text{Li}_2\text{SO}_4(s) \)[/tex]
This equation involves the formation of 8 moles of [tex]\( \text{Li}_2\text{SO}_4 \)[/tex] from elements in their standard states, but we need the formation of exactly 1 mole. We'll need to divide everything by 8 for it to properly show the formation of 1 mole, so this is close but not a perfect match.
3. [tex]\( 2 \text{Li}(s) + \frac{1}{8} \text{S}_8(s,\text{rhombic}) + 2 \text{O}_2(g) \rightarrow \text{Li}_2\text{SO}_4(s) \)[/tex]
This accurately shows 1 mole of [tex]\( \text{Li}_2\text{SO}_4 \)[/tex] being formed from its elements in their standard states (solid lithium, solid sulfur, and gaseous oxygen). This satisfies the requirement for [tex]\( \Delta H_f \)[/tex].
4. [tex]\( \text{Li}_2\text{SO}_4(aq) \rightarrow 2 \text{Li}^+(aq) + \text{SO}_4^{2-}(aq) \)[/tex]
This equation represents the dissociation, not the formation, of [tex]\( \text{Li}_2\text{SO}_4 \)[/tex].
5. [tex]\( 8 \text{Li}_2\text{SO}_4(s) \rightarrow 16 \text{Li}(s) + \text{S}_8(s,\text{rhombic}) + 16 \text{O}_2(g) \)[/tex]
This equation is the reverse of the formation reaction and deals with the breakdown of [tex]\( \text{Li}_2\text{SO}_4 \)[/tex], not its formation.
Hence, the correct thermochemical equation that illustrates the formation enthalpy ([tex]\( \Delta H_f \)[/tex]) for [tex]\( \text{Li}_2\text{SO}_4 \)[/tex] is:
[tex]\[ 2 \text{Li}(s) + \frac{1}{8} \text{S}_8(s,\text{rhombic}) + 2 \text{O}_2(g) \rightarrow \text{Li}_2\text{SO}_4(s) \][/tex]
And thus, the correct answer is:
[tex]\[ Option\ 3 \][/tex]
So, the correct choice is:
[tex]\[ 3 \][/tex]
