What is [tex]\(\Delta S^{\circ}\)[/tex] for the following reaction?

[tex]\[ 4 \text{Cr} (s) + 3 \text{O}_2 (g) \rightarrow 2 \text{Cr}_2 \text{O}_3 (s) \][/tex]

[tex]\[
\begin{array}{lll}
\text{Substance} & \text{S}^{\circ} \, (\text{J} / \text{K} \cdot \text{mol}) \\
\text{Cr} (s) & 23.77 \\
\text{O}_2 (g) & 205.138 \\
\text{Cr}_2 \text{O}_3 (s) & 80.65 \\
\end{array}
\][/tex]

Select one:
A. [tex]\(+148.26 \, \text{J} / \text{K} \cdot \text{mol}\)[/tex]
B. [tex]\(-148.26 \, \text{J} / \text{K} \cdot \text{mol}\)[/tex]
C. [tex]\(+66.22 \, \text{J} / \text{K} \cdot \text{mol}\)[/tex]
D. [tex]\(+871.80 \, \text{J} / \text{K} \cdot \text{mol}\)[/tex]
E. [tex]\(-549.19 \, \text{J} / \text{K} \cdot \text{mol}\)[/tex]



Answer :

To determine the standard entropy change, [tex]\(\Delta S^{\circ}\)[/tex], for the reaction:
[tex]\[ 4 \, \text{Cr (s)} + 3 \, \text{O}_2 \, \text{(g)} \rightarrow 2 \, \text{Cr}_2 \text{O}_3 \, \text{(s)} \][/tex]

we need to follow these steps:

1. List the standard molar entropies [tex]\(S^{\circ}\)[/tex] of the reactants and products:
[tex]\[ \begin{align*} S^{\circ} (\text{Cr (s)}) &= 23.77 \, \text{J/K·mol} \\ S^{\circ} (\text{O}_2 \, \text{(g)}) &= 205.138 \, \text{J/K·mol} \\ S^{\circ} (\text{Cr}_2 \text{O}_3 \, \text{(s)}) &= 80.65 \, \text{J/K·mol} \end{align*} \][/tex]

2. Identify the coefficients in the balanced chemical equation:
[tex]\[ \begin{align*} n_{\text{Cr}} &= 4 \\ n_{\text{O}_2} &= 3 \\ n_{\text{Cr}_2 \text{O}_3} &= 2 \end{align*} \][/tex]

3. Calculate the total entropy of the products:
The only product is [tex]\( \text{Cr}_2 \text{O}_3 \)[/tex], and there are 2 moles of it.
[tex]\[ \sum S^{\circ}(\text{products}) = 2 \times S^{\circ}(\text{Cr}_2 \text{O}_3) = 2 \times 80.65 = 161.3 \, \text{J/K·mol} \][/tex]

4. Calculate the total entropy of the reactants:
There are 4 moles of [tex]\( \text{Cr (s)} \)[/tex] and 3 moles of [tex]\( \text{O}_2 \, \text{(g)} \)[/tex].
[tex]\[ \sum S^{\circ}(\text{reactants}) = 4 \times S^{\circ}(\text{Cr (s)}) + 3 \times S^{\circ}(\text{O}_2 \, \text{(g)}) = 4 \times 23.77 + 3 \times 205.138 = 95.08 + 615.414 = 710.494 \, \text{J/K·mol} \][/tex]

5. Calculate the standard entropy change, [tex]\(\Delta S^{\circ}\)[/tex]:
[tex]\[ \Delta S^{\circ} = \sum S^{\circ} (\text{products}) - \sum S^{\circ} (\text{reactants}) \][/tex]
[tex]\[ \Delta S^{\circ} = 161.3 \, \text{J/K·mol} - 710.494 \, \text{J/K·mol} = -549.194 \, \text{J/K·mol} \][/tex]

Therefore, the standard entropy change for the reaction is [tex]\(-549.19 \, \text{J/K·mol}\)[/tex], which corresponds to option E:

E. [tex]\(-549.19 \, \text{J/K·mol}\)[/tex]