Chapter 20, Problem 12Q

The major industrial use of hydrogen is in the production of ammonia by the Haber process:

${\text{3H}}_{\text{2}}\left(g\right)+{\text{N}}_{\text{2}}\left(g\right)\to {\text{2NH}}_{\text{3}}\left(g\right)$

a. Using data from Appendix 4, calculate ∆H°, ∆S°, and ∆G° for the Haber process reaction.

b. Is the reaction spontaneous at standard conditions?

c. At what temperatures is the reaction spontaneous at standard conditions? Assume ∆H° and ∆S° do not depend on temperature.

Interpretation Introduction

Interpretation: Reaction of commercial production of ammonia is given. The value of $\Delta H°$ and $\Delta S°$ is to be calculated for the given reaction. If the reaction is spontaneous is to be identified. The temperature at which this reaction is spontaneous is to be calculated.

Concept introduction: The expression to calculate $\Delta H°$ is,

$\Delta H°=\sum n_{\text{p}}\Delta H{°}_{\left(\text{product}\right)}-\sum n_{\text{f}}\Delta H{°}_{\left(\text{reactant}\right)}$

The expression to calculate $\Delta S°$ is,

$\Delta S°=\sum n_{\text{p}}\Delta S{°}_{\left(\text{product}\right)}-\sum n_{\text{f}}\Delta S{°}_{\left(\text{reactant}\right)}$

A reaction is said to be spontaneous if the value of $\Delta G°$ is negative.

Explanation of Solution

The stated reaction is,

$3{\text{H}}_{\text{2}}\left(g\right)+{\text{N}}_{\text{2}}\left(g\right)\stackrel{}{\rightleftharpoons }{\text{2NH}}_{\text{3}}\left(g\right)$

Refer to Appendix $4$

The value of $\Delta H°\left(\text{kJ/mol}\right)$ for the given reactant and product is,

Molecules	$\Delta H°\left(\text{kJ/mol}\right)$
${\text{NH}}_{\text{3}}(g)$	$-\text{46}$
${\text{N}}_{\text{2}}(g)$	$\text{0}$
${\text{H}}_{\text{2}}(g)$	$\text{0}$

The formula of $\Delta H°$ is,

$\Delta H°=\sum n_{\text{p}}\Delta H{°}_{\left(\text{product}\right)}-\sum n_{\text{f}}\Delta H{°}_{\left(\text{reactant}\right)}$

Where,

• $\Delta H°$ the standard enthalpy of reaction.
• $n_{\text{p}}$ is the number of moles of each product.
• $n_{\text{r}}$ is the number of moles each reactant.
• $\Delta H{°}_{\left(\text{product}\right)}$ is the standard enthalpy of product at a pressure of $\text{1}\text{atm}$.
• $\Delta H{°}_{\left(\text{reactant}\right)}$ is the standard enthalpy of reactant at a pressure of $\text{1}\text{atm}$.

Substitute all the values from the table in the above equation.

$\begin{array}{c}\Delta H°=\sum n_{\text{p}}\Delta H{°}_{\left(\text{product}\right)}-\sum n_{\text{f}}\Delta H{°}_{\left(\text{reactant}\right)}\\ =\left[\text{2}\left(-\text{46}\right)-\left\{\text{3}\left(\text{0}\right)+\left(\text{0}\right)\right\}\right]\text{kJ}\\ =\underset{\_}{-92kJ}\end{array}$

The value of $\Delta S°$ for the given reaction is $\underset{\_}{199J/K}$.

Refer to Appendix $4$

The value of $\Delta S°\left(\text{J/K}\cdot \text{mol}\right)$ for the given reactant and product is,

Molecules	$\Delta S°\left(\text{kJ/mol}\right)$
${\text{NH}}_{\text{3}}(g)$	$\text{193}$
${\text{N}}_{\text{2}}(g)$	$\text{192}$
${\text{H}}_{\text{2}}(g)$	$\text{131}$

The formula of $\Delta S°$ is,

$\Delta S°=\sum n_{\text{p}}\Delta S{°}_{\left(\text{product}\right)}-\sum n_{\text{f}}\Delta S{°}_{\left(\text{reactant}\right)}$

Where,

• $\Delta S°$ is the standard entropy of reaction.
• $n_{\text{p}}$ is the number of moles of each product.
• $n_{\text{r}}$ is the number of moles each reactant.
• $\Delta S{°}_{\left(\text{product}\right)}$ is the standard entropy of product at a pressure of $\text{1}\text{atm}$.
• $\Delta S{°}_{\left(\text{reactant}\right)}$ is the standard entropy of reactant at a pressure of $\text{1}\text{atm}$.

Substitute all the values from the table in the above equation.

$\begin{array}{c}\Delta S°=\sum n_{\text{p}}\Delta S{°}_{\left(\text{product}\right)}-\sum n_{\text{f}}\Delta S{°}_{\left(\text{reactant}\right)}\\ =\left[\text{2}\left(\text{193}\right)-\left\{\text{3}\left(\text{131}\right)+\left(\text{192}\right)\right\}\right]\text{J/K}\\ =\underset{\_}{199J/K}\end{array}$

Interpretation Introduction

Interpretation: Reaction of commercial production of ammonia is given. The value of $\Delta H°$ and $\Delta S°$ is to be calculated for the given reaction. If the reaction is spontaneous is to be identified. The temperature at which this reaction is spontaneous is to be calculated.

Concept introduction: The expression to calculate $\Delta H°$ is,

$\Delta H°=\sum n_{\text{p}}\Delta H{°}_{\left(\text{product}\right)}-\sum n_{\text{f}}\Delta H{°}_{\left(\text{reactant}\right)}$

The expression to calculate $\Delta S°$ is,

$\Delta S°=\sum n_{\text{p}}\Delta S{°}_{\left(\text{product}\right)}-\sum n_{\text{f}}\Delta S{°}_{\left(\text{reactant}\right)}$

A reaction is said to be spontaneous if the value of $\Delta G°$ is negative.

Explanation of Solution

Refer to Appendix $\text{4}$.

The value of $\Delta G°\left(\text{kJ/mol}\right)$ for the given reactant and product is,

Molecules	$\Delta G°\left(\text{kJ/mol}\right)$
${\text{NH}}_{\text{3}}(g)$	$-\text{17}$
${\text{N}}_{\text{2}}(g)$	$\text{0}$
${\text{H}}_{\text{2}}(g)$	$\text{0}$

The formula of $\Delta G°$ is,

$\Delta G°=\sum n_{\text{p}}\Delta G{°}_{\left(\text{product}\right)}-\sum n_{\text{f}}\Delta G{°}_{\left(\text{reactant}\right)}$

Where,

• $n_{\text{p}}$ is the number of moles of each product.
• $n_{\text{r}}$ is the number of moles each reactant.
• $\Delta G{°}_{\left(\text{product}\right)}$ is the free energy of product at a pressure of $\text{1}\text{atm}$.
• $\Delta G{°}_{\left(\text{reactant}\right)}$ is the free energy of reactant at a pressure of $\text{1}\text{atm}$.

Substitute all the values from the table in the above equation.

$\begin{array}{c}\Delta G°=\sum n_{\text{p}}\Delta G{°}_{\left(\text{product}\right)}-\sum n_{\text{f}}\Delta G{°}_{\left(\text{reactant}\right)}\\ =\left[\text{2}\left(-\text{17}\right)-\left\{\left(\text{0}\right)+\left(\text{0}\right)\right\}\right]\text{kJ}\\ =\underset{\_}{-34kJ}\end{array}$

Since, the value of $\Delta G°$ is negative, hence, the given reaction is spontaneous.

Interpretation Introduction

Interpretation: Reaction of commercial production of ammonia is given. The value of $\Delta H°$ and $\Delta S°$ is to be calculated for the given reaction. If the reaction is spontaneous is to be identified. The temperature at which this reaction is spontaneous is to be calculated.

Concept introduction: The expression to calculate $\Delta H°$ is,

$\Delta H°=\sum n_{\text{p}}\Delta H{°}_{\left(\text{product}\right)}-\sum n_{\text{f}}\Delta H{°}_{\left(\text{reactant}\right)}$

The expression to calculate $\Delta S°$ is,

$\Delta S°=\sum n_{\text{p}}\Delta S{°}_{\left(\text{product}\right)}-\sum n_{\text{f}}\Delta S{°}_{\left(\text{reactant}\right)}$

A reaction is said to be spontaneous if the value of $\Delta G°$ is negative.

Explanation of Solution

The value of $\Delta H$ is $-92\text{kJ}$.

The value of $\Delta S$ is $199\text{J/K}$.

The conversion of kilo-joule $\left(\text{kJ}\right)$ into joule $\left(\text{J}\right)$ is done as,

$\text{1}\text{kJ}={\text{10}}^{\text{3}}\text{J}$

Hence,

The conversion of $-92\text{kJ}$ into joule is,

$\begin{array}{c}-\text{92}\text{kJ}=\left(-\text{92}\times {\text{10}}^{\text{3}}\right)\text{J}\\ =-\text{92}\times {\text{10}}^{\text{3}}\text{J}\end{array}$

Formula

The formula of $\Delta G$ is,

$\Delta G=\Delta H°-T\Delta S°$

Where,

• $\Delta H°$ is the standard enthalpy of reaction.
• $\Delta G°$ is the free energy change.
• $T$ is the given temperature.
• $\Delta S°$ is the standard entropy of reaction.

At equilibrium, the value of $\Delta G$ is zero.

Substitute the values of $\Delta G°,\Delta H°$ and $\Delta S°$ in the above equation.

$\begin{array}{c}\Delta G=\Delta H°-T\Delta S°\\ \text{0}=\left(-\text{92}\times {\text{10}}^{\text{3}}\text{J}\right)-T\left(\text{199}\text{J/K}\right)\\ T\approx \underset{\_}{462K}\end{array}$

The reaction will be spontaneous if the temperature is greater than $\underset{\_}{462K}$.