Chapter 18, Problem 18.79QP

(a)

Interpretation Introduction

Interpretation: The values of standard Gibbs free energy $\left(\Delta G_{\text{rxn}}^{\text{ο}}\right)$ for the given reactions are to be calculated. The balanced chemical equation of the overall reaction obtained by coupling the given two reactions is to be stated. Also, whether the coupled reaction is spontaneous under standard condition is to be explained.

Concept introduction: Gibbs free energy is a thermodynamic quantity that is used to calculate the maximum work of reversible reaction performed by a system. It is equal to the difference between the enthalpy and the product of entropy at absolute temperature.

To determine: The values of $\Delta G_{\text{rxn}}^{\text{ο}}$ for the given reactions.

Answer to Problem 18.79QP

Solution

The value of $\Delta G_{\text{rxn}}^{\text{ο}}$ for the decomposition reaction of methane is $\underset{\_}{50.8kJ}$.

The value of $\Delta G_{\text{rxn}}^{\text{ο}}$ for oxidation reaction of carbon is $\underset{\_}{-394.4kJ}$.

Explanation of Solution

Explanation

Given

The decomposition reaction of methane is,

${\text{CH}}_{\text{4}}\left(g\right)\stackrel{}{\to }\text{C}\left(s\right)+2{\text{H}}_{\text{2}}\left(g\right)$

The reaction consumes one mole of methane to produce one mole of carbon and two moles of hydrogen.

The $\Delta G_{\text{f}}^{\text{ο}}$ value of ${\text{CH}}_{\text{4}}\left(g\right)$ is $-50.8\text{kJ/mol}$.

The $\Delta G_{\text{f}}^{\text{ο}}$ value of $\text{C}\left(s\right)$ is $0.0\text{kJ/mol}$.

The $\Delta G_{\text{f}}^{\text{ο}}$ value of ${\text{H}}_2\left(g\right)$ is $0.0\text{kJ/mol}$.

The change in standard free energy of the reaction, $\Delta G_{\text{rxn}}^{\text{ο}}$, using the standard free energy of formation is calculated from the given formula,

$\Delta G_{\text{rxn}}^{\text{ο}}={\displaystyle \sum n_{\text{products}}\Delta G_{\text{f}\text{.products}}^{\text{ο}}-}{\displaystyle \sum n_{\text{reactants}}\Delta G_{\text{f}\text{.reactants}}^{\text{ο}}}$

Where,

• $n$ is the number of moles of for products and reactants.
• $\Delta G_{\text{f}}^{\text{ο}}$ is the standard free energy of formation of reactants and products.

Substitute the values of $n_{\text{products}}$, $\Delta G_{\text{f}\text{.products}}^{\text{ο}}$, $n_{\text{reactants}}$ and $\Delta G_{\text{f}\text{.reactants}}^{\text{ο}}$ in the above formula to calculate the $\Delta G_{\text{rxn}}^{\text{ο}}$ for the reaction.

$\begin{array}{c}\Delta G_{\text{rxn}}^{\text{ο}}=\left[\left(n_{\text{C}}\times \Delta G_{\text{C}}^{\text{ο}}+n_{{\text{H}}_{\text{2}}}\times \Delta G_{{\text{H}}_{\text{2}}}^{\text{ο}}\right)\right]-\left[n_{{\text{CH}}_4}\times \Delta G_{{\text{CH}}_4}^{\text{ο}}\right]\\ =\left[\left(1\times \left(-0.0\text{kJ/mol}\right)+2\times 0.0\text{kJ/mol}\right)\right]-\\ \left[1\times \left(-50.8\text{kJ/mol}\right)\right]\\ =\left[0.0\text{kJ}+0.0\text{kJ}\right]-\left(-50.8\text{kJ}\right)\\ =0.0\text{kJ}+50.8\text{kJ}\end{array}$

Simplify the above equation,

$\Delta G_{\text{rxn}}^{\text{ο}}=\underset{\_}{50.8kJ}$

Hence, the value of $\Delta G_{\text{rxn}}^{\text{ο}}$ for the reaction is $\underset{\_}{50.8kJ}$.

The oxidation reaction of carbon is,

$\text{C}\left(s\right)+{\text{O}}_{\text{2}}\left(g\right)\stackrel{}{\to }{\text{CO}}_{\text{2}}\left(g\right)$

The reaction consumes one mole of carbon and one mole of oxygen gas to produce one mole of carbon dioxide.

The $\Delta G_{\text{f}}^{\text{ο}}$ value of ${\text{CO}}_2\left(g\right)$ is $-394.4\text{kJ/mol}$.

The $\Delta G_{\text{f}}^{\text{ο}}$ value of $\text{C}\left(s\right)$ is $0.0\text{kJ/mol}$.

The $\Delta G_{\text{f}}^{\text{ο}}$ value of ${\text{O}}_2\left(g\right)$ is $0.0\text{kJ/mol}$.

The change in standard free energy of the reaction, $\Delta G_{\text{rxn}}^{\text{ο}}$, using the standard free energy of formation is calculated from the given formula,

$\Delta G_{\text{rxn}}^{\text{ο}}={\displaystyle \sum n_{\text{products}}\Delta G_{\text{f}\text{.products}}^{\text{ο}}-}{\displaystyle \sum n_{\text{reactants}}\Delta G_{\text{f}\text{.reactants}}^{\text{ο}}}$

Where,

• $n$ is the number of moles of for products and reactants.
• $\Delta G_{\text{f}}^{\text{ο}}$ is the standard free energy of formation of reactants and products.

Substitute the values of $n_{\text{products}}$, $\Delta G_{\text{f}\text{.products}}^{\text{ο}}$, $n_{\text{reactants}}$ and $\Delta G_{\text{f}\text{.reactants}}^{\text{ο}}$ in the above formula to calculate the $\Delta G_{\text{rxn}}^{\text{ο}}$ for the reaction.

$\begin{array}{c}\Delta G_{\text{rxn}}^{\text{ο}}=\left[n_{{\text{CO}}_2}\times \Delta G_{{\text{CO}}_2}^{\text{ο}}\right]-\left[\left(n_{\text{C}}\times \Delta G_{\text{C}}^{\text{ο}}+n_{{\text{O}}_{\text{2}}}\times \Delta G_{{\text{O}}_{\text{2}}}^{\text{ο}}\right)\right]\\ =\left[1\times \left(-394.4\text{kJ/mol}\right)\right]-\\ \left[\left(1\times \left(-0.0\text{kJ/mol}\right)+1\times 0.0\text{kJ/mol}\right)\right]\\ =-394.4\text{kJ}-0.0\text{kJ}\\ =\underset{\_}{-394.4kJ}\end{array}$

Hence, the value of $\Delta G_{\text{rxn}}^{\text{ο}}$ for the reaction is $\underset{\_}{-394.4kJ}$.

(b)

Interpretation Introduction

To determine: The balanced chemical equation of the overall reaction obtained by coupling the given two reactions and if the coupled reaction is spontaneous under standard condition.

Answer to Problem 18.79QP

Solution

The overall balanced chemical equation is,

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

The value of $\Delta G_{\text{overall}}^{\text{ο}}$ is $\underset{\_}{-343.6kJ}$. The value of $\Delta G_{\text{overall}}^{\text{ο}}<0$, the reaction is spontaneous under standard condition.

Explanation of Solution

Explanation

The decomposition reaction of methane is,

${\text{CH}}_{\text{4}}\left(g\right)\stackrel{}{\to }\text{C}\left(s\right)+2{\text{H}}_{\text{2}}\left(g\right)$ (1)

The oxidation reaction of carbon is,

$\text{C}\left(s\right)+{\text{O}}_{\text{2}}\left(g\right)\stackrel{}{\to }{\text{CO}}_{\text{2}}\left(g\right)$ (2)

The two reactions are combined to give overall reaction as,

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

Hence, the overall reaction is,

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

The $\Delta G_{\text{rxn}}^{\text{ο}}$ values of the coupled reactions are additive. Hence, their values are added to give $\Delta G_{\text{rxn}}^{\text{ο}}$ of the overall reaction. The change in free energy of the overall reaction is considered to be $\Delta G_{\text{overall}}^{\text{ο}}$.

The change in free energy of the decomposition reaction of methane is considered to $\Delta G_1^{\text{ο}}$ and the change in free energy of the oxidation reaction of carbon is considered to be $\Delta G_2^{\text{ο}}$. The value of $\Delta G_1^{\text{ο}}$ is $50.8\text{kJ}$ and the value of $\Delta G_2^{\text{ο}}$ is $-394.4\text{kJ}$ as calculated in part (a) and (b) respectively. The overall change in free energy is calculated as,

$\Delta G_{\text{overall}}^{\text{ο}}=\Delta G_{\text{1}}^{\text{ο}}+\Delta G_2^{\text{ο}}$

Substitute the values of $\Delta G_1^{\text{ο}}$ and $\Delta G_2^{\text{ο}}$ in the above expression to calculate the value of $\Delta G_{\text{overall}}^{\text{ο}}$.

$\begin{array}{c}\Delta G_{\text{overall}}^{\text{ο}}=50.8\text{kJ}+\left(-394.4\text{kJ}\right)\\ =50.8\text{kJ}-394.4\text{kJ}\\ =\underset{\_}{-343.6kJ}\end{array}$

When $\Delta G<0$ then the reaction is spontaneous and when $\Delta G>0$ the reaction is non spontaneous. If $\Delta G=0$ then the reaction is at equilibrium.

Hence, the value of $\Delta G_{\text{overall}}^{\text{ο}}$ is $\underset{\_}{-343.6kJ}$. The value of $\Delta G_{\text{overall}}^{\text{ο}}<0$, the reaction is spontaneous under standard condition.

Conclusion

1. a. The value of $\Delta G_{\text{rxn}}^{\text{ο}}$ for the decomposition reaction of methane is $\underset{\_}{50.8kJ}$ and the value of $\Delta G_{\text{rxn}}^{\text{ο}}$ for oxidation reaction of carbon is $\underset{\_}{-394.4kJ}$.
2. b. The value of $\Delta G_{\text{overall}}^{\text{ο}}$ is $\underset{\_}{-343.6kJ}$ which is less than zero shows the reaction is spontaneous under standard condition