Chapter 17, Problem 17.31QP

(a)

Interpretation Introduction

Interpretation: The $\Delta S^{\text{ο}}$ value for each of the given atmospheric reaction that contributes to the formation of photochemical smog 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 atmospheric reaction that contributes to the formation of photochemical smog.

Answer to Problem 17.31QP

Solution

The $\Delta S^{\text{ο}}$ value for the given atmospheric reaction is $\underset{\_}{24.9J{K}^{-1}}$.

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

The standard molar entropy of $\text{NO}\left(g\right)$ is $210.7{\text{Jmol}}^{-1}{\text{K}}^{-1}$.

The given reaction is,

${\text{N}}_2\left(g\right)+{\text{O}}_2\left(g\right)\to 2\text{NO}\left(g\right)$ (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 $\text{NO}\left(g\right)$ is $2\text{mol}$.
• The stoichiometry coefficient of reactant ${\text{N}}_2\left(g\right)$ 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,

${\displaystyle \sum n_{\text{products}}\times }{S}^{\text{ο}}\left(\text{products}\right)=2\text{mol}\times S_{\text{NO}\left(g\right)}^{\text{ο}}$ (3)

Where,

• $S_{\text{NO}\left(g\right)}^{\text{ο}}$ is the standard molar entropy of ${\text{N}}_2\left(g\right)$.

Substitute the value of $S_{\text{NO}\left(g\right)}^{\text{ο}}$ in equation (3).

$\begin{array}{c}{\displaystyle \sum n_{\text{products}}\times }{S}^{\text{ο}}\left(\text{products}\right)=2\text{mol}\times 210.7{\text{Jmol}}^{-1}{\text{K}}^{-1}\\ =421.4{\text{JK}}^{-1}\end{array}$

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

${\displaystyle \sum m_{\text{reactants}}\times }{S}^{\text{ο}}\left(\text{reactants}\right)=\left(1\text{mol}\times S_{{\text{N}}_2\left(g\right)}^{\text{ο}}\right)+\left(1\text{mol}\times S_{{\text{O}}_{\text{2}}\left(g\right)}^{\text{ο}}\right)$ (4)

Where,

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

Substitute the values of $S_{{\text{N}}_2\left(g\right)}^{\text{ο}}$ and $S_{{\text{O}}_{\text{2}}\left(g\right)}^{\text{ο}}$ in equation (4).

$\begin{array}{c}{\displaystyle \sum m_{\text{reactants}}\times }{S}^{\text{ο}}\left(\text{reactants}\right)=\left(1\text{mol}\times 191.5{\text{Jmol}}^{-1}{\text{K}}^{-1}\right)+\left(1\text{mol}\times 205.0{\text{Jmol}}^{-1}{\text{K}}^{-1}\right)\\ =191.5{\text{JK}}^{-1}+205.0{\text{JK}}^{-1}\\ =396.5{\text{JK}}^{-1}\end{array}$

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).

$\begin{array}{c}\Delta S^{\text{ο}}=421.4{\text{JK}}^{-1}-396.5{\text{JK}}^{-1}\\ =24.9{\text{JK}}^{-1}\end{array}$

Thus, the $\Delta S^{\text{ο}}$ value for the given atmospheric reaction is $\underset{\_}{24.9J{K}^{-1}}$.

(b)

Interpretation Introduction

To determine: The $\Delta S^{\text{ο}}$ value for the given atmospheric reaction that contributes to the formation of photochemical smog.

Answer to Problem 17.31QP

Solution

The $\Delta S^{\text{ο}}$ value for the given atmospheric reaction is $\underset{\_}{-146.4J{K}^{-1}}$.

Explanation of Solution

Explanation

Refer Appendix $4$.

The standard molar entropy of ${\text{NO}}_2\left(g\right)$ is $240.0{\text{Jmol}}^{-1}{\text{K}}^{-1}$.

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 $\text{NO}\left(g\right)$ is $210.7{\text{Jmol}}^{-1}{\text{K}}^{-1}$.

The given reaction is,

$2\text{NO}\left(g\right)+{\text{O}}_2\left(g\right)\to 2{\text{NO}}_2\left(g\right)$ (5)

In equation (5),

• The stoichiometry coefficient of reactant $\text{NO}\left(g\right)$ is $2\text{mol}$.
• The stoichiometry coefficient of reactant ${\text{O}}_2\left(g\right)$ is $1\text{mol}$.
• The stoichiometry coefficient of product ${\text{NO}}_2\left(g\right)$ is $2\text{mol}$.

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

${\displaystyle \sum n_{\text{products}}\times }{S}^{\text{ο}}\left(\text{products}\right)=2\text{mol}\times S_{{\text{NO}}_2\left(g\right)}^{\text{ο}}$ (6)

Where,

• $S_{{\text{NO}}_2\left(g\right)}^{\text{ο}}$ is the standard molar entropy of ${\text{NO}}_2\left(g\right)$.

Substitute the value of $S_{{\text{NO}}_2\left(g\right)}^{\text{ο}}$ in equation (6).

$\begin{array}{c}{\displaystyle \sum n_{\text{products}}\times }{S}^{\text{ο}}\left(\text{products}\right)=2\text{mol}\times 240.0{\text{Jmol}}^{-1}{\text{K}}^{-1}\\ =480.0{\text{JK}}^{-1}\end{array}$

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

${\displaystyle \sum m_{\text{reactants}}\times }{S}^{\text{ο}}\left(\text{reactants}\right)=\left(2\text{mol}\times S_{\text{NO}\left(g\right)}^{\text{ο}}\right)+\left(1\text{mol}\times S_{{\text{O}}_2\left(g\right)}^{\text{ο}}\right)$ (7)

Where,

• $S_{\text{NO}\left(g\right)}^{\text{ο}}$ is the standard molar entropy of $\text{NO}\left(g\right)$.
• $S_{{\text{O}}_2\left(g\right)}^{\text{ο}}$  is the standard molar entropy of ${\text{O}}_2\left(g\right)$.

Substitute the values of $S_{\text{NO}\left(g\right)}^{\text{ο}}$ and $S_{{\text{O}}_2\left(g\right)}^{\text{ο}}$ in equation (7).

$\begin{array}{c}{\displaystyle \sum m_{\text{reactants}}\times }{S}^{\text{ο}}\left(\text{reactants}\right)=\left(2\text{mol}\times 210.7{\text{Jmol}}^{-1}{\text{K}}^{-1}\right)+\left(1\text{mol}\times 205.0{\text{Jmol}}^{-1}{\text{K}}^{-1}\right)\\ =421.4{\text{JK}}^{-1}+205.0{\text{JK}}^{-1}\\ =626.4{\text{JK}}^{-1}\end{array}$

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).

$\begin{array}{c}\Delta S^{\text{ο}}=480.0{\text{JK}}^{-1}-626.4{\text{JK}}^{-1}\\ =-146.4{\text{JK}}^{-1}\end{array}$

Thus, the $\Delta S^{\text{ο}}$ value for the given atmospheric reaction is $\underset{\_}{-146.4J{K}^{-1}}$.

(c)

Interpretation Introduction

To determine: The $\Delta S^{\text{ο}}$ value for the given atmospheric reaction that contributes to the formation of photochemical smog.

Answer to Problem 17.31QP

Solution

The $\Delta S^{\text{ο}}$ value for the given atmospheric reaction is $\underset{\_}{-73.2J{K}^{-1}}$.

Explanation of Solution

Explanation

Refer Appendix $4$.

The standard molar entropy of ${\text{NO}}_2\left(g\right)$ is $240.0{\text{Jmol}}^{-1}{\text{K}}^{-1}$.

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 $\text{NO}\left(g\right)$ is $210.7{\text{Jmol}}^{-1}{\text{K}}^{-1}$.

The given reaction is,

$\text{NO}\left(g\right)+\frac{1}{2}{\text{O}}_2\left(g\right)\to {\text{NO}}_2\left(g\right)$ (8)

In equation (8),

• The stoichiometry coefficient of reactant $\text{NO}\left(g\right)$ is $1\text{mol}$.
• The stoichiometry coefficient of reactant ${\text{O}}_2\left(g\right)$ is $\frac{1}{2}\text{mol}$.
• The stoichiometry coefficient of product ${\text{NO}}_2\left(g\right)$ is $1\text{mol}$.

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

${\displaystyle \sum n_{\text{products}}\times }{S}^{\text{ο}}\left(\text{products}\right)=1\text{mol}\times S_{{\text{NO}}_2\left(g\right)}^{\text{ο}}$ (9)

Where,

• $S_{{\text{NO}}_2\left(g\right)}^{\text{ο}}$ is the standard molar entropy of ${\text{NO}}_2\left(g\right)$.

Substitute the value of $S_{{\text{NO}}_2\left(g\right)}^{\text{ο}}$ in equation (9).

$\begin{array}{c}{\displaystyle \sum n_{\text{products}}\times }{S}^{\text{ο}}\left(\text{products}\right)=1\text{mol}\times 240.0{\text{Jmol}}^{-1}{\text{K}}^{-1}\\ =240.0{\text{JK}}^{-1}\end{array}$

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

${\displaystyle \sum m_{\text{reactants}}\times }{S}^{\text{ο}}\left(\text{reactants}\right)=\left(1\text{mol}\times S_{\text{NO}\left(g\right)}^{\text{ο}}\right)+\left(\frac{1}{2}\text{mol}\times S_{{\text{O}}_2\left(g\right)}^{\text{ο}}\right)$ (10)

Where,

• $S_{\text{NO}\left(g\right)}^{\text{ο}}$ is the standard molar entropy of $\text{NO}\left(g\right)$.
• $S_{{\text{O}}_2\left(g\right)}^{\text{ο}}$ is the standard molar entropy of ${\text{O}}_2\left(g\right)$

Substitute the values of $S_{\text{NO}\left(g\right)}^{\text{ο}}$ and $S_{{\text{O}}_2\left(g\right)}^{\text{ο}}$ in equation (10).

$\begin{array}{c}{\displaystyle \sum m_{\text{reactants}}\times }{S}^{\text{ο}}\left(\text{reactants}\right)=\left(1\text{mol}\times 210.7{\text{Jmol}}^{-1}{\text{K}}^{-1}\right)+\left(\frac{1}{2}\text{mol}\times 205.0{\text{Jmol}}^{-1}{\text{K}}^{-1}\right)\\ =210.7{\text{JK}}^{-1}+102.5{\text{JK}}^{-1}\\ =313.2{\text{JK}}^{-1}\end{array}$

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).

$\begin{array}{c}\Delta S^{\text{ο}}=240.0{\text{JK}}^{-1}-313.2{\text{JK}}^{-1}\\ =-73.2{\text{JK}}^{-1}\end{array}$

Thus, the $\Delta S^{\text{ο}}$ value for the given atmospheric reaction is $\underset{\_}{-73.2J{K}^{-1}}$.

(d)

Interpretation Introduction

To determine: The $\Delta S^{\text{ο}}$ value for the given atmospheric reaction that contributes to the formation of photochemical smog.

Answer to Problem 17.31QP

Solution

The $\Delta S^{\text{ο}}$ value for the given atmospheric reaction is $\underset{\_}{-175.8J{K}^{-1}}$.

Explanation of Solution

Explanation

Refer Appendix $4$.

The standard molar entropy of ${\text{NO}}_2\left(g\right)$ is $240.0{\text{Jmol}}^{-1}{\text{K}}^{-1}$.

The standard molar entropy of ${\text{N}}_2{\text{O}}_4\left(g\right)$ is $304.2{\text{Jmol}}^{-1}{\text{K}}^{-1}$.

The given reaction is,

${\text{2NO}}_2\left(g\right)\to {\text{N}}_2{\text{O}}_4\left(g\right)$ (11)

In equation (11),

• The stoichiometry coefficient of reactant ${\text{NO}}_2\left(g\right)$ is $2\text{mol}$.
• The stoichiometry coefficient of product ${\text{N}}_2{\text{O}}_4\left(g\right)$ is $1\text{mol}$.

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

${\displaystyle \sum n_{\text{products}}\times }{S}^{\text{ο}}\left(\text{products}\right)=1\text{mol}\times S_{{\text{N}}_2{\text{O}}_4\left(g\right)}^{\text{ο}}$ (12)

Where,

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

Substitute the value of $S_{{\text{N}}_2{\text{O}}_4\left(g\right)}^{\text{ο}}$ in equation (12).

$\begin{array}{c}{\displaystyle \sum n_{\text{products}}\times }{S}^{\text{ο}}\left(\text{products}\right)=1\text{mol}\times 304.2{\text{Jmol}}^{-1}{\text{K}}^{-1}\\ =304.2{\text{JK}}^{-1}\end{array}$

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

${\displaystyle \sum m_{\text{reactants}}\times }{S}^{\text{ο}}\left(\text{reactants}\right)=\left(2\text{mol}\times S_{{\text{NO}}_2\left(g\right)}^{\text{ο}}\right)$ (13)

Where,

• $S_{{\text{NO}}_2\left(g\right)}^{\text{ο}}$ is the standard molar entropy of ${\text{NO}}_2\left(g\right)$.

Substitute the value of $S_{{\text{NO}}_2\left(g\right)}^{\text{ο}}$ in equation (13).

$\begin{array}{c}{\displaystyle \sum m_{\text{reactants}}\times }{S}^{\text{ο}}\left(\text{reactants}\right)=2\text{mol}\times 240.0{\text{Jmol}}^{-1}{\text{K}}^{-1}\\ =480.0{\text{JK}}^{-1}\end{array}$

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).

$\begin{array}{c}\Delta S^{\text{ο}}=304.2{\text{JK}}^{-1}-480.0{\text{JK}}^{-1}\\ =-175.8{\text{JK}}^{-1}\end{array}$

Thus, the $\Delta S^{\text{ο}}$ value for the given atmospheric reaction is $\underset{\_}{-175.8J{K}^{-1}}$.

Conclusion

1. a. The $\Delta S^{\text{ο}}$ value for the given atmospheric reaction is $\underset{\_}{24.9J{K}^{-1}}$.
2. b. The $\Delta S^{\text{ο}}$ value for the given atmospheric reaction is $\underset{\_}{-146.4J{K}^{-1}}$.
3. c. The $\Delta S^{\text{ο}}$ value for the given atmospheric reaction is $\underset{\_}{-73.2J{K}^{-1}}$.
4. d. The $\Delta S^{\text{ο}}$ value for the given atmospheric reaction is $\underset{\_}{-175.8J{K}^{-1}}$