Chapter 14.7, Problem 15P

Interpretation Introduction

Interpretation:

Plot the equilibrium mole fraction of ozone in the air as a function of temperature from $0°\text{F}$ to $100°\text{F}$.

Concept Introduction:

The expression of the standard change in Gibbs free energy at $298.15\text{K}$ is,

$\begin{array}{c}\Delta {\underset{\_}{G}}_{298.15\text{K}}^0={\displaystyle \sum _i{v}_i{\underset{\_}{G}}_i^0}\\ =v_{O_3}{\underset{\_}{G}}_{f,O_3}^0+v_{O_2}{\underset{\_}{G}}_{f,O_2}^0\end{array}$

Here, stoichiometric coefficient of compound $i$ is $v_i$, standard Gibbs free energy of compound $i$ is ${\underset{\_}{G}}_i^0$, stoichiometric coefficient of $O_3$ is $v_{O_3}$, standard Gibbs free energy of $O_3$ is ${\underset{\_}{G}}_{f,O_3}^0$, stoichiometric coefficient of $O_2$ is $v_{O_2}$, and standard Gibbs free energy of $O_2$ is ${\underset{\_}{G}}_{f,O_2}^0$.

The expression of the standard enthalpy of reaction at $298.15\text{K}$ is,

$\begin{array}{c}\Delta {\underset{\_}{H}}_{298.15\text{K}}^0={\displaystyle \sum _i{v}_i{\underset{\_}{H}}_i^0}\\ =v_{O_3}{\underset{\_}{H}}_{f,O_3}^0+v_{O_2}{\underset{\_}{H}}_{f,O_2}^0\end{array}$

Here, standard enthalpy of reaction of compound $i$ is ${\underset{\_}{H}}_i^0$, standard enthalpy of reaction of $O_3$ is ${\underset{\_}{H}}_{f,O_3}^0$ and standard enthalpy of reaction of $O_2$ is ${\underset{\_}{H}}_{f,O_2}^0$.

The expression of the standard change in Gibbs free energy as a function of temperature is,

$\frac{\Delta {\underset{\_}{G}}_T^0}{RT}=\left\{\begin{array}{l}\frac{\Delta {\underset{\_}{G}}_R^0}{R{T}_R}+\frac{J}{R}\left(\frac{1}{T}-\frac{1}{T_R}\right)-\frac{\Delta A}{R}\ln \left(\frac{T}{T_R}\right)-\frac{\Delta B}{2R}\ln \left(T-T_R\right)-\frac{\Delta C}{6R}\ln \left(T^2-T_R^2\right)\\ -\frac{\Delta D}{12R}\ln \left(T^3-T_R^3\right)-\frac{\Delta E}{20R}\ln \left(T^4-T_R^4\right)\end{array}\right\}$

Here, gas constant is $R$, temperature is $T$, reference temperature is $T_R$, $J$, $A$, $B$, $C$, $D$, and $E$ are constants.

The expression of the constant $J$ is,

$J=\Delta {\underset{\_}{H}}_R^0-\Delta A{T}_R-\frac{\Delta B}{2}{T}_R^2-\frac{\Delta C}{3}{T}_R^3-\frac{\Delta D}{4}{T}_R^4-\frac{\Delta E}{5}{T}_R^5$

The expression of the constant $\Delta A$ is,

$\begin{array}{c}\Delta A={\displaystyle \sum _i{v}_{i}A}\\ =v_{O_3}{A}_{O_3}+v_{O_2}{A}_{O_2}\end{array}$

The expression of the equilibrium constant is,

$K_T=\exp \left(-\frac{\Delta {\underset{\_}{G}}_T^0}{RT}\right)$

The expression of the equilibrium constant in terms of mole fractions is,

$K_T=\frac{y_{O_3}^2}{y_{O_2}^3}$

Here, mole fraction of $O_3$ is $y_{O_3}$, and mole fraction of $O_2$ is $y_{O_2}$.