Chapter 18, Problem 58GQ

Wet limestone is used to scrub SO₂ gas from the exhaust gases of power plants. One possible reaction gives hydrated calcium sulfite:

CaCO₃(s) + SO₂(g) + ½ H₂O(ℓ) ⇄ CaSO₃ · ½ H₂O(s) + CO₂(g)

Another reaction gives hydrated calcium sulfate:

CaCO₃(s) + SO₂(g) + ½ H₂O(ℓ) + ½ O₂(g) ⇄ CaSO₄ · ½H₂O(s) + CO₂(g)

(a) Which reaction is more product-favored at equilibrium? Use the data in the table below and any other information needed in Appendix L to calculate Δ_rG° for each reaction at 25 °C.

(b) Calculate Δ_rG° for the reaction

CaSO₃ · ½ H₂O(s) + ½ O₂(g) ⇄ CaSO₄ · ½ H₂O(s)

Is this reaction product- or reactant-favored at equilibrium?

Interpretation Introduction

Interpretation:

Among the given reactions the reactions that is more favoured at product side should be determined.

Concept introduction:

The Gibbs free energy or the free energy change is a thermodynamic quantity represented by ${\text{Δ}}_{\text{r}}{\text{G}}^{\text{o}}$. It is related to entropy and entropy by the following expression,

  ${\text{ΔG}}^{\text{o}}{\text{= ΔH}}^{\text{o}}{\text{- TΔS}}^{\text{o}}$

Here, ${\text{ΔH}}^{\text{o}}$ is the change in enthalpy and ${\text{ΔS}}^{\text{o}}$ is the change in entropy.

Answer to Problem 58GQ

The reaction which is more product-favored at equilibrium is reaction $2$ because it has greater negative free energy change.

Explanation of Solution

The value of ${\text{ΔG}}^{\text{o}}$ for the given reaction is calculated below.

Given:

The Appendix L referred for the values of standard entropies and enthalpies.

For reaction $1$,

$\begin{array}{cccccccccc}& {\text{CaCO}}_{\text{3}}\left(\text{s}\right)& \text{+}& {\text{SO}}_{\text{2}}\left(\text{g}\right)& \text{+}& \frac{\text{1}}{\text{2}}{\text{H}}_{\text{2}}\text{O}\left(\text{l}\right)& \rightleftarrows & \begin{array}{l}{\text{ CaSO}}_{\text{3}}\text{×}\\ \left(\frac{\text{1}}{\text{2}}{\text{H}}_{\text{2}}\text{O}\left(\text{s}\right)\right)\end{array}& \text{+}& {\text{CO}}_{\text{2}}\left(\text{g}\right)\\ \begin{array}{l}{\text{Δ}}_{\text{f}}{\text{H}}^{\text{°}}\\ \left(\text{kJ/mol}\right)\end{array}& \text{-1207}\text{.6}& & \text{-296}\text{.84}& & \text{-285}\text{.83}& & \text{-1311}\text{.7}& & \text{-393}\text{.50}\\ \begin{array}{l}{\text{S}}^{\text{o}}\\ \left(\text{J/K×mol}\right)\end{array}& \text{91}\text{.7}& & \text{248}\text{.21}& & \text{69}\text{.95}& & \text{121}\text{.3}& & \text{213}\text{.74}\end{array}$

The standard enthalpy change is expressed as,

$\begin{array}{c}{\text{Δ}}_{\text{r}}{\text{H}}^{\text{°}}=\sum {\text{nΔ}}_{\text{f}}{\text{H}}^{\text{°}}\left(\text{products}\right)-\sum {\text{nΔ}}_{\text{f}}{\text{H}}^{\text{°}}\left(\text{reactants}\right)\\ \text{=}\left[\begin{array}{l}\left[\begin{array}{l}\left({\text{1 mol CaSO}}_{\text{3}}\text{×}\frac{\text{1}}{\text{2}}{\text{H}}_{\text{2}}\text{O}\left(\text{s}\right)\text{/mol-rxn}\right){\text{Δ}}_{\text{f}}{\text{H}}^{\text{°}}\left[{\text{CaSO}}_{\text{3}}\text{×}\frac{\text{1}}{\text{2}}{\text{H}}_{\text{2}}\text{O}\left(\text{s}\right)\right]\text{+}\\ \left({\text{1 mol CO}}_{\text{2}}\left(\text{g}\right)\text{/mol-rxn}\right){\text{Δ}}_{\text{f}}{\text{H}}^{\text{°}}\left[{\text{CO}}_{\text{2}}\left(\text{g}\right)\right]\end{array}\right]\text{-}\\ \left[\begin{array}{l}\left({\text{1 mol CaCO}}_{\text{3}}\left(\text{s}\right)\text{/mol-rxn}\right){\text{Δ}}_{\text{f}}{\text{H}}^{\text{°}}\left[{\text{CaCO}}_{\text{3}}\left(\text{s}\right)\right]\text{+}\\ \left(\frac{\text{1}}{\text{2}}{\text{mol H}}_{\text{2}}\text{O}\left(\text{l}\right)\text{/mol-rxn}\right){\text{Δ}}_{\text{f}}{\text{H}}^{\text{°}}\left[{\text{H}}_{\text{2}}\text{O}\left(\text{l}\right)\right]\text{+}\\ \left({\text{1 mol SO}}_{\text{2}}\left(\text{g}\right)\text{/mol-rxn}\right){\text{Δ}}_{\text{f}}{\text{H}}^{\text{°}}\left[{\text{SO}}_{\text{2}}\left(\text{g}\right)\right]\end{array}\right]\end{array}\right]\end{array}$

Substituting the respective values

$\begin{array}{c}{\text{Δ}}_{\text{r}}{\text{H}}^{\text{°}}\text{=}\left[\begin{array}{l}\left[\begin{array}{l}\left({\text{1 mol CaSO}}_{\text{3}}\text{×}\frac{\text{1}}{\text{2}}{\text{H}}_{\text{2}}\text{O}\left(\text{s}\right)\text{/mol-rxn}\right)\left(\text{-1311}\text{.7 kJ/mol}\right)\text{+}\\ \left({\text{1 mol CO}}_{\text{2}}\left(\text{g}\right)\text{/mol-rxn}\right)\left(\text{-393}\text{.50 kJ/mol}\right)\end{array}\right]\text{-}\\ \left[\begin{array}{l}\left({\text{1 mol CaCO}}_{\text{3}}\left(\text{s}\right)\text{/mol-rxn}\right)\left(\text{-1207}\text{.6 kJ/mol}\right)\text{+}\\ \left(\text{0}{\text{.5 mol H}}_{\text{2}}\text{O}\left(\text{l}\right)\text{/mol-rxn}\right)\left(\text{-285}\text{.83 kJ/mol}\right)\text{+}\\ \left({\text{1 mol SO}}_{\text{2}}\left(\text{g}\right)\text{/mol-rxn}\right)\left(\text{-296}\text{.84 kJ/mol}\right)\end{array}\right]\end{array}\right]\\ \text{=-57}\text{.845 kJ/mol-rxn }\end{array}$

Also,

$\begin{array}{c}{\text{Δ}}_{\text{r}}{\text{S}}^{\text{°}}\text{= }\sum {\text{nS}}^{\text{°}}\left(\text{products}\right)\text{-}\sum {\text{nS}}^{\text{°}}\left(\text{reactants}\right)\\ \text{=}\left[\begin{array}{l}\left[\begin{array}{l}\left({\text{1 mol CaSO}}_{\text{3}}\text{×}\frac{\text{1}}{\text{2}}{\text{H}}_{\text{2}}\text{O}\left(\text{s}\right)\text{/mol-rxn}\right){\text{S}}^{\text{°}}\left[{\text{CaSO}}_{\text{3}}\text{×}\frac{\text{1}}{\text{2}}{\text{H}}_{\text{2}}\text{O}\left(\text{s}\right)\right]\text{+}\\ \left({\text{1 mol CO}}_{\text{2}}\left(\text{g}\right)\text{/mol-rxn}\right){\text{S}}^{\text{°}}\left[{\text{CO}}_{\text{2}}\left(\text{g}\right)\right]\end{array}\right]\text{-}\\ \left[\begin{array}{l}\left({\text{1 mol CaCO}}_{\text{3}}\left(\text{s}\right)\text{/mol-rxn}\right){\text{S}}^{\text{°}}\left[{\text{CaCO}}_{\text{3}}\left(\text{s}\right)\right]\text{+}\\ \left(\frac{\text{1}}{\text{2}}{\text{mol H}}_{\text{2}}\text{O}\left(\text{l}\right)\text{/mol-rxn}\right){\text{S}}^{\text{°}}\left[{\text{H}}_{\text{2}}\text{O}\left(\text{l}\right)\right]\text{+}\\ \left({\text{1 mol SO}}_{\text{2}}\left(\text{g}\right)\text{/mol-rxn}\right){\text{S}}^{\text{°}}\left[{\text{SO}}_{\text{2}}\left(\text{g}\right)\right]\end{array}\right]\end{array}\right]\end{array}$

Substituting the respective values

$\begin{array}{c}{\text{Δ}}_{\text{r}}{\text{S}}^{\text{°}}\text{=}\left[\begin{array}{l}\left[\begin{array}{l}\left({\text{1 mol CaSO}}_{\text{3}}\text{×}\frac{\text{1}}{\text{2}}{\text{H}}_{\text{2}}\text{O}\left(\text{s}\right)\text{/mol-rxn}\right)\left(\text{121}\text{.3 J/K×mol}\right)\text{+}\\ \left({\text{1 mol CO}}_{\text{2}}\left(\text{g}\right)\text{/mol-rxn}\right)\left(\text{213}\text{.74 J/K×mol}\right)\end{array}\right]\text{-}\\ \left[\begin{array}{l}\left({\text{1 mol CaCO}}_{\text{3}}\left(\text{s}\right)\text{/mol-rxn}\right)\left(\text{91}\text{.7 J/K×mol}\right)\text{+}\\ \left(\frac{\text{1}}{\text{2}}{\text{mol H}}_{\text{2}}\text{O}\left(\text{l}\right)\text{/mol-rxn}\right)\left(\text{69}\text{.95 J/K×mol}\right)\text{+}\\ \left({\text{1 mol SO}}_{\text{2}}\left(\text{g}\right)\text{/mol-rxn}\right)\left(\text{248}\text{.21 J/K×mol}\right)\end{array}\right]\end{array}\right]\\ \text{=-39}\text{.84 J/K×mol-rxn }\end{array}$

Now, ${\text{ΔG}}^{\text{o}}{\text{= ΔH}}^{\text{o}}{\text{- TΔS}}^{\text{o}}$

Substituting the value of ${\text{ΔH}}^{\text{o}}$ and ${\text{ΔS}}^{\text{o}}$.

$\begin{array}{c}{\text{ΔG}}^{\text{o}}\text{=-57}\text{.845 kJ/mol-rxn-}\left[\left(\text{298K}\right)\left(\text{-39}\text{.84 J/K×mol- rxn}\right)\right]\left(\frac{\text{1 kJ}}{\text{1000 J}}\right)\\ \text{=-45}\text{.97 kJ/mol-rxn}\end{array}$

For reaction $2$,

$\begin{array}{cccccccccccc}& \begin{array}{l}{\text{CaCO}}_{\text{3}}\\ \left(\text{s}\right)\end{array}& \text{+}& \begin{array}{l}{\text{SO}}_{\text{2}}\\ \left(\text{g}\right)\end{array}& \text{+}& \begin{array}{l}\frac{\text{1}}{\text{2}}{\text{H}}_{\text{2}}\text{O}\\ \left(\text{l}\right)\end{array}& \text{+}& \begin{array}{l}\frac{\text{1}}{\text{2}}{\text{O}}_{\text{2}}\\ \left(\text{g}\right)\end{array}& \rightleftarrows & \begin{array}{l}{\text{CaSO}}_{\text{4}}\text{×}\\ \frac{\text{1}}{\text{2}}{\text{H}}_{\text{2}}\text{O}\\ \left(\text{s}\right)\end{array}& \text{+}& \begin{array}{l}{\text{CO}}_{\text{2}}\\ \left(\text{g}\right)\end{array}\\ \begin{array}{l}{\text{Δ}}_{\text{f}}{\text{H}}^{\text{°}}\\ \left(\text{kJ/mol}\right)\end{array}& \text{-1207}\text{.6}& & \text{-296}\text{.8}& & \text{-285}\text{.8}& & \text{0}& & \text{-1574}\text{.6}& & \text{-393}\text{.5}\\ \begin{array}{l}{\text{S}}^{\text{o}}\\ \left(\text{J/K×mol}\right)\end{array}& \text{91}\text{.7}& & \text{248}\text{.2}& & \text{69}\text{.9}& & \text{205}\text{.0}& & \text{134}\text{.8}& & \text{213}\text{.7}\end{array}$

The standard enthalpy change is expressed as,

$\begin{array}{c}{\text{Δ}}_{\text{r}}{\text{H}}^{\text{°}}\text{=}\sum {\text{nΔ}}_{\text{f}}{\text{H}}^{\text{°}}\left(\text{products}\right)\text{- }\sum {\text{nΔ}}_{\text{f}}{\text{H}}^{\text{°}}\left(\text{reactants}\right)\\ \text{=}\left[\begin{array}{l}\left[\begin{array}{l}\left({\text{1 mol CaSO}}_{\text{4}}\text{×}\frac{\text{1}}{\text{2}}{\text{H}}_{\text{2}}\text{O}\left(\text{s}\right)\text{/mol-rxn}\right){\text{Δ}}_{\text{f}}{\text{H}}^{\text{°}}\left[{\text{CaSO}}_{\text{4}}\text{×}\frac{\text{1}}{\text{2}}{\text{H}}_{\text{2}}\text{O}\left(\text{s}\right)\right]\text{+}\\ \left({\text{1 mol CO}}_{\text{2}}\left(\text{g}\right)\text{/mol-rxn}\right){\text{Δ}}_{\text{f}}{\text{H}}^{\text{°}}\left[{\text{CO}}_{\text{2}}\left(\text{g}\right)\right]\end{array}\right]\text{-}\\ \left[\begin{array}{l}\left({\text{1 mol CaCO}}_{\text{3}}\left(\text{s}\right)\text{/mol-rxn}\right){\text{Δ}}_{\text{f}}{\text{H}}^{\text{°}}\left[{\text{CaCO}}_{\text{3}}\left(\text{s}\right)\right]\text{+}\\ \left(\frac{\text{1}}{\text{2}}{\text{mol H}}_{\text{2}}\text{O}\left(\text{l}\right)\text{/mol-rxn}\right){\text{Δ}}_{\text{f}}{\text{H}}^{\text{°}}\left[{\text{H}}_{\text{2}}\text{O}\left(\text{l}\right)\right]\text{+}\\ \left({\text{1 mol SO}}_{\text{2}}\left(\text{g}\right)\text{/mol-rxn}\right){\text{Δ}}_{\text{f}}{\text{H}}^{\text{°}}\left[{\text{SO}}_{\text{2}}\left(\text{g}\right)\right]\text{+}\\ \left(\text{0}{\text{.5 mol O}}_{\text{2}}\left(\text{g}\right)\text{/mol-rxn}\right){\text{Δ}}_{\text{f}}{\text{H}}^{\text{°}}\left[{\text{O}}_{\text{2}}\left(\text{g}\right)\right]\end{array}\right]\end{array}\right]\end{array}$

Substituting the respective values

$\begin{array}{c}{\text{Δ}}_{\text{r}}{\text{H}}^{\text{°}}\text{=}\left[\begin{array}{l}\left[\begin{array}{l}\left({\text{1 mol CaSO}}_{\text{3}}\text{×}\frac{\text{1}}{\text{2}}{\text{H}}_{\text{2}}\text{O}\left(\text{s}\right)\text{/mol-rxn}\right)\left(\text{-1574}\text{.6 kJ/mol}\right)\text{+}\\ \left({\text{1 mol CO}}_{\text{2}}\left(\text{g}\right)\text{/mol-rxn}\right)\left(\text{-393}\text{.50 kJ/mol}\right)\end{array}\right]\text{-}\\ \left[\begin{array}{l}\left({\text{1 mol CaCO}}_{\text{3}}\left(\text{s}\right)\text{/mol-rxn}\right)\left(\text{-1207}\text{.6 kJ/mol}\right)\text{+}\\ \left(\text{0}{\text{.5 mol H}}_{\text{2}}\text{O}\left(\text{l}\right)\text{/mol-rxn}\right)\left(\text{-285}\text{.83 kJ/mol}\right)\text{+}\\ \left({\text{1 mol SO}}_{\text{2}}\left(\text{g}\right)\text{/mol-rxn}\right)\left(\text{-296}\text{.84 kJ/mol}\right)\text{+}\\ \left(\text{0}{\text{.5 mol O}}_{\text{2}}\left(\text{g}\right)\text{/mol-rxn}\right)\left(\text{0 kJ/mol}\right)\end{array}\right]\end{array}\right]\\ \text{=-320}\text{.75 kJ/mol-rxn }\end{array}$

Also,

$\begin{array}{c}{\text{Δ}}_{\text{r}}{\text{S}}^{\text{°}}\text{= }\sum {\text{nS}}^{\text{°}}\left(\text{products}\right)\text{-}\sum {\text{nS}}^{\text{°}}\left(\text{reactants}\right)\\ \text{=}\left[\begin{array}{l}\left[\begin{array}{l}\left({\text{1 mol CaSO}}_{\text{4}}\text{×}\frac{\text{1}}{\text{2}}{\text{H}}_{\text{2}}\text{O}\left(\text{s}\right)\text{/mol-rxn}\right){\text{S}}^{\text{°}}\left[{\text{CaSO}}_{\text{4}}\text{×}\frac{\text{1}}{\text{2}}{\text{H}}_{\text{2}}\text{O}\left(\text{s}\right)\right]\text{+}\\ \left({\text{1 mol CO}}_{\text{2}}\left(\text{g}\right)\text{/mol-rxn}\right){\text{S}}^{\text{°}}\left[{\text{CO}}_{\text{2}}\left(\text{g}\right)\right]\end{array}\right]\text{-}\\ \left[\begin{array}{l}\left({\text{1 mol CaCO}}_{\text{3}}\left(\text{s}\right)\text{/mol-rxn}\right){\text{S}}^{\text{°}}\left[{\text{CaCO}}_{\text{3}}\left(\text{s}\right)\right]\text{+}\\ \left(\frac{\text{1}}{\text{2}}{\text{mol H}}_{\text{2}}\text{O}\left(\text{l}\right)\text{/mol-rxn}\right){\text{S}}^{\text{°}}\left[{\text{H}}_{\text{2}}\text{O}\left(\text{l}\right)\right]\text{+}\\ \left({\text{1 mol SO}}_{\text{2}}\left(\text{g}\right)\text{/mol-rxn}\right){\text{S}}^{\text{°}}\left[{\text{SO}}_{\text{2}}\left(\text{g}\right)\right]\text{+}\\ \left(\text{0}{\text{.5 mol O}}_{\text{2}}\left(\text{g}\right)\text{/mol-rxn}\right){\text{S}}^{\text{°}}\left[{\text{O}}_{\text{2}}\left(\text{g}\right)\right]\end{array}\right]\end{array}\right]\end{array}$

Substituting the respective values

$\begin{array}{c}{\text{Δ}}_{\text{r}}{\text{S}}^{\text{°}}\text{=}\left[\begin{array}{l}\left[\begin{array}{l}\left({\text{1 mol CaSO}}_{\text{3}}\text{×}\frac{\text{1}}{\text{2}}{\text{H}}_{\text{2}}\text{O}\left(\text{s}\right)\text{/mol-rxn}\right)\left(\text{134}\text{.8 J/K×mol}\right)\text{+}\\ \left({\text{1 mol CO}}_{\text{2}}\left(\text{g}\right)\text{/mol-rxn}\right)\left(\text{213}\text{.7 J/K×mol}\right)\end{array}\right]\text{-}\\ \left[\begin{array}{l}\left({\text{1 mol CaCO}}_{\text{3}}\left(\text{s}\right)\text{/mol-rxn}\right)\left(\text{91}\text{.7 J/K×mol}\right)\text{+}\\ \left(\text{0}{\text{.5 mol H}}_{\text{2}}\text{O}\left(\text{l}\right)\text{/mol-rxn}\right)\left(\text{69}\text{.9 J/K×mol}\right)\text{+}\\ \left({\text{1 mol SO}}_{\text{2}}\left(\text{g}\right)\text{/mol-rxn}\right)\left(\text{ 248}\text{.2 J/K×mol}\right)\text{+}\\ \left(\text{0}{\text{.5 mol O}}_{\text{2}}\left(\text{g}\right)\text{/mol-rxn}\right)\left(\text{205}\text{.08 J/K×mol}\right)\end{array}\right]\end{array}\right]\\ \text{=-219}\text{.61 kJ/mol-rxn }\end{array}$

Now, ${\text{ΔG}}^{\text{o}}{\text{= ΔH}}^{\text{o}}{\text{- TΔS}}^{\text{o}}$

Substituting the value of ${\text{ΔH}}^{\text{o}}$ and ${\text{ΔS}}^{\text{o}}$.

$\begin{array}{c}{\text{ΔG}}^{\text{o}}\text{= -320}\text{.75 kJ/mol-rxn-}\left[\left(\text{298K}\right)\left(\text{-219}\text{.61 J/K×mol- rxn}\right)\right]\left(\frac{\text{1 kJ}}{\text{1000 J}}\right)\\ \text{= -255}\text{.30 kJ/mol-rxn}\end{array}$

Interpretation Introduction

Interpretation:

The ${\text{Δ}}_{\text{r}}{\text{G}}^{\text{o}}$ value for the given reaction should be calculated and identified that the reaction is product or reactant favoured.

Concept introduction:

The Gibbs free energy or the free energy change is a thermodynamic quantity represented by ${\text{Δ}}_{\text{r}}{\text{G}}^{\text{o}}$. It is related to entropy and entropy by the following expression,

  ${\text{ΔG}}^{\text{o}}{\text{= ΔH}}^{\text{o}}{\text{- TΔS}}^{\text{o}}$

Here, ${\text{ΔH}}^{\text{o}}$ is the change in enthalpy and ${\text{ΔS}}^{\text{o}}$ is the change in entropy.

Answer to Problem 58GQ

The ${\text{Δ}}_{\text{r}}{G}^{\text{°}}$ for the given reaction is $-236.41kJ\text{/}mol\text{-}rxn$.

Thus, the reaction is product-favored at equilibrium.

Explanation of Solution

The value of $\Delta {\text{G}}^{\text{o}}$ is calculated below.

Given:

The Appendix L referred for the values of standard entropies and enthalpies.

$\begin{array}{cccccc}& {\text{CaSO}}_{\text{3}}\text{×}\frac{\text{1}}{\text{2}}{\text{H}}_{\text{2}}\text{O}\left(\text{s}\right)& \text{+}& \frac{\text{1}}{\text{2}}{\text{O}}_{\text{2}}\left(\text{g}\right)& \to & {\text{CaSO}}_{\text{4}}\text{×}\frac{\text{1}}{\text{2}}{\text{H}}_{\text{2}}\text{O}\left(\text{s}\right)\\ {\text{Δ}}_{\text{f}}{\text{H}}^{\text{°}}\left(\text{kJ/mol}\right)& \text{-1311}\text{.7}& & \text{0}& & \text{-1574}\text{.65}\\ {\text{S}}^{\text{o}}\left(\text{J/K×mol}\right)& \text{121}\text{.3}& & \text{205}\text{.07}& & \text{134}\text{.8}\end{array}$

The standard enthalpy change is expressed as,

$\begin{array}{c}{\text{Δ}}_{\text{r}}{\text{H}}^{\text{°}}=\sum {\text{nΔ}}_{\text{f}}{\text{H}}^{\text{°}}\left(\text{products}\right)-\sum {\text{nΔ}}_{\text{f}}{\text{H}}^{\text{°}}\left(\text{reactants}\right)\\ \text{=}\left[\begin{array}{l}\left({\text{1 mol CaSO}}_{\text{4}}\text{×}\frac{\text{1}}{\text{2}}{\text{H}}_{\text{2}}\text{O}\left(\text{s}\right)\text{/mol-rxn}\right){\text{Δ}}_{\text{f}}{\text{H}}^{\text{°}}\left[{\text{CaSO}}_{\text{4}}\text{×}\frac{\text{1}}{\text{2}}{\text{H}}_{\text{2}}\text{O}\left(\text{s}\right)\right]\text{-}\\ \left[\begin{array}{l}\left({\text{1 mol CaSO}}_{\text{3}}\text{×}\frac{\text{1}}{\text{2}}{\text{H}}_{\text{2}}\text{O}\left(\text{s}\right)\text{/mol-rxn}\right){\text{Δ}}_{\text{f}}{\text{H}}^{\text{°}}\left[{\text{CaSO}}_{\text{3}}\text{×}\frac{\text{1}}{\text{2}}{\text{H}}_{\text{2}}\text{O}\left(\text{s}\right)\right]\text{+}\\ \left(\text{0}{\text{.5 mol O}}_{\text{2}}\left(\text{g}\right)\text{/mol-rxn}\right){\text{Δ}}_{\text{f}}{\text{H}}^{\text{°}}\left[{\text{O}}_{\text{2}}\left(\text{g}\right)\right]\end{array}\right]\end{array}\right]\end{array}$

Substituting the respective values

$\begin{array}{c}{\text{Δ}}_{\text{r}}{\text{H}}^{\text{°}}\text{=}\left[\begin{array}{l}\left({\text{1 mol CaSO}}_{\text{4}}\text{×}\frac{\text{1}}{\text{2}}{\text{H}}_{\text{2}}\text{O}\left(\text{s}\right)\text{/mol-rxn}\right)\left(\text{-1574}\text{.65 kJ/mol}\right)\text{-}\\ \left[\begin{array}{l}\left({\text{1 mol CaSO}}_{\text{3}}\text{×}\frac{\text{1}}{\text{2}}{\text{H}}_{\text{2}}\text{O}\left(\text{s}\right)\text{/mol-rxn}\right)\left(\text{-1311}\text{.7 kJ/mol}\right)\text{+}\\ \left(\text{0}{\text{.5 mol O}}_{\text{2}}\left(\text{g}\right)\text{/mol-rxn}\right)\left(\text{0 kJ/mol}\right)\end{array}\right]\end{array}\right]\\ \text{=-262}\text{.95 kJ/mol-rxn}\end{array}$

Also,

$\begin{array}{c}{\text{Δ}}_{\text{r}}{\text{S}}^{\text{°}}=\sum {\text{nS}}^{\text{°}}\left(\text{products}\right)-\sum {\text{nS}}^{\text{°}}\left(\text{reactants}\right)\\ \text{=}\left[\begin{array}{l}\left({\text{1 mol CaSO}}_{\text{4}}\text{×}\frac{\text{1}}{\text{2}}{\text{H}}_{\text{2}}\text{O}\left(\text{s}\right)\text{/mol-rxn}\right){\text{S}}^{\text{°}}\left[{\text{CaSO}}_{\text{4}}\text{×}\frac{\text{1}}{\text{2}}{\text{H}}_{\text{2}}\text{O}\left(\text{s}\right)\right]\text{-}\\ \left[\begin{array}{l}\left({\text{1 mol CaSO}}_{\text{3}}\text{×}\frac{\text{1}}{\text{2}}{\text{H}}_{\text{2}}\text{O}\left(\text{s}\right)\text{/mol-rxn}\right){\text{S}}^{\text{°}}\left[{\text{CaSO}}_{\text{3}}\text{×}\frac{\text{1}}{\text{2}}{\text{H}}_{\text{2}}\text{O}\left(\text{s}\right)\right]\text{+}\\ \left(\text{0}{\text{.5 mol O}}_{\text{2}}\left(\text{g}\right)\text{/mol-rxn}\right){\text{S}}^{\text{°}}\left[{\text{O}}_{\text{2}}\left(\text{g}\right)\right]\end{array}\right]\end{array}\right]\end{array}$

Substituting the values,

$\begin{array}{c}{\text{Δ}}_{\text{r}}{\text{S}}^{\text{°}}\text{=}\left[\begin{array}{l}\left({\text{1 mol CaSO}}_{\text{4}}\text{×}\frac{\text{1}}{\text{2}}{\text{H}}_{\text{2}}\text{O}\left(\text{s}\right)\text{/mol-rxn}\right)\left(\text{134}\text{.8 J/K×mol}\right)\text{-}\\ \left[\begin{array}{l}\left({\text{1 mol CaSO}}_{\text{3}}\text{×}\frac{\text{1}}{\text{2}}{\text{H}}_{\text{2}}\text{O}\left(\text{s}\right)\text{/mol-rxn}\right)\left(\text{121}\text{.3 J/K×mol}\right)\text{+}\\ \left(\text{0}{\text{.5 mol O}}_{\text{2}}\left(\text{g}\right)\text{/mol-rxn}\right)\left(\text{205}\text{.07 J/K×mol}\right)\end{array}\right]\end{array}\right]\\ \text{=-89}\text{.03 J/K×mol-rxn}\end{array}$

Now, ${\text{ΔG}}^{\text{o}}{\text{= ΔH}}^{\text{o}}{\text{- TΔS}}^{\text{o}}$

Substitute the value of ${\text{ΔH}}^{\text{o}}$ and ${\text{ΔS}}^{\text{o}}$.

$\begin{array}{c}{\text{ΔG}}^{\text{o}}\text{= -262}\text{.95 kJ/mol-rxn-}\left[\left(\text{298K}\right)\left(\text{-89}\text{.03 J/K×mol- rxn}\right)\right]\left(\frac{\text{1 kJ}}{\text{1000 J}}\right)\\ \text{= -236}\text{.41 kJ/mol-rxn}\end{array}$