On the basic of the following `Delta_(r)G^(Theta)` values at `1073K`:
`S_(1)(s) +2O_(2)(g) rarr 2SO_(2)(g) Delta_(r)G^(Theta) =- 544 kJ mol^(-1)`
`2Zn(s) +O_(2)(g) rarr 2ZnO(s) Delta_(r)G^(Theta) =- 480 kJ mol^(-1)`
`2Zn(s) +S_(2)(s) rarr 2ZnS(s) Delta_(r)G^(Theta) =- 293 KJ mol^(-1)`
Show that roasting of zinc sulphide to zinc oxide is a spontaneous process.

Options

A. `-357 KJ`
B. `-731 KJ`
C. `-773 KJ`
D. `-229 KJ`

Explanation

To show that the roasting of zinc sulfide (ZnS) to zinc oxide (ZnO) is a spontaneous process, we will use the provided Gibbs free energy change (ΔG) values for the relevant reactions. A spontaneous process occurs when the change in Gibbs free energy (ΔG) is negative.

### Step-by-Step Solution:

1. **Write the reactions and their ΔG values:**
- Reaction 1: \( S_2(s) + 2O_2(g) \rightarrow 2SO_2(g) \)
\( \Delta_rG^\Theta = -544 \, \text{kJ/mol} \)
- Reaction 2: \( 2Zn(s) + O_2(g) \rightarrow 2ZnO(s) \)
\( \Delta_rG^\Theta = -480 \, \text{kJ/mol} \)
- Reaction 3: \( 2Zn(s) + S_2(s) \rightarrow 2ZnS(s) \)
\( \Delta_rG^\Theta = -293 \, \text{kJ/mol} \)

2. **Reverse Reaction 3:**
To find the ΔG for the conversion of ZnS to Zn and S2, we reverse Reaction 3:
\[ 2ZnS(s) \rightarrow 2Zn(s) + S_2(s) \Delta_rG^\Theta = +293 \, \text{kJ/mol} \]

3. **Combine the reversed Reaction 3 with Reaction 2:**
Now, we add the reversed Reaction 3 and Reaction 2:
\[ 2ZnS(s) + O_2(g) \rightarrow 2ZnO(s) + S_2(s) \]
The ΔG for this combined reaction is:
\[ \Delta G = \Delta G_{(reversed \, Reaction \, 3)} + \Delta G_{(Reaction \, 2)} \]
\[ \Delta G = +293 \, \text{kJ/mol} + (-480 \, \text{kJ/mol}) = -187 \, \text{kJ/mol} \]

4. **Add Reaction 1 to the combined reaction:**
Next, we add Reaction 1 to the combined reaction:
\[ S_2(s) + 2O_2(g) \rightarrow 2SO_2(g) \Delta_rG^\Theta = -544 \, \text{kJ/mol} \]
The overall reaction becomes:
\[ 2ZnS(s) + 2O_2(g) \rightarrow 2ZnO(s) + 2SO_2(g) \]
The total ΔG for this overall reaction is:
\[ \Delta G_{total} = -187 \, \text{kJ/mol} + (-544 \, \text{kJ/mol}) = -731 \, \text{kJ/mol} \]

5. **Conclusion:**
Since the overall ΔG for the roasting of zinc sulfide to zinc oxide is \(-731 \, \text{kJ/mol}\), which is negative, we conclude that the process is spontaneous.