For a chemical reaction:

$a_1{A}_1+a_2{A}_2+a_3{A}_3+\cdot \cdot \cdot =b_1{B}_1+b_2{B}_2+b_3{B}_3+\cdot \cdot \cdot$ (12)

of which:

$A_i$, $B_j$—labels for chemical substances;

$a_i$—stoichiometric coefficients for reactants;

$b_j$—stoichiometric coefficients for products.

Thermodynamic functions $\Delta H,\Delta S,\Delta G$ at 298 K and 1.013 ∙ 10⁵ Pa are defined by the expression [18] :

$\Delta H={\displaystyle \underset{j}{\sum }{b}_j\cdot \Delta h_j-{\displaystyle \underset{i}{\sum }{a}_i\cdot \Delta h_i}}$ (13)

$\Delta S={\displaystyle \underset{j}{\sum }{b}_j\cdot s_j-{\displaystyle \underset{i}{\sum }{a}_i\cdot s_i}}$ (14)

$\Delta G={\displaystyle \underset{j}{\sum }{b}_j\cdot \Delta g_j-{\displaystyle \underset{i}{\sum }{a}_i\cdot \Delta g_i}}$ (15)

of which:

$a_i$—the number of kilomoles of the i-th reactant components;

$b_{ј}$—the number of kilomoles of the j-th component for products;

$\Delta h_i$—bond enthalpy of the i-th component;

$\Delta h_j$—bond enthalpy of the j-th component;

$s_i$—specificentropies and connections of the i-th component;

$s_j$—specific entropies and connections of the j-th component;

$\Delta g_i$—specific free enthalpies of the i-th component;

$\Delta g_j$—specific free enthalpies of the j-th component.

The dependence of enthalpy, entropy and free enthalpy of reaction (12) on temperature are defined by the expressions:

$\Delta H_T=\Delta H_{298}+{\displaystyle \underset{298}{\overset{T}{\int }}\Delta c_{mp}\left(T\right)\text{d}T}$ (16)

$\Delta S_T=\Delta S_{298}+{\displaystyle \underset{298}{\overset{T}{\int }}\frac{\Delta c_{mp}\left(T\right)}{T}\text{d}T}$ (17)

$\Delta G=\Delta H-T\cdot \Delta S$ (18)

of which:

$\Delta c_{mp}={\displaystyle \underset{j}{\sum }{b}_j\cdot c_{mp}{}_j}-{\displaystyle \underset{i}{\sum }{b}_i\cdot c_{m{p}_i}}$ (19)

the sum of the specific molar heat capacities of the components.

$c_{mp}\left(T\right)=a+b\cdot T+c\cdot T^2+d\cdot T^3+e\cdot T^4,\text{kJ}/\left(\text{kmol}\cdot \text{K}\right)$ (20)

dependence of molar heat capacity on temperature.

$a,b,c,d,e,f$—polynomial coefficients $c_{mp}\left(T\right)$.

If $\Delta G>0$, the reaction proceeds from right to left, i.e. in the direction of the formation of reactants of the reaction. If $\Delta G<0$ the reaction proceeds from left to right, i.e. in the direction of the formation of reaction product.

Values of enthalpy, entropy and free enthalpy of reaction components $\text{CO}+{\text{H}}_{\text{2}}\text{O}={\text{CO}}_2+{\text{H}}_{\text{2}}$ at 298 K and 1.013 × 10⁵ Pa are shown in Table 1 , in Table 2 the values of the polynomial coefficients (20) are shown.

For a chemical reaction:

${\displaystyle \underset{i}{\sum }{a}_i\cdot A_i}={\displaystyle \underset{j}{\sum }{b}_j\cdot B_j}$ (21)

The chemical equilibrium constant expressed in terms of partial pressures is:

$K_p=\frac{{\displaystyle \underset{j}{\prod }{\left(p_{B_j}\right)}^{b_j}}}{{\displaystyle \underset{i}{\prod }{\left(p_{A_i}\right)}^{a_i}}}$ (22)

Value of chemical equilibrium constant ${K^{\prime }}_p$ reduced to pressure $p_0=1.013\times {10}^5\text{Pa}$ is determined by the expression:

${K^{\prime }}_p={\text{e}}^{-\frac{\Delta G}{R_u\cdot T}}=K_p\cdot p_0^{-\left({\displaystyle \underset{j}{\sum }{b}_j-{\displaystyle \underset{i}{\sum }{a}_i}}\right)}$ (23)

of which:

$R_u=8.314\text{kJ}/\left(\text{kmol}\cdot \text{K}\right)$—universal gas constant.

Using numerical thermodynamic data for the pure components involved in the reaction $\text{CO}+{\text{H}}_{\text{2}}\text{O}={\text{CO}}_2+{\text{H}}_{\text{2}}$ at 298 K and 1.013 × 10⁵ Pa ( Table 1 and Table 2 ) and using expressions from (13) to (23) the values of the thermodynamic functions $\Delta H,\Delta S,\Delta G,{K^{\prime }}_p$ reaction can be calculated considered depending on the reaction temperature ( Table 3 ).

Reaction equilibrium constant $\text{CO}+{\text{H}}_{\text{2}}\text{O}={\text{CO}}_2+{\text{H}}_{\text{2}}$ (reaction (10)) can also be determined by combining Equations (2) and (3), i.e.

$K_p=\frac{K_{p3}}{K_{p2}}.$ (24)

Substituting Equations (6) and (5) into Equation (24) gives:

$\begin{array}{c}\log K_p=-36.72508+3994.704/T-4.462408\times {10}^{-3}\cdot T\\ +0.6718146\times {10}^{-6}\cdot T^2+12.220277\cdot \log T\end{array}$ (25)

concluding that the equilibrium reaction constant is $\text{CO}+{\text{H}}_{\text{2}}\text{O}={\text{CO}}_2+{\text{H}}_{\text{2}}$:

$K_p={10}^{-36.72508+3994.704/T-4.462408\times {10}^{-3}\cdot T+0.6718146\times {10}^{-6}\cdot T^2+12.220277\cdot \log T}$ (26)

Values of reaction equilibrium constant $\text{CO}+{\text{H}}_{\text{2}}\text{O}={\text{CO}}_2+{\text{H}}_{\text{2}}$ obtained using expression (23) is in agreement with the values of the equilibrium constant presented in the literature (Equation (26)) ( Table 3 ).

In the temperature interval 298 K to 1500 K, thermodynamic functions $\Delta H$ and $\Delta S$ are negative, so the sign is $\Delta G$ determined by the relative ratio of the enthalpy and entropy terms (Equation (18)) ( Figure 1 ). This means that the reaction temperature is the decisive factor for the thermodynamic equilibrium of the reaction under consideration. In the temperature range 298 K to ≈1090 K, the free reaction enthalpy is less than zero ( $\Delta G<0$) and the equilibrium reaction constant under consideration is very large $K_p\gg 1$ ( Figure 2 ), which means that the reaction is shifted towards the formation of reaction products. Above 1090 K, the reaction enters an unfavorable area and the reaction equilibrium shifts towards the formation of reaction reactants.

The values of the thermodynamic functions of the considered reaction shown in Table 3 are in agreement with data from the literature [21] [22] .

Balance of reaction components:

$\text{CO}+{\text{H}}_{\text{2}}\text{O}={\text{CO}}_2+{\text{H}}_{\text{2}}$ (27)

can be formulated as follows.

● $\text{CO}$ $n_{\text{CO}}=a-y,\text{kmol}$ (28)

● ${\text{H}}_{\text{2}}\text{O}$ $n_{{\text{H}}_{\text{2}}\text{O}}=b-y,\text{kmol}$ (29)

● ${\text{CO}}_{\text{2}}$ $n_{{\text{CO}}_{\text{2}}}=y,\text{kmol}$ (30)

● ${\text{H}}_{\text{2}}$ $n_{{\text{H}}_{\text{2}}}=y,\text{kmol}$ (31)

of which:

a—the number of kilomoles of carbon monoxide that enters the reaction (27);

b—the number of kilomoles of water (water vapor) that enters the reaction (27);

y—the number of kilomoles of carbon dioxide or hydrogen in the mixture after establishing chemical equilibrium balance.

The total number of kilomoles in the mixture at any conversion time is equal to:

$n_{\sum }=n_{\text{CO}}+n_{{\text{H}}_{\text{2}}\text{O}}+n_{{\text{CO}}_{\text{2}}}+n_{{\text{H}}_{\text{2}}}$

$n_{\sum }=\left(a-y\right)+\left(b-y\right)+y+y=a+b,\text{kmol}$ (32)

The molar fraction of the components in the mixture after establishing chemical equilibrium is:

$y_{\text{CO}}=\frac{n_{\text{CO}}}{n_{\sum }}=\frac{a-y}{a+b},\text{kmol}/\text{kmol}$ (33)

$y_{{\text{H}}_{\text{2}}\text{O}}=\frac{n_{{\text{H}}_{\text{2}}\text{O}}}{n_{\sum }}=\frac{b-y}{a+b},\text{kmol}/\text{kmol}$ (34)

$y_{{\text{CO}}_{\text{2}}}=\frac{n_{{\text{CO}}_{\text{2}}}}{n_{\sum }}=\frac{y}{a+b},\text{kmol}/\text{kmol}$ (35)

$y_{{\text{H}}_{\text{2}}}=\frac{n_{{\text{H}}_{\text{2}}}}{n_{\sum }}=\frac{y}{a+b},\text{kmol}/\text{kmol}$ (36)

The partial pressures of the components in an equilibrium mixture are:

$p_{\text{CO}}=y_{\text{CO}}\cdot p=\frac{a-y}{a+b}\cdot p,\text{Pa}$ (37)

$p_{{\text{H}}_{\text{2}}\text{O}}=y_{{\text{H}}_{\text{2}}\text{O}}\cdot p=\frac{b-y}{a+b}\cdot p,\text{Pa}$ (38)

$p_{{\text{CO}}_{\text{2}}}=y_{{\text{CO}}_{\text{2}}}\cdot p=\frac{y}{a+b}\cdot p,\text{Pa}$ (39)

$p_{{\text{H}}_{\text{2}}}=y_{{\text{H}}_{\text{2}}}\cdot p=\frac{y}{a+b}\cdot p,\text{Pa}$ (40)

of which:

p—total pressure in the reactor space after equilibrium is established, Pa.

By changing partial pressures $p_{{\text{CO}}_2}$, $p_{{\text{H}}_{\text{2}}}$, $p_{\text{CO}}$ and $p_{{\text{H}}_{\text{2}}\text{O}}$ into equation (22) the equilibrium reaction constant $\text{CO}+{\text{H}}_{\text{2}}\text{O}={\text{CO}}_2+{\text{H}}_2$ is given by the expression:

$K_p=\frac{p_{{\text{CO}}_{\text{2}}}\cdot p_{{\text{H}}_{\text{2}}}}{p_{\text{CO}}\cdot p_{{\text{H}}_{\text{2}}\text{O}}}=\frac{\frac{y}{a+b}\cdot p\cdot \frac{y}{a+b}\cdot p}{\frac{a-y}{a+b}\cdot p\cdot \frac{b-y}{a+b}\cdot p}$

and after rearranging the previous expression, we obtain a quadratic equation of the form:

$\left(K_p-1\right)\cdot y^2-K_p\cdot \left(a+b\right)\cdot y+ab\cdot K_p=0$ (41)

By solving equation (40) for the unknown quantity y two solutions are obtained:

$y_{1/2}=\frac{\left(a+b\right)\cdot K_p\pm \sqrt{{\left(a+b\right)}^2\cdot K_p^2-4\cdot ab\cdot K_p\cdot \left(K_p-1\right)}}{2\cdot \left(K_p-1\right)},\left(K_p\ne 1\right)$ (42)

That solution is taken y for which mole fractions $y_{\text{CO}}$, $y_{{\text{CO}}_2}$, $y_{{\text{H}}_{\text{2}}\text{O}}$ and $y_{{\text{H}}_{\text{2}}}$ they make physical sense. At an equimolar ratio $a=b=1\text{kmol}$ from equation (42) we get:

$y_{1/2}=\frac{K_p\pm \sqrt{K_p}}{K_p-1},\left(K_p\ne 1\right)$ (43)

Under the given conditions $0<y<1$ equation (42) takes the form:

$y=\frac{K_p-\sqrt{K_p}}{K_p-1},\left(K_p\ne 1\right)$ (44)

Degree of conversion of reactants CO and H₂O is determined using the expres-sion:

${\eta }_{\text{CO}}=\frac{a-\left(a-y\right)}{a}=\frac{y}{a},\text{kmol}/\text{kmol}$ (45)

${\eta }_{{\text{H}}_{\text{2}}\text{O}}=\frac{b-\left(b-y\right)}{b}=\frac{y}{b},\text{kmol}/\text{kmol}$ (46)

Results of the calculation of the composition of the equilibrium reaction mixture $\text{CO}+{\text{H}}_{\text{2}}\text{O}={\text{CO}}_2+{\text{H}}_{\text{2}}$ at an equimolar ratio of reactants in the temperature range 298 K to 1500 K and pressure $p=1.013\times {10}^5\text{Pa}$ are shown in Table 4 and graphical dependence on Figure 3 . It can be seen that at an ambient temperature of 298K, the degree of conversion of reactants into products is 99.69% ( Figure 4 ). Since the reaction $\text{CO}+{\text{H}}_{\text{2}}\text{O}={\text{CO}}_2+{\text{H}}_{\text{2}}$ occurs without changing the number of moles Δn = 0 (Equation (23)), changing the pressure in the system has no effect on changing the equilibrium composition. The equilibrium constant depends only on temperature (Equation (26)).