Chapter 18, Problem 18.32QP

(a)

Interpretation Introduction

Interpretation: The $\Delta S^{\text{ο}}$ value for each of the given reaction is to be calculated.

Concept introduction: The standard entropy change $\left(\Delta S^{\text{ο}}\right)$ is calculated by the formula,

$\Delta S^{\text{ο}}={\displaystyle \sum n_{\text{products}}\times }{S}^{\text{ο}}\left(\text{products}\right)-{\displaystyle \sum m_{\text{reactants}}\times }{S}^{\text{ο}}\left(\text{reactants}\right)$

To determine: The $\Delta S^{\text{ο}}$ value for the given reaction.

Answer to Problem 18.32QP

Solution

The $\Delta S^{\text{ο}}$ value for the given reaction is $\underset{\_}{16.61J{K}^{-1}}$.

Explanation of Solution

Explanation

Refer Appendix $4$.

The standard molar entropy of ${\text{N}}_2\left(g\right)$ is $191.5{\text{Jmol}}^{-1}{\text{K}}^{-1}$.

The standard molar entropy of ${\text{CH}}_4\left(g\right)$ is $186.2{\text{Jmol}}^{-1}{\text{K}}^{-1}$.

The standard molar entropy of $\text{HCN}\left(g\right)$ is $201.81{\text{Jmol}}^{-1}{\text{K}}^{-1}$.

The standard molar entropy of ${\text{NH}}_3\left(g\right)$ is $192.5{\text{Jmol}}^{-1}{\text{K}}^{-1}$.

The given reaction is,

${\text{CH}}_4\left(g\right)+{\text{N}}_2\left(g\right)\to \text{HCN}\left(g\right)+{\text{NH}}_3\left(g\right)$ (1)

The standard entropy change $\left(\Delta S^{\text{ο}}\right)$ is calculated by the formula,

$\Delta S^{\text{ο}}={\displaystyle \sum n_{\text{products}}\times }{S}^{\text{ο}}\left(\text{products}\right)-{\displaystyle \sum m_{\text{reactants}}\times }{S}^{\text{ο}}\left(\text{reactants}\right)$ (2)

Where,

• $n_{\text{products}}$ is the stoichiometry coefficient of products.
• $m_{\text{reactants}}$ is the stoichiometry coefficient of reactants.
• $S^{\text{ο}}\left(\text{products}\right)$ is the standard molar entropy of products.
• $S^{\text{ο}}\left(\text{reactants}\right)$ is the standard molar entropy of reactants.

In equation (1),

• The stoichiometry coefficient of product $\text{HCN}\left(g\right)$ is $1\text{mol}$.
• The stoichiometry coefficient of product ${\text{NH}}_3\left(g\right)$ is $1\text{mol}$.
• The stoichiometry coefficient of reactant ${\text{N}}_2\left(g\right)$ is $1\text{mol}$.
• The stoichiometry coefficient of reactant ${\text{CH}}_4\left(g\right)$ is $1\text{mol}$.

The ${\displaystyle \sum n_{\text{products}}\times }{S}^{\text{ο}}\left(\text{products}\right)$ is calculated by,

${\displaystyle \sum n_{\text{products}}\times }{S}^{\text{ο}}\left(\text{products}\right)=\left(1\text{mol}\times S_{\text{HCN}\left(g\right)}^{\text{ο}}\right)+\left(1\text{mol}\times S_{{\text{NH}}_3\left(g\right)}^{\text{ο}}\right)$ (3)

Where,

• $S_{\text{HCN}\left(g\right)}^{\text{ο}}$ is the standard molar entropy of $\text{HCN}\left(g\right)$.
• $S_{{\text{NH}}_3\left(g\right)}^{\text{ο}}$ is the standard molar entropy of ${\text{NH}}_3\left(g\right)$.

Substitute the value of $S_{\text{HCN}\left(g\right)}^{\text{ο}}$ and $S_{{\text{NH}}_3\left(g\right)}^{\text{ο}}$ in equation (3).

$\begin{array}{c}{\displaystyle \sum n_{\text{products}}\times }{S}^{\text{ο}}\left(\text{products}\right)=\left(1\text{mol}\times 201.81{\text{Jmol}}^{-1}{\text{K}}^{-1}\right)+\left(1\text{mol}\times 192.5{\text{Jmol}}^{-1}{\text{K}}^{-1}\right)\\ =201.81{\text{JK}}^{-1}+192.5{\text{JK}}^{-1}\\ =394.31{\text{JK}}^{-1}\end{array}$

The ${\displaystyle \sum m_{\text{reactants}}\times }{S}^{\text{ο}}\left(\text{reactants}\right)$ is calculated by,

${\displaystyle \sum m_{\text{reactants}}\times }{S}^{\text{ο}}\left(\text{reactants}\right)=\left(1\text{mol}\times S_{{\text{N}}_2\left(g\right)}^{\text{ο}}\right)+\left(1\text{mol}\times S_{{\text{CH}}_{\text{4}}\left(g\right)}^{\text{ο}}\right)$ (4)

Where,

• $S_{{\text{N}}_2\left(g\right)}^{\text{ο}}$ is the standard molar entropy of ${\text{N}}_2\left(g\right)$.
• $S_{{\text{CH}}_{\text{4}}\left(g\right)}^{\text{ο}}$ is the standard molar entropy of ${\text{CH}}_4\left(g\right)$.

Substitute the values of $S_{{\text{N}}_2\left(g\right)}^{\text{ο}}$ and $S_{{\text{CH}}_{\text{4}}\left(g\right)}^{\text{ο}}$ in equation (4).

$\begin{array}{c}{\displaystyle \sum m_{\text{reactants}}\times }{S}^{\text{ο}}\left(\text{reactants}\right)=\left(1\text{mol}\times 191.5{\text{Jmol}}^{-1}{\text{K}}^{-1}\right)+\left(1\text{mol}\times 186.0{\text{Jmol}}^{-1}{\text{K}}^{-1}\right)\\ =191.5{\text{JK}}^{-1}+186.2{\text{JK}}^{-1}\\ =377.7{\text{JK}}^{-1}\end{array}$

Substitute the values of ${\displaystyle \sum m_{\text{reactants}}\times }{S}^{\text{ο}}\left(\text{reactants}\right)$ and ${\displaystyle \sum n_{\text{products}}\times }{S}^{\text{ο}}\left(\text{products}\right)$ in equation (2).

$\begin{array}{c}\Delta S^{\text{ο}}=394.31{\text{JK}}^{-1}-377.7{\text{JK}}^{-1}\\ =16.61{\text{JK}}^{-1}\end{array}$

Thus, the $\Delta S^{\text{ο}}$ value for the given reaction is $\underset{\_}{16.61J{K}^{-1}}$.

(b)

Interpretation Introduction

To determine: The $\Delta S^{\text{ο}}$ value for the given reaction.

Answer to Problem 18.32QP

Solution

The $\Delta S^{\text{ο}}$ value for the given reaction is $\underset{\_}{-11.3J{K}^{-1}}$.

Explanation of Solution

Explanation

Refer Appendix $4$.

The standard molar entropy of ${\text{Cu}}_2\text{S}\left(s\right)$ is $120.9{\text{Jmol}}^{-1}{\text{K}}^{-1}$.

The standard molar entropy of ${\text{O}}_2\left(g\right)$ is $205.0{\text{Jmol}}^{-1}{\text{K}}^{-1}$.

The standard molar entropy of ${\text{SO}}_2\left(g\right)$ is $248.2{\text{Jmol}}^{-1}{\text{K}}^{-1}$.

The standard molar entropy of $\text{Cu}\left(s\right)$ is $33.2{\text{Jmol}}^{-1}{\text{K}}^{-1}$.

The given reaction is,

${\text{Cu}}_2\text{S}\left(s\right)+{\text{O}}_2\left(g\right)\to 2\text{Cu}\left(s\right)+{\text{SO}}_2\left(g\right)$ (5)

In equation (5),

• The stoichiometry coefficient of reactant ${\text{Cu}}_2\text{S}\left(s\right)$ is $1\text{mol}$.
• The stoichiometry coefficient of reactant ${\text{O}}_2\left(g\right)$ is $1\text{mol}$.
• The stoichiometry coefficient of product ${\text{SO}}_2\left(g\right)$ is $1\text{mol}$.
• The stoichiometry coefficient of product $\text{Cu}\left(s\right)$ is $2\text{mol}$.

The ${\displaystyle \sum n_{\text{products}}\times }{S}^{\text{ο}}\left(\text{products}\right)$ is calculated by,

${\displaystyle \sum n_{\text{products}}\times }{S}^{\text{ο}}\left(\text{products}\right)=\left(1\text{mol}\times S_{\text{Cu}\left(s\right)}^{\text{ο}}\right)+\left(1\text{mol}\times S_{{\text{SO}}_{\text{2}}\left(g\right)}^{\text{ο}}\right)$ (6)

Where,

• $S_{\text{Cu}\left(s\right)}^{\text{ο}}$ is the standard molar entropy of $\text{Cu}\left(s\right)$.
• $S_{{\text{SO}}_{\text{2}}\left(g\right)}^{\text{ο}}$ is the standard molar entropy of ${\text{SO}}_2\left(g\right)$.

Substitute the values of $S_{\text{Cu}\left(s\right)}^{\text{ο}}$ and $S_{{\text{SO}}_{\text{2}}\left(g\right)}^{\text{ο}}$ in equation (6).

$\begin{array}{c}{\displaystyle \sum n_{\text{products}}\times }{S}^{\text{ο}}\left(\text{products}\right)=\left(2\text{mol}\times 33.2{\text{Jmol}}^{-1}{\text{K}}^{-1}\right)+\left(1\text{mol}\times 248.2{\text{Jmol}}^{-1}{\text{K}}^{-1}\right)\\ =66.4{\text{JK}}^{-1}+248.2{\text{JK}}^{-1}\\ =314.6{\text{JK}}^{-1}\end{array}$

The ${\displaystyle \sum m_{\text{reactants}}\times }{S}^{\text{ο}}\left(\text{reactants}\right)$ is calculated by,

${\displaystyle \sum m_{\text{reactants}}\times }{S}^{\text{ο}}\left(\text{reactants}\right)=\left(1\text{mol}\times S_{{\text{Cu}}_2\text{S}\left(s\right)}^{\text{ο}}\right)+\left(1\text{mol}\times S_{{\text{O}}_2\left(g\right)}^{\text{ο}}\right)$ (7)

Where,

• $S_{{\text{Cu}}_2\text{S}\left(s\right)}^{\text{ο}}$ is the standard molar entropy of ${\text{Cu}}_2\text{S}\left(s\right)$.
• $S_{{\text{O}}_2\left(g\right)}^{\text{ο}}$ is the standard molar entropy of ${\text{O}}_2\left(g\right)$.

Substitute the values of $S_{{\text{Cu}}_2\text{S}\left(s\right)}^{\text{ο}}$ and $S_{{\text{O}}_2\left(g\right)}^{\text{ο}}$ in equation (7).

$\begin{array}{c}{\displaystyle \sum m_{\text{reactants}}\times }{S}^{\text{ο}}\left(\text{reactants}\right)=\left(1\text{mol}\times 120.9{\text{Jmol}}^{-1}{\text{K}}^{-1}\right)+\left(1\text{mol}\times 205.0{\text{Jmol}}^{-1}{\text{K}}^{-1}\right)\\ =120.9{\text{JK}}^{-1}+205.0{\text{JK}}^{-1}\\ =325.9{\text{JK}}^{-1}\end{array}$

Substitute the values of ${\displaystyle \sum m_{\text{reactants}}\times }{S}^{\text{ο}}\left(\text{reactants}\right)$ and ${\displaystyle \sum n_{\text{products}}\times }{S}^{\text{ο}}\left(\text{products}\right)$ in equation (2).

$\begin{array}{c}\Delta S^{\text{ο}}=314.6{\text{JK}}^{-1}-325.9{\text{JK}}^{-1}\\ =-11.3{\text{JK}}^{-1}\end{array}$

Thus, the $\Delta S^{\text{ο}}$ value for the given atmospheric reaction is $\underset{\_}{-11.3J{K}^{-1}}$.

(c)

Interpretation Introduction

To determine: The $\Delta S^{\text{ο}}$ value for the given reaction.

Answer to Problem 18.32QP

Solution

The $\Delta S^{\text{ο}}$ value for the given reaction is $\underset{\_}{-306.6J{K}^{-1}}$.

Explanation of Solution

Explanation

Refer Appendix $4$.

The standard molar entropy of ${\text{SO}}_3\left(g\right)$ is $256.8{\text{Jmol}}^{-1}{\text{K}}^{-1}$.

The standard molar entropy of ${\text{H}}_2\text{O}\left(l\right)$ is $69.9{\text{Jmol}}^{-1}{\text{K}}^{-1}$.

The standard molar entropy of ${\text{H}}_2{\text{SO}}_4\left(aq\right)$ is $20.1{\text{Jmol}}^{-1}{\text{K}}^{-1}$.

The given reaction is,

${\text{SO}}_3\left(g\right)+{\text{H}}_2\text{O}\left(l\right)\to {\text{H}}_2{\text{SO}}_4\left(aq\right)$ (8)

In equation (8),

• The stoichiometry coefficient of reactant ${\text{SO}}_3\left(g\right)$ is $1\text{mol}$.
• The stoichiometry coefficient of reactant ${\text{H}}_2\text{O}\left(l\right)$ is $1\text{mol}$.
• The stoichiometry coefficient of product ${\text{H}}_2{\text{SO}}_4\left(aq\right)$ is $1\text{mol}$.

The ${\displaystyle \sum n_{\text{products}}\times }{S}^{\text{ο}}\left(\text{products}\right)$ is calculated by,

${\displaystyle \sum n_{\text{products}}\times }{S}^{\text{ο}}\left(\text{products}\right)=1\text{mol}\times S_{{\text{H}}_2{\text{SO}}_4\left(aq\right)}^{\text{ο}}$ (9)

Where,

• $S_{{\text{H}}_2{\text{SO}}_4\left(aq\right)}^{\text{ο}}$ is the standard molar entropy of ${\text{H}}_2{\text{SO}}_4\left(aq\right)$.

Substitute the value of $S_{{\text{H}}_2{\text{SO}}_4\left(aq\right)}^{\text{ο}}$ in equation (9).

$\begin{array}{c}{\displaystyle \sum n_{\text{products}}\times }{S}^{\text{ο}}\left(\text{products}\right)=1\text{mol}\times 20.1{\text{Jmol}}^{-1}{\text{K}}^{-1}\\ =20.1{\text{JK}}^{-1}\end{array}$

The ${\displaystyle \sum m_{\text{reactants}}\times }{S}^{\text{ο}}\left(\text{reactants}\right)$ is calculated by,

${\displaystyle \sum m_{\text{reactants}}\times }{S}^{\text{ο}}\left(\text{reactants}\right)=\left(1\text{mol}\times S_{{\text{SO}}_3\left(g\right)}^{\text{ο}}\right)+\left(1\text{mol}\times S_{{\text{H}}_2\text{O}\left(l\right)}^{\text{ο}}\right)$ (10)

Where,

• $S_{{\text{SO}}_3\left(g\right)}^{\text{ο}}$ is the standard molar entropy of ${\text{SO}}_3\left(g\right)$.
• $S_{{\text{H}}_2\text{O}\left(l\right)}^{\text{ο}}$ is the standard molar entropy of ${\text{H}}_2\text{O}\left(l\right)$

Substitute the values of $S_{{\text{SO}}_3\left(g\right)}^{\text{ο}}$ and $S_{{\text{H}}_2\text{O}\left(l\right)}^{\text{ο}}$ in equation (10). $\begin{array}{c}{\displaystyle \sum m_{\text{reactants}}\times }{S}^{\text{ο}}\left(\text{reactants}\right)=\left(1\text{mol}\times 256.8{\text{Jmol}}^{-1}{\text{K}}^{-1}\right)+\left(1\text{mol}\times 69.9{\text{Jmol}}^{-1}{\text{K}}^{-1}\right)\\ =256.8{\text{JK}}^{-1}+69.9{\text{JK}}^{-1}\\ =326.7{\text{JK}}^{-1}\end{array}$

Substitute the values of ${\displaystyle \sum m_{\text{reactants}}\times }{S}^{\text{ο}}\left(\text{reactants}\right)$ and ${\displaystyle \sum n_{\text{products}}\times }{S}^{\text{ο}}\left(\text{products}\right)$ in equation (2).

$\begin{array}{c}\Delta S^{\text{ο}}=20.1{\text{JK}}^{-1}-326.7{\text{JK}}^{-1}\\ =-306.6{\text{JK}}^{-1}\end{array}$

Thus, the $\Delta S^{\text{ο}}$ value for the given atmospheric reaction is $\underset{\_}{-306.6J{K}^{-1}}$.

(d)

Interpretation Introduction

To determine: The $\Delta S^{\text{ο}}$ value for the given reaction.

Answer to Problem 18.32QP

Solution

The $\Delta S^{\text{ο}}$ value for the given reaction is $\underset{\_}{-124.6J{K}^{-1}}$.

Explanation of Solution

Explanation

Refer Appendix $4$.

The standard molar entropy of $\text{S}\left(g\right)$ is $167.8{\text{Jmol}}^{-1}{\text{K}}^{-1}$.

The standard molar entropy of ${\text{O}}_2\left(g\right)$ is $205.0{\text{Jmol}}^{-1}{\text{K}}^{-1}$.

The standard molar entropy of ${\text{SO}}_2\left(g\right)$ is $248.2{\text{Jmol}}^{-1}{\text{K}}^{-1}$.

The given reaction is,

$\text{S}\left(g\right)+{\text{O}}_2\left(g\right)\to {\text{SO}}_2\left(g\right)$ (11)

In equation (11),

• The stoichiometry coefficient of product ${\text{SO}}_2\left(g\right)$ is $1\text{mol}$.
• The stoichiometry coefficient of reactant $\text{S}\left(g\right)$ is $1\text{mol}$.
• The stoichiometry coefficient of reactant ${\text{O}}_2\left(g\right)$ is $1\text{mol}$.

The ${\displaystyle \sum n_{\text{products}}\times }{S}^{\text{ο}}\left(\text{products}\right)$ is calculated by,

${\displaystyle \sum n_{\text{products}}\times }{S}^{\text{ο}}\left(\text{products}\right)=1\text{mol}\times S_{{\text{SO}}_2\left(g\right)}^{\text{ο}}$ (12)

Where,

• $S_{{\text{SO}}_2\left(g\right)}^{\text{ο}}$ is the standard molar entropy of ${\text{SO}}_2\left(g\right)$.

Substitute the value of $S_{{\text{SO}}_2\left(g\right)}^{\text{ο}}$ in equation (12).

$\begin{array}{c}{\displaystyle \sum n_{\text{products}}\times }{S}^{\text{ο}}\left(\text{products}\right)=1\text{mol}\times 248.2{\text{Jmol}}^{-1}{\text{K}}^{-1}\\ =248.2{\text{JK}}^{-1}\end{array}$

The ${\displaystyle \sum m_{\text{reactants}}\times }{S}^{\text{ο}}\left(\text{reactants}\right)$ is calculated by,

${\displaystyle \sum m_{\text{reactants}}\times }{S}^{\text{ο}}\left(\text{reactants}\right)=\left(1\text{mol}\times S_{\text{S}\left(g\right)}^{\text{ο}}\right)+\left(1\text{mol}\times S_{{\text{O}}_2\left(g\right)}^{\text{ο}}\right)$ (13)

Where,

• $S_{\text{S}\left(g\right)}^{\text{ο}}$ is the standard molar entropy of $\text{S}\left(g\right)$.
• $S_{{\text{O}}_2\left(g\right)}^{\text{ο}}$ is the standard molar entropy of ${\text{O}}_2\left(g\right)$.

Substitute the values of $S_{\text{S}\left(g\right)}^{\text{ο}}$ and $S_{{\text{O}}_2\left(g\right)}^{\text{ο}}$ in equation (13).

$\begin{array}{c}{\displaystyle \sum m_{\text{reactants}}\times }{S}^{\text{ο}}\left(\text{reactants}\right)=\left(1\text{mol}\times 167.8{\text{Jmol}}^{-1}{\text{K}}^{-1}\right)+\left(1\text{mol}\times 205.0{\text{Jmol}}^{-1}{\text{K}}^{-1}\right)\\ =167.8{\text{JK}}^{-1}+205.0{\text{JK}}^{-1}\\ =372.8{\text{JK}}^{-1}\end{array}$

Substitute the values of ${\displaystyle \sum m_{\text{reactants}}\times }{S}^{\text{ο}}\left(\text{reactants}\right)$ and ${\displaystyle \sum n_{\text{products}}\times }{S}^{\text{ο}}\left(\text{products}\right)$ in equation (2).

$\begin{array}{c}\Delta S^{\text{ο}}=248.2{\text{JK}}^{-1}-372.8{\text{JK}}^{-1}\\ =-124.6{\text{JK}}^{-1}\end{array}$

Thus, the $\Delta S^{\text{ο}}$ value for the given atmospheric reaction is $\underset{\_}{-124.6J{K}^{-1}}$.

Conclusion

1. a. The $\Delta S^{\text{ο}}$ value for the given reaction is $\underset{\_}{16.61J{K}^{-1}}$.
2. b. The $\Delta S^{\text{ο}}$ value for the given reaction is $\underset{\_}{-11.3J{K}^{-1}}$.
3. c. The $\Delta S^{\text{ο}}$ value for the given reaction is $\underset{\_}{-306.6J{K}^{-1}}$.
4. d. The $\Delta S^{\text{ο}}$ value for the given reaction is $\underset{\_}{-124.6J{K}^{-1}}$