Chapter 21, Problem 18PS

Interpretation Introduction

Interpretation:

To determine the ${\text{Δ}}_{\text{r}}{H}^{\text{°}}$,${\text{Δ}}_{\text{r}}{G}^{\text{°}}$ and ${\text{Δ}}_{\text{r}}{S}^{\text{°}}$ for the production of hydrogen by reaction of carbon and water.

Concept introduction:

The Gibbs free energy or the free energy change is a thermodynamic quantity represented by ${\Delta }_{\text{r}}{G}^{\circ }$. The expression for the free energy change is:

${\text{Δ}}_{\text{r}}{G}^{\text{°}}=\sum n{\Delta }_f{G}^{\text{°}}\left(\text{products}\right)-\sum n{\Delta }_f{G}^{\text{°}}\left(\text{reactants}\right)$

The expression for the enthalpy change is:

${\text{Δ}}_{\text{r}}{H}^{\text{°}}=\sum n{\Delta }_f{H}^{\text{°}}\left(\text{products}\right)-\sum n{\Delta }_f{H}^{\text{°}}\left(\text{reactants}\right)$

The expression for the entropy change is:

${\text{Δ}}_{\text{r}}{S}^{\text{°}}=\sum n{S}^{\text{°}}\left(\text{products}\right)-\sum n{S}^{\text{°}}\left(\text{reactants}\right)$

Answer to Problem 18PS

The value of ${\text{Δ}}_{\text{r}}{G}^{\text{°}}$ for the production of hydrogen by reaction of carbon and water is

$+91.42kJ\text{/}mol\text{-}rxn$.

The value of ${\text{Δ}}_{\text{r}}{H}^{\text{°}}$ for the production of hydrogen by reaction of carbon and water is

$+130.30kJ\text{/}mol\text{-}rxn$.

The value of ${\text{Δ}}_{\text{r}}{S}^{\text{°}}$ for the production of hydrogen by reaction of carbon and water is

$+133.93J\text{/}K\cdot mol\text{-}rxn$.

Explanation of Solution

The ${\text{Δ}}_{\text{r}}{G}^{\text{°}}$,${\text{Δ}}_{\text{r}}{H}^{\text{°}}$ and ${\text{Δ}}_{\text{r}}{S}^{\text{°}}$ for the given reaction is calculated below.

Given:

Refer to Appendix L for the values of standard free energy values.

The given reaction is,

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

The ${\text{Δ}}_{\text{r}}{G}^{\text{°}}$ for $\text{C}\left(\text{s}\right)$ is $\text{0 kJ/mol}$.

The ${\text{Δ}}_{\text{r}}{G}^{\text{°}}$ for ${\text{H}}_{\text{2}}\text{O}\left(\text{g}\right)$ is $-\text{228}\text{.59 kJ/mol}$.

The ${\text{Δ}}_{\text{r}}{G}^{\text{°}}$ for $\text{CO}\left(\text{g}\right)$ is $-137.168\text{ kJ/mol}$.

The ${\text{Δ}}_{\text{r}}{G}^{\text{°}}$ for ${\text{H}}_{\text{2}}\left(\text{g}\right)$ is $\text{0 kJ/mol}$.

The expression for the free energy change is:

$\begin{array}{c}{\text{Δ}}_{\text{r}}{G}^{\text{°}}=\sum n{\Delta }_f{G}^{\text{°}}\left(\text{products}\right)-\sum n{\Delta }_f{G}^{\text{°}}\left(\text{reactants}\right)\\ =\left[\begin{array}{l}\left[\begin{array}{l}\left(\text{1 mol CO}\left(\text{g}\right)\text{/mol-rxn}\right){\Delta }_f{G}^{°}\left[\text{CO}\left(\text{g}\right)\right]+\\ \left({\text{1 mol H}}_2\left(\text{g}\right)\text{/mol-rxn}\right){\Delta }_f{G}^{°}\left[{\text{H}}_2\left(\text{g}\right)\right]\end{array}\right]-\\ \left[\begin{array}{l}\left(\text{1 mol C}\left(\text{s}\right)\text{/mol-rxn}\right){\Delta }_f{G}^{°}\left[\text{C}\left(\text{s}\right)\right]+\\ \left({\text{1 mol H}}_{\text{2}}\text{O}\left(\text{g}\right)\text{/mol-rxn}\right){\Delta }_f{G}^{°}\left[{\text{H}}_{\text{2}}\text{O}\left(\text{g}\right)\right]\end{array}\right]\end{array}\right]\end{array}$

Substitute the values,

$\begin{array}{c}{\text{Δ}}_{\text{r}}{G}^{\text{°}}=\left[\begin{array}{l}\left[\begin{array}{l}\left(\text{1 mol CO}\left(\text{g}\right)\text{/mol-rxn}\right)\left(-137.168\text{ kJ/mol}\right)+\\ \left({\text{1 mol H}}_2\left(\text{g}\right)\text{/mol-rxn}\right)\left(\text{0 kJ/mol}\right)\end{array}\right]-\\ \left[\begin{array}{l}\left(\text{1 mol C}\left(\text{s}\right)\text{/mol-rxn}\right)\left(\text{0 kJ/mol}\right)+\\ \left({\text{1 mol H}}_{\text{2}}\text{O}\left(\text{g}\right)\text{/mol-rxn}\right)\left(-\text{228}\text{.59 kJ/mol}\right)\end{array}\right]\end{array}\right]\\ =+91.42\text{ kJ/mol-rxn}\end{array}$

The ${\text{Δ}}_{\text{r}}{H}^{\text{°}}$ for $\text{C}\left(\text{s}\right)$ is $\text{0 kJ/mol}$.

The ${\text{Δ}}_{\text{r}}{H}^{\text{°}}$ for ${\text{H}}_{\text{2}}\text{O}\left(\text{g}\right)$ is $-\text{241}\text{.83 kJ/mol}$.

The ${\text{Δ}}_{\text{r}}{H}^{\text{°}}$ for $\text{CO}\left(\text{g}\right)$ is $-110.525\text{ kJ/mol}$.

The ${\text{Δ}}_{\text{r}}{H}^{\text{°}}$ for ${\text{H}}_{\text{2}}\left(\text{g}\right)$ is $\text{0 kJ/mol}$.

The expression for the enthalpy change is:

$\begin{array}{c}{\text{Δ}}_{\text{r}}{H}^{\text{°}}=\sum n{\Delta }_f{H}^{\text{°}}\left(\text{products}\right)-\sum n{\Delta }_f{H}^{\text{°}}\left(\text{reactants}\right)\\ =\left[\begin{array}{l}\left[\begin{array}{l}\left(\text{1 mol CO}\left(\text{g}\right)\text{/mol-rxn}\right){\Delta }_f{H}^{°}\left[\text{CO}\left(\text{g}\right)\right]+\\ \left({\text{1 mol H}}_2\left(\text{g}\right)\text{/mol-rxn}\right){\Delta }_f{H}^{°}\left[{\text{H}}_2\left(\text{g}\right)\right]\end{array}\right]-\\ \left[\begin{array}{l}\left(\text{1 mol C}\left(\text{s}\right)\text{/mol-rxn}\right){\Delta }_f{H}^{°}\left[\text{C}\left(\text{s}\right)\right]+\\ \left({\text{1 mol H}}_{\text{2}}\text{O}\left(\text{g}\right)\text{/mol-rxn}\right){\Delta }_f{H}^{°}\left[{\text{H}}_{\text{2}}\text{O}\left(\text{g}\right)\right]\end{array}\right]\end{array}\right]\end{array}$

Substitute the values,

$\begin{array}{c}\text{Δ}{H}^{\text{°}}=\left[\begin{array}{l}\left[\begin{array}{l}\left(\text{1 mol CO}\left(\text{g}\right)\text{/mol-rxn}\right)\left(-110.525\text{ kJ/mol}\right)+\\ \left({\text{1 mol H}}_2\left(\text{g}\right)\text{/mol-rxn}\right)\left(\text{0 kJ/mol}\right)\end{array}\right]-\\ \left[\begin{array}{l}\left(\text{1 mol C}\left(\text{s}\right)\text{/mol-rxn}\right)\left(\text{0 kJ/mol}\right)+\\ \left({\text{1 mol H}}_{\text{2}}\text{O}\left(\text{g}\right)\text{/mol-rxn}\right)\left(-\text{241}\text{.83 kJ/mol}\right)\end{array}\right]\end{array}\right]\\ =+131.30\text{ kJ/mol-rxn}\end{array}$

The $S^{\text{°}}$ for $\text{C}\left(\text{s}\right)$ is $\text{5}\text{.6 J/K}\cdot \text{mol}$.

The $S^{\text{°}}$ for ${\text{H}}_{\text{2}}\text{O}\left(\text{g}\right)$ is $\text{188}\text{.84 J/K}\cdot \text{mol}$.

The $S^{\text{°}}$ for $\text{CO}\left(\text{g}\right)$ is $197.674\text{ J/K}\cdot \text{mol}$.

The $S^{\text{°}}$ for ${\text{H}}_{\text{2}}\left(\text{g}\right)$ is $\text{130}\text{.7 J/K}\cdot \text{mol}$.

The expression for the entropy change is:

$\begin{array}{c}{\text{Δ}}_{\text{r}}{S}^{\text{°}}=\sum n{S}^{\text{°}}\left(\text{products}\right)-\sum n{S}^{\text{°}}\left(\text{reactants}\right)\\ =\left[\begin{array}{l}\left[\begin{array}{l}\left(\text{1 mol CO}\left(\text{g}\right)\text{/mol-rxn}\right)S^{°}\left[\text{CO}\left(\text{g}\right)\right]+\\ \left({\text{1 mol H}}_2\left(\text{g}\right)\text{/mol-rxn}\right)S^{°}\left[{\text{H}}_2\left(\text{g}\right)\right]\end{array}\right]-\\ \left[\begin{array}{l}\left(\text{1 mol C}\left(\text{s}\right)\text{/mol-rxn}\right)S^{°}\left[\text{C}\left(\text{s}\right)\right]+\\ \left({\text{1 mol H}}_{\text{2}}\text{O}\left(\text{g}\right)\text{/mol-rxn}\right)S^{°}\left[{\text{H}}_{\text{2}}\text{O}\left(\text{g}\right)\right]\end{array}\right]\end{array}\right]\end{array}$

Substitute the values,

$\begin{array}{c}{\text{Δ}}_{\text{r}}{S}^{\text{°}}=\left[\begin{array}{l}\left[\begin{array}{l}\left(\text{1 mol CO}\left(\text{g}\right)\text{/mol-rxn}\right)\left(197.674\text{ J/K}\cdot \text{mol}\right)+\\ \left({\text{1 mol H}}_2\left(\text{g}\right)\text{/mol-rxn}\right)\left(\text{130}\text{.7 J/K}\cdot \text{mol}\right)\end{array}\right]-\\ \left[\begin{array}{l}\left(\text{1 mol C}\left(\text{s}\right)\text{/mol-rxn}\right)\left(\text{5}\text{.6 J/K}\cdot \text{mol}\right)+\\ \left({\text{1 mol H}}_{\text{2}}\text{O}\left(\text{g}\right)\text{/mol-rxn}\right)\left(\text{188}\text{.84 J/K}\cdot \text{mol}\right)\end{array}\right]\end{array}\right]\\ =133.93\text{ J/K}\cdot \text{mol-rxn}\end{array}$

Conclusion

The value of ${\text{Δ}}_{\text{r}}{G}^{\text{°}}$ for the production of hydrogen by reaction of carbon and water is

$+91.42kJ\text{/}mol\text{-}rxn$.

The value of ${\text{Δ}}_{\text{r}}{H}^{\text{°}}$ for the production of hydrogen by reaction of carbon and water is

$+130.30kJ\text{/}mol\text{-}rxn$.

The value of ${\text{Δ}}_{\text{r}}{S}^{\text{°}}$ for the production of hydrogen by reaction of carbon and water is

$+133.93J\text{/}K\cdot mol\text{-}rxn$.