Chapter 18, Problem 18.89QP

(a)

Interpretation Introduction

Interpretation: The information regarding fuel cells having alkali metal carbonates as electrolytes that can make use of Methane as a fuel and the conversion of Methane into Hydrogen in a two step process is given. Various questions based on this concept have to be answered.

Concept introduction: The device that converts the chemical energy from fuel into electrical energy is called fuel cell. Here reaction occurs because of the oxidation of fuel.

To determine: The oxidation number of Carbon and Hydrogen in reactants and products.

Answer to Problem 18.89QP

Solution

The oxidation state of Carbon and Hydrogen in equation ${\text{CH}}_4\left(g\right)+{\text{H}}_2\text{O}\left(g\right)\to \text{CO}\left(g\right)+3{\text{H}}_2\left(g\right)$ is,

• The oxidation number of Hydrogen is $\underset{\_}{+1}$ and $\text{C}$ is $\underset{\_}{-4}$ in ${\text{CH}}_4$.
• The oxidation number of Hydrogen is $\underset{\_}{+1}$ in ${\text{H}}_2\text{O}$.
• The oxidation number of $\text{C}$ is $\underset{\_}{+2}$ in $\text{CO}$
• As there is no difference of electronegativity in ${\text{H}}_2$, the oxidation number of Hydrogen is zero.

The oxidation state of Carbon and Hydrogen in equation $\text{CO}\left(g\right)+{\text{H}}_2\text{O}\left(g\right)\to {\text{H}}_2\left(g\right)+{\text{CO}}_2\left(g\right)$ is,

• The oxidation number of $\text{C}$ is $\underset{\_}{+2}$ in $\text{CO}$.
• The oxidation number of Hydrogen is $\underset{\_}{+1}$ in ${\text{H}}_2\text{O}$.
• As there is no difference of electronegativity in ${\text{H}}_2$, the oxidation number of Hydrogen is zero.
• The oxidation number of $\text{C}$ is $\underset{\_}{+4}$ in ${\text{CO}}_{\text{2}}$.

Explanation of Solution

Explanation

The given reaction is,

${\text{CH}}_4\left(g\right)+{\text{H}}_2\text{O}\left(g\right)\to \text{CO}\left(g\right)+3{\text{H}}_2\left(g\right)$

The oxidation number of Carbon in ${\text{CH}}_4$ is,

$\begin{array}{c}x+4\left(+1\right)=0\\ x=0-4\\ x=\underset{\_}{-4}\end{array}$

The oxidation number of Hydrogen is $\underset{\_}{+1}$ and $\text{C}$ is $\underset{\_}{-4}$.

The oxidation number of Hydrogen in ${\text{H}}_2\text{O}$ is,

$\begin{array}{c}2x+1\left(-2\right)=0\\ 2x=0+2\\ 2x=2\\ x=\underset{\_}{1}\end{array}$

The oxidation number of Hydrogen is $\underset{\_}{+1}$.

The oxidation number of Carbon in $\text{CO}$ is,

$\begin{array}{r}x+1\left(-2\right)=0\\ x=0+2\\ x=\underset{\_}{2}\end{array}$

The oxidation number of $\text{C}$ is $\underset{\_}{+2}$.

As there is no difference of electronegativity in ${\text{H}}_2$, the oxidation number of Hydrogen is zero.

The given reaction is,

$\text{CO}\left(g\right)+{\text{H}}_2\text{O}\left(g\right)\to {\text{H}}_2\left(g\right)+{\text{CO}}_2\left(g\right)$

The oxidation number of Carbon in $\text{CO}$ is,

$\begin{array}{c}x+1\left(-2\right)=0\\ x=0+2\\ x=\underset{\_}{2}\end{array}$

The oxidation number of $\text{C}$ is $\underset{\_}{+2}$.

The oxidation number of Hydrogen in ${\text{H}}_2\text{O}$ is,

$\begin{array}{c}2x+1\left(-2\right)=0\\ 2x=0+2\\ 2x=2\\ x=\underset{\_}{1}\end{array}$

The oxidation number of Hydrogen is $\underset{\_}{+1}$.

As there is no difference of electronegativity in ${\text{H}}_2$, the oxidation number of Hydrogen is zero.

The oxidation number of Carbon in ${\text{CO}}_{\text{2}}$ is,

$\begin{array}{c}x+2\left(-2\right)=0\\ x=0+4\\ x=\underset{\_}{4}\end{array}$

The oxidation number of $\text{C}$ is $\underset{\_}{+4}$.

As there is no difference of electronegativity in ${\text{H}}_2$, the oxidation number of Hydrogen is zero.

(b)

Interpretation Introduction

To determine: The value of $\Delta G{°}_{\text{rxn}}$ for each of the given reactions and overall reaction.

Answer to Problem 18.89QP

Solution

The value of $\Delta G{°}_{\text{rxn}}$ for equation (1) is $\underset{\_}{142.2kJ}$.

The value of $\Delta G{°}_{\text{rxn}}$ for equation (2) is $\underset{\_}{-28.6kJ}$.

The value of $\Delta G{°}_{\text{rxn}}$ for overall reaction is $\underset{\_}{113.6kJ}$ $\underset{\_}{-28.6kJ}$.

Explanation of Solution

Explanation

The given reaction is,

${\text{CH}}_4\left(g\right)+{\text{H}}_2\text{O}\left(g\right)\to \text{CO}\left(g\right)+3{\text{H}}_2\left(g\right)$ (1)

The standard free energy change is given as,

$\Delta G{°}_{\text{rxn}}=\sum m\Delta G{°}_{\text{Products}}-\sum n\Delta G{°}_{\text{Reactants}}$

Where,

• $m$ is the coefficient of reactants.
• $n$ is the coefficients of products.
• $\Delta G{°}_{\text{Reactants}}$ is the standard free energy change of reactants.
• $\Delta G{°}_{\text{Products}}$ is the standard free energy change of products.

The heat of reaction for the above reaction is,

$\Delta G{°}_{\text{rxn}}=\left({\left(\Delta G_{\text{f}}°\right)}_{\text{CO}}+3\times {\left(\Delta G_{\text{f}}°\right)}_{{\text{H}}_2}\right)-\left(\left(\Delta {\left(G_{\text{f}}°\right)}_{{\text{CH}}_4}\right)+\left({\left(\Delta G_{\text{f}}°\right)}_{{\text{H}}_2\text{O}}\right)\right)$

The value of ${\left(\Delta G_{\text{f}}°\right)}_{\text{CO}}$ is $-137.2\text{kJ}/\text{mol}$, ${\left(\Delta G_{\text{f}}°\right)}_{{\text{H}}_2}$ is $0.0\text{kJ}/\text{mol}$, $\left(\Delta {\left(G_{\text{f}}°\right)}_{{\text{CH}}_4}\right)$ is $-50.8\text{kJ}/\text{mol}$ and ${\left(\Delta G_{\text{f}}°\right)}_{{\text{H}}_2\text{O}}$ is $-228.6\text{kJ}/\text{mol}$.

Substitute these values in above equation as,

$\begin{array}{c}\Delta G{°}_{{\text{rxn}}_1}=\left({\left(\Delta G_{\text{f}}°\right)}_{\text{CO}}+3\times {\left(\Delta G_{\text{f}}°\right)}_{{\text{H}}_2}\right)-\left(\left(\Delta {\left(G_{\text{f}}°\right)}_{{\text{CH}}_4}\right)+\left({\left(\Delta G_{\text{f}}°\right)}_{{\text{H}}_2\text{O}}\right)\right)\\ =\left(\begin{array}{c}\left(1\text{mol}\left(-137.2\text{kJ}/\text{mol}\right)+3\text{mol}\times 0\text{kJ}/\text{mol}\right)-\\ \left(1\text{mol}\left(-50.8\text{kJ}/\text{mol}\right)+1\text{mol}\left(-228.6\text{kJ}/\text{mol}\right)\right)\end{array}\right)\\ =-137.2\text{kJ}+\text{279}\text{.4}\text{kJ}\\ =\underset{\_}{142.2kJ}\end{array}$

The given reaction is,

$\text{CO}\left(g\right)+{\text{H}}_2\text{O}\left(g\right)\to {\text{H}}_2\left(g\right)+{\text{CO}}_2\left(g\right)$ (2)

The heat of reaction for the above reaction is,

$\Delta G{°}_{\text{rxn}}=\left({\left(\Delta G_{\text{f}}°\right)}_{\text{CO}}+{\left(\Delta G_{\text{f}}°\right)}_{{\text{H}}_2}\right)-\left(\left(\Delta {\left(G_{\text{f}}°\right)}_{{\text{CO}}_2}\right)+\left({\left(\Delta G_{\text{f}}°\right)}_{{\text{H}}_2\text{O}}\right)\right)$

The value of ${\left(\Delta G_{\text{f}}°\right)}_{\text{CO}}$ is $-137.2\text{kJ}/\text{mol}$, ${\left(\Delta G_{\text{f}}°\right)}_{{\text{H}}_2}$ is $0.0\text{kJ}/\text{mol}$, $\left(\Delta {\left(G_{\text{f}}°\right)}_{{\text{CO}}_2}\right)$ is $-394.4\text{kJ}/\text{mol}$ and ${\left(\Delta G_{\text{f}}°\right)}_{{\text{H}}_2\text{O}}$ is $-228.6\text{kJ}/\text{mol}$.

Substitute these values in above equation as,

$\begin{array}{c}\Delta G{°}_{{\text{rxn}}_2}=\left({\left(\Delta G_{\text{f}}°\right)}_{{\text{CO}}_{\text{2}}}+{\left(\Delta G_{\text{f}}°\right)}_{{\text{H}}_2}\right)-\left(\left(\Delta {\left(G_{\text{f}}°\right)}_{\text{CO}}\right)+\left({\left(\Delta G_{\text{f}}°\right)}_{{\text{H}}_2\text{O}}\right)\right)\\ =\left(\begin{array}{c}\left(1\text{mol}\left(-394.4\text{kJ}/\text{mol}\right)+1\text{mol}\times 0\text{kJ}/\text{mol}\right)-\\ \left(1\text{mol}\left(-137.2\text{kJ}/\text{mol}\right)+1\text{mol}\left(-228.6\text{kJ}/\text{mol}\right)\right)\end{array}\right)\\ =-394.4\text{kJ}+\text{365}\text{.8}\text{kJ}\\ =\underset{\_}{-28.6kJ}\end{array}$

Add the equation (1) and (20 and cancel the similar terms on both the sides of the reaction,

${\text{CH}}_4\left(g\right)+2{\text{H}}_2\text{O}\left(g\right)\to {\text{CO}}_2\left(g\right)+4{\text{H}}_2\left(g\right)$

The value of $\Delta G{°}_{\text{rxn}}$ for the above reaction is given as,

$\begin{array}{c}\Delta G{°}_{\text{rxn}}=\Delta G{°}_{{\text{rxn}}_1}+\Delta G{°}_{{\text{rxn}}_2}\\ =142.2\text{kJ}-28.6\text{kJ}\\ =\underset{\_}{113.6kJ}\end{array}$

Conclusion

1. a. The oxidation state of Carbon and Hydrogen in equation ${\text{CH}}_4\left(g\right)+{\text{H}}_2\text{O}\left(g\right)\to \text{CO}\left(g\right)+3{\text{H}}_2\left(g\right)$ is,

• The oxidation number of Hydrogen is $\underset{\_}{+1}$ and $\text{C}$ is $\underset{\_}{-4}$ in ${\text{CH}}_4$.
• The oxidation number of Hydrogen is $\underset{\_}{+1}$ in ${\text{H}}_2\text{O}$.
• The oxidation number of $\text{C}$ is $\underset{\_}{+2}$ in $\text{CO}$
• As there is no difference of electronegativity in ${\text{H}}_2$, the oxidation number of Hydrogen is zero.

The oxidation state of Carbon and Hydrogen in equation $\text{CO}\left(g\right)+{\text{H}}_2\text{O}\left(g\right)\to {\text{H}}_2\left(g\right)+{\text{CO}}_2\left(g\right)$ is,

• The oxidation number of $\text{C}$ is $\underset{\_}{+2}$ in $\text{CO}$.
• The oxidation number of Hydrogen is $\underset{\_}{+1}$ in ${\text{H}}_2\text{O}$.
• As there is no difference of electronegativity in ${\text{H}}_2$, the oxidation number of Hydrogen is zero.
• The oxidation number of $\text{C}$ is $\underset{\_}{+4}$ in ${\text{CO}}_{\text{2}}$.

1. b. The value of $\Delta G{°}_{\text{rxn}}$ for equation (1) is $\underset{\_}{142.2kJ}$.

   The value of $\Delta G{°}_{\text{rxn}}$ for equation (2) is $\underset{\_}{-28.6kJ}$.

   The value of $\Delta G{°}_{\text{rxn}}$ for overall reaction is $\underset{\_}{113.6kJ}$