Chapter 16.6, Problem 38P

Estimate KP for the following equilibrium reaction at 2500 K:

${\text{CO + H}}_{\text{2}}\text{O}\rightleftharpoons {\text{CO}}_{\text{2}}{\text{+H}}_{\text{2}}$

At 2000 K it is known that the enthalpy of reaction is –26,176 kJ/kmol and KP is 0.2209. Compare your result with the value obtained from the definition of the equilibrium constant.

To determine

Calculate $K_P$ for the below equilibrium reaction at 2500 K.

$\text{CO}+{\text{H}}_{\text{2}}\text{O}\rightleftharpoons {\text{CO}}_{\text{2}}+{\text{H}}_{\text{2}}$

Compare the results for the values of $K_P$ obtained from the definition of the equilibrium constant and calculated $K_P$.

Answer to Problem 38P

The equilibrium constant obtained from the equilibrium reaction at 2500 K is $\underset{\_}{0.1644}$.

Explanation of Solution

Express the standard-state Gibbs function change.

$\begin{array}{c}\Delta G^{*}\left(T\right)=v_{{\text{CO}}_{\text{2}}}{\overline{g}}^{*}{}_{{\text{CO}}_{\text{2}}}\left(T\right)+v_{{\text{H}}_{\text{2}}}{\overline{g}}^{*}{}_{{\text{H}}_{\text{2}}}\left(T\right)-v_{\text{CO}}{\overline{g}}^{*}{}_{\text{CO}}\left(T\right)-v_{{\text{H}}_{\text{2}}\text{O}}{\overline{g}}^{*}{}_{{\text{H}}_{\text{2}}\text{O}}\left(T\right)\\ =v_{{\text{CO}}_{\text{2}}}{\left(\overline{h}-T\overline{s}\right)}_{{\text{CO}}_{\text{2}}}+v_{{\text{H}}_{\text{2}}}{\left(\overline{h}-T\overline{s}\right)}_{{\text{H}}_{\text{2}}}-v_{\text{CO}}{\left(\overline{h}-T\overline{s}\right)}_{\text{CO}}-v_{{\text{H}}_{\text{2}}\text{O}}{\left(\overline{h}-T\overline{s}\right)}_{{\text{H}}_{\text{2}}\text{O}}\\ =\left[\begin{array}{c}{v}_{{\text{CO}}_{\text{2}}}{\left({\overline{h}}_f^o+\left(\overline{h}-{\overline{h}}^o\right)-T\overline{s}\right)}_{{\text{CO}}_{\text{2}}}+v_{{\text{H}}_{\text{2}}}{\left({\overline{h}}_f^o+\left(\overline{h}-{\overline{h}}^o\right)-T\overline{s}\right)}_{{\text{H}}_{\text{2}}}-\\ v_{{\text{H}}_{\text{2}}\text{O}}{\left({\overline{h}}_f^o+\left(\overline{h}-{\overline{h}}^o\right)-T\overline{s}\right)}_{{\text{H}}_{\text{2}}\text{O}}-v_{\text{CO}}{\left({\overline{h}}_f^o+\left(\overline{h}-{\overline{h}}^o\right)-T\overline{s}\right)}_{\text{CO}}\end{array}\right]\end{array}$ (I)

Here, the Gibbs function of components ${\text{H}}_{\text{2}}\text{O,}\text{CO,}{\text{H}}_{\text{2}}\text{,}\text{and}{\text{CO}}_{\text{2}}$ at 1 atm pressure and temperature T are ${\overline{g}}^{*}{}_{{\text{H}}_{\text{2}}\text{O}}\left(T\right),{\overline{g}}^{*}{}_{\text{CO}}\left(T\right),{\overline{g}}^{*}{}_{{\text{H}}_{\text{2}}}\left(T\right),\text{and}{\overline{g}}^{*}{}_{{\text{CO}}_{\text{2}}}\left(T\right)$ respectively, enthalpy on the unit mole basis of ${\text{H}}_{\text{2}}\text{O,}\text{CO,}{\text{H}}_{\text{2}}\text{,}\text{and}{\text{CO}}_{\text{2}}$ are ${\overline{h}}_{{\text{H}}_{\text{2}}\text{O}},{\overline{h}}_{\text{CO}},{\overline{h}}_{{\text{H}}_{\text{2}}},\text{and}{\overline{h}}_{{\text{CO}}_{\text{2}}}$ respectively, absolute entropy of ${\text{H}}_{\text{2}}\text{O,}\text{CO,}{\text{H}}_{\text{2}}\text{,}\text{and}{\text{CO}}_{\text{2}}$ are ${\overline{s}}_{{\text{H}}_{\text{2}}\text{O}},{\overline{s}}_{\text{CO}},{\overline{s}}_{{\text{H}}_{\text{2}}},\text{and}{\overline{s}}_{{\text{CO}}_{\text{2}}}$, temperature of ${\text{H}}_{\text{2}}\text{O,}\text{CO,}{\text{H}}_{\text{2}}\text{,}\text{and}{\text{CO}}_{\text{2}}$ are $T_{{\text{H}}_{\text{2}}\text{O}},T_{\text{CO}},T_{{\text{H}}_{\text{2}}},\text{and}{T}_{{\text{CO}}_{\text{2}}}$, sensible enthalpy at the specified state of ${\text{H}}_{\text{2}}\text{O,}\text{CO,}{\text{H}}_{\text{2}}\text{,}\text{and}{\text{CO}}_{\text{2}}$ are ${\overline{h}}_{{\text{H}}_{\text{2}}\text{O}},{\overline{h}}_{\text{CO}},{\overline{h}}_{{\text{H}}_{\text{2}}},\text{and}{\overline{h}}_{{\text{CO}}_{\text{2}}}$, the sensible enthalpy at the standard reference state of $25°\text{C}$ and 1 atm of ${\text{H}}_{\text{2}}\text{O,}\text{CO,}{\text{H}}_{\text{2}}\text{,}\text{and}{\text{CO}}_{\text{2}}$ are ${\overline{h}}_{{\text{H}}_{\text{2}}\text{O}}^o,{\overline{h}}_{\text{CO}}^o,{\overline{h}}_{{\text{H}}_{\text{2}}}^o,\text{and}{\overline{h}}_{{\text{CO}}_{\text{2}}}^o$, the enthalpy of formation at a specified states of $25°\text{C}$ and 1 atm for the components ${\text{H}}_{\text{2}}\text{O,}\text{CO,}{\text{H}}_{\text{2}}\text{,}\text{and}{\text{CO}}_{\text{2}}$ are ${\overline{h}}_{f,{\text{H}}_{\text{2}}\text{O}}^o,{\overline{h}}_{f,\text{CO}}^o,{\overline{h}}_{f,{\text{H}}_{\text{2}}}^o,\text{and}{\overline{h}}_{f,{\text{CO}}_{\text{2}}}^o$, stoichiometric coefficients of components ${\text{H}}_{\text{2}}\text{O,}\text{CO,}{\text{H}}_{\text{2}}\text{,}\text{and}{\text{CO}}_{\text{2}}$ are $v_{{\text{H}}_{\text{2}}\text{O}},v_{\text{CO}},v_{{\text{H}}_{\text{2}}},\text{and}{v}_{{\text{CO}}_{\text{2}}}$ respectively.

Write the equation to calculate the equilibrium constant for the chemical equilibrium of ideal-gas mixtures.

$K_P=e^{-\Delta G^{*}\left(T\right)/R_{u}T}$ (II)

Here, universal gas constant is $R_u$ and temperature of Gibbs function of formation is T.

Write the equation of van’t Hoff to estimate the equilibrium constant.

$\ln \left(\frac{K_{P,\text{est}}}{K_{P1}}\right)=\frac{{\overline{h}}_R}{R_u}\left(\frac{1}{T_R}-\frac{1}{T}\right)$ (III)

Here, temperature of reaction is $T_R$, equilibrium constant at temperature 2000 K is $K_{P1}$, and enthalpy of reaction at 2000 K is ${\overline{h}}_R$.

Conclusion:

From the equilibrium reaction, the values of $v_{{\text{H}}_{\text{2}}\text{O}},v_{\text{CO}},v_{{\text{H}}_{\text{2}}},\text{and}{v}_{{\text{CO}}_{\text{2}}}$ are 1, 1, 1, and 1 respectively.

Refer to Table A-26, obtain the values of ${\overline{h}}_{f,{\text{H}}_{\text{2}}\text{O}}^o,{\overline{h}}_{f,\text{CO}}^o,{\overline{h}}_{f,{\text{H}}_{\text{2}}}^o,\text{and}{\overline{h}}_{f,{\text{CO}}_{\text{2}}}^o$ as below:

$\begin{array}{l}{\overline{h}}_{f,{\text{H}}_{\text{2}}\text{O}}^o=-241,820\text{kJ/kmol}\\ {\overline{h}}_{f,\text{CO}}^o=-110,520\text{kJ/kmol}\\ {\overline{h}}_{f,{\text{H}}_{\text{2}}}^o=0\\ {\overline{h}}_{f,{\text{CO}}_{\text{2}}}^o=-393,520\text{kJ/kmol}\end{array}$

Refer to Table A-23, obtain the value of ${\overline{h}}_{{\text{H}}_{\text{2}}\text{O}}^o$ at temperature of 298 K.

${\overline{h}}_{{\text{H}}_{\text{2}}\text{O}}^o=\text{9904}\text{kJ/kmol}$

Refer to Table A-23, obtain the value of ${\overline{h}}_{{\text{H}}_{\text{2}}\text{O}}$ and ${\overline{s}}_{{\text{H}}_{\text{2}}\text{O}}$ at temperature of 2500 K.

$\begin{array}{l}{\overline{h}}_{{\text{H}}_{\text{2}}\text{O}}=\text{108,868}\text{kJ/kmol}\\ {\overline{s}}_{{\text{H}}_{\text{2}}\text{O}}=276.286\text{kJ/kmol}\cdot \text{K}\end{array}$

Refer to Table A-21, obtain the value of ${\overline{h}}_{\text{CO}}^o$ at temperature of 298 K.

${\overline{h}}_{\text{CO}}^o=\text{8669}\text{kJ/kmol}$

Refer to Table A-21, obtain the value of ${\overline{h}}_{\text{CO}}$ and ${\overline{s}}_{\text{CO}}$ at temperature of 2500 K.

$\begin{array}{l}{\overline{h}}_{\text{CO}}=\text{83692}\text{kJ/kmol}\\ {\overline{s}}_{\text{CO}}=266.755\text{kJ/kmol}\cdot \text{K}\end{array}$

Refer to Table A-22, obtain the value of ${\overline{h}}_{{\text{H}}_{\text{2}}}^o$ at temperature of 298 K.

${\overline{h}}_{{\text{H}}_{\text{2}}}^o=\text{8468}\text{kJ/kmol}$

Refer to Table A-22, obtain the value of ${\overline{h}}_{{\text{H}}_{\text{2}}}$ and ${\overline{s}}_{{\text{H}}_{\text{2}}}$ at temperature of 2500 K.

$\begin{array}{l}{\overline{h}}_{{\text{H}}_{\text{2}}}=\text{78960}\text{kJ/kmol}\\ {\overline{s}}_{{\text{H}}_{\text{2}}}=196.125\text{kJ/kmol}\cdot \text{K}\end{array}$

Refer to Table A-20, obtain the value of ${\overline{h}}_{{\text{CO}}_{\text{2}}}^o$ at temperature of 298 K.

${\overline{h}}_{{\text{CO}}_{\text{2}}}^o=\text{9364}\text{kJ/kmol}$

Refer to Table A-20, obtain the value of ${\overline{h}}_{{\text{CO}}_{\text{2}}}$ and ${\overline{s}}_{{\text{CO}}_{\text{2}}}$ at temperature of 2500 K.

$\begin{array}{l}{\overline{h}}_{{\text{CO}}_{\text{2}}}=\text{131,290}\text{kJ/kmol}\\ {\overline{s}}_{{\text{CO}}_{\text{2}}}=322.808\text{kJ/kmol}\cdot \text{K}\end{array}$

Substitute 1 for $v_{{\text{CO}}_{\text{2}}}$, 1 for $v_{{\text{H}}_{\text{2}}}$, 1 for $v_{{\text{H}}_{\text{2}}\text{O}}$, 1 for $v_{\text{CO}}$, $\text{9904}\text{kJ/kmol}$ for ${\overline{h}}_{{\text{H}}_{\text{2}}\text{O}}^o$, $\text{108,868}\text{kJ/kmol}$ for ${\overline{h}}_{{\text{H}}_{\text{2}}\text{O}}$, $\text{8669}\text{kJ/kmol}$ for ${\overline{h}}_{\text{CO}}^o$, $\text{83692}\text{kJ/kmol}$ for ${\overline{h}}_{\text{CO}}$, $\text{8468}\text{kJ/kmol}$ for ${\overline{h}}_{{\text{H}}_{\text{2}}}^o$, $\text{78960}\text{kJ/kmol}$ for ${\overline{h}}_{{\text{H}}_{\text{2}}}$, $\text{9364}\text{kJ/kmol}$ for ${\overline{h}}_{{\text{CO}}_{\text{2}}}^o$, $\text{131,290}\text{kJ/kmol}$ for ${\overline{h}}_{{\text{CO}}_{\text{2}}}$, $-241,820\text{kJ/kmol}$ for ${\overline{h}}_{f,{\text{H}}_{\text{2}}\text{O}}^o$, $-110,520\text{kJ/kmol}$ for ${\overline{h}}_{f,\text{CO}}^o$, 0 for ${\overline{h}}_{f,{\text{H}}_{\text{2}}}^o$, $-393,520\text{kJ/kmol}$ for ${\overline{h}}_{f,{\text{CO}}_{\text{2}}}^o$, 2500 K for $T_{{\text{H}}_{\text{2}}\text{O}},T_{\text{CO}},T_{{\text{H}}_{\text{2}}},\text{and}{T}_{{\text{CO}}_{\text{2}}}$, $276.286\text{kJ/kmol}\cdot \text{K}$ for ${\overline{s}}_{{\text{H}}_{\text{2}}\text{O}}$, $266.755\text{kJ/kmol}\cdot \text{K}$ for ${\overline{s}}_{\text{CO}}$, $196.125\text{kJ/kmol}\cdot \text{K}$ for ${\overline{s}}_{{\text{H}}_{\text{2}}}$, and $322.808\text{kJ/kmol}\cdot \text{K}$ for ${\overline{s}}_{{\text{CO}}_{\text{2}}}$ in Equation (I).

$\begin{array}{c}\Delta G^{*}\left(T\right)=\left[\begin{array}{c}1\left(\begin{array}{l}-393,520\text{kJ/kmol}+\left(\text{131,290}\text{kJ/kmol}-\text{9364}\text{kJ/kmol}\right)-\\ 2500\text{K}\left(322.808\text{kJ/kmol}\cdot \text{K}\right)\end{array}\right)+\\ 1\left(\begin{array}{l}0+\left(\text{78960}\text{kJ/kmol}-\text{8468}\text{kJ/kmol}\right)-\\ 2500\text{K}\left(196.125\text{kJ/kmol}\cdot \text{K}\right)\end{array}\right)-\\ 1\left(\begin{array}{l}-241,820\text{kJ/kmol}+\left(\text{108,868}\text{kJ/kmol}-\text{9904}\text{kJ/kmol}\right)-\\ 2500\text{K}\left(276.286\text{kJ/kmol}\cdot \text{K}\right)\end{array}\right)-\\ 1\left(\begin{array}{l}-110,520\text{kJ/kmol}+\left(\text{83692}\text{kJ/kmol}-\text{8669}\text{kJ/kmol}\right)-\\ 2500\text{K}\left(266.755\text{kJ/kmol}\cdot \text{K}\right)\end{array}\right)\end{array}\right]\\ =37,531\text{kJ/kmol}\end{array}$

Substitute $37,531\text{kJ/kmol}$ for $\Delta G^{*}\left(T\right)$, $2500\text{K}$ for T, and $8.314\text{kJ/kmol}\cdot \text{K}$ for $R_u$ in Equation (II).

$\begin{array}{c}{K}_P=e^{-37,531\text{kJ/kmol}/\left[\left(8.314\text{kJ/kmol}\cdot \text{K}\right)2500\text{K}\right]}\\ =0.1644\end{array}$

Thus, the equilibrium constant obtained from the equilibrium reaction at 2500 K is $\underset{\_}{0.1644}$.

Substitute $-26,176\text{kJ/kmol}$ for ${\overline{h}}_R$, $8.314\text{kJ/kmol}\cdot \text{K}$ for $R_u$, 2500 K for T, 2000 K for $T_R$, and 0.2209 for $K_{P1}$ in Equation (III).

$\begin{array}{l}\ln \left(\frac{K_{P,\text{est}}}{0.2209}\right)=\frac{-26,176\text{kJ/kmol}}{8.314\text{kJ/kmol}\cdot \text{K}}\left(\frac{1}{2000\text{K}}-\frac{1}{2500\text{K}}\right)\\ \ln \left(\frac{K_{P,\text{est}}}{0.2209}\right)=-0.31484\\ \left(\frac{K_{P,\text{est}}}{0.2209}\right)=e^{-0.31484}\\ \left(\frac{K_{P,\text{est}}}{0.2209}\right)=0.7299\end{array}$

$\begin{array}{l}{K}_{P,\text{est}}=0.7299\times 0.2209\\ K_{P,\text{est}}=0.1612\end{array}$

The value obtained for equilibrium constant at 2000 K from the definition of the equilibrium constant is 0.1612 which is smaller than the value obtained for equilibrium constant at 2500 K from the equilibrium reaction.