Thermodynamic Analysis of Ammonia Synthesis

The paper researches the effect of temperature on ammonia synthesis. The ammonia synthesis reaction ${\text{N}}_2{}_{\left(\text{g}\right)}+3{\text{H}}_2{}_{\left(\text{g}\right)}=2{\text{NH}}_3{}_{\left(\text{g}\right)}$ is exothermic with a negative entropy change. $\Delta G<0$ condition is fulfilled at lower temperatures up to 464 K. It is a constant equilibrium of the ammonia synthesis reaction ${K^{\prime }}_p\gg 1$ at lower temperatures, which means that the NH _{3( g) )} synthesis reaction is shifted in the direction of NH _3(g) formation (higher production of ammonia). The downside of lowering the temperature is that more ammonia is obtained, but the reaction rate slows down. Above 464K, the free enthalpy of the NH _3(g) synthesis reaction is greater than zero, so the reaction enters thermodynamically unfavorable conditions. By increasing the reaction temperature, the ammonia yield NH _3(g) decreases in the equilibrium mixture. At 400 K, it is 0.5128 kmol/kmol (51.28%) and at 900 K, the synthesis process NH _3(g) is practically complete.

• Temperature
• Thermodynamic Functions
• Synthesis
• Ammonia

1. Introduction

Ammonia is a colorless gas with a pungent odor. It dissolves well in water. At ambient temperature, ammonia NH_3(g) is stable. Ammonia is mainly used for the production of nitric acid and mineral fertilizers, carbamide, ammonium sulfate and others. Ammonia gas reacts quickly with sulfuric acid vapors SO_3(g) and H₂O_(g) creating ammonium sulfate (NH₄)₂SO_4(s) which is a high-quality artificial fertilizer. Gaseous ammonia is also used for the reduction of SO_3(g) and NO_x(g) in flue gases, and the reduction ranges from 77% - 97% [1] . In the literature [2] - [7] there are data on the synthesis of ammonia

${\text{N}}_2{}_{\left(\text{g}\right)}+3{\text{H}}_2{}_{\left(\text{g}\right)}=2{\text{NH}}_3{}_{\left(\text{g}\right)}$. (1)

These data are not complete. They relate to a certain temperature and pressure. Data on the effect of temperature on the composition of the equilibrium reaction mixture (1) are also incomplete. In engineering practice, the Haber-Bosch process is most often used for the production of ammonia [8] [9] .

Knowledge of the value of thermodynamic functions $\Delta H,\Delta S,\Delta G$ and $K_p$, as well as knowing the composition of the equilibrium reaction mixture (1) is the starting point in the design phase of devices and apparatus for the reduction of sulfur and nitrogen oxides in flue gases, the production of mineral fertilizers and polymer materials, which gives significance to this paper.

2. Mathematical Formulation

Thermodynamic Functions

The enthalpy of a chemical reaction is defined as follows [10] [11] :

$\Delta H={\displaystyle \underset{j}{\sum }{b}_j\cdot \Delta h_j}-{\displaystyle \underset{i}{\sum }{b}_i\cdot \Delta h_i},$ (2)

of which:

$b_i$—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.

The dependence of reaction enthalpy on temperature is given by the expression:

$\Delta H_T=\Delta H_{298}+{\displaystyle \underset{298}{\overset{T}{\int }}\Delta c_{mp}\left(T\right)\text{d}T},$ (3)

of which:

$\Delta c_{mp}={\displaystyle \underset{j}{\sum }{b}_j\cdot c_{mp}{}_j}-{\displaystyle \underset{i}{\sum }{b}_i\cdot c_{m{p}_i}},$ (4)

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

The dependence of molar heat capacity on temperature can be shown in the form of a polynomial:

$c_{mp}\left(T\right)=a+b\cdot {10}^{-3}\cdot T+c\cdot {10}^5\cdot T^{-2},$ (5)

of which:

$a,b,c$—polynomial coefficients that are determined experimentally,

T-absolute temperature.

The entropy of a reaction is defined by the expression:

$\Delta S={\displaystyle \underset{j}{\sum }{b}_j\cdot s_j}-{\displaystyle \underset{i}{\sum }{b}_i\cdot s_i},$ (6)

of which:

$s_i$—specific entropies and connections of the i-th component.

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

The dependence of reaction entropy on temperature is given by the expression:

$\Delta S_T=\Delta S_{298}+{\displaystyle \underset{298}{\overset{T}{\int }}\frac{\Delta c_{mp}\left(T\right)}{T}\text{d}T}$ (7)

The free enthalpy of reaction is defined by the expression:

$\Delta G={\displaystyle \underset{j}{\sum }{b}_j\cdot \Delta g_j}-{\displaystyle \underset{i}{\sum }{b}_i\cdot \Delta g_i}$ (8)

of which:

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

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

The dependence of the free reaction enthalpy on the reaction enthalpy, temperature and entropy of the reaction is given by the expression:

$\Delta G=\Delta H-T\cdot \Delta S$ (9)

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

For a chemical reaction:

${\displaystyle \underset{i}{\sum }{a}_i\cdot A_i}={\displaystyle \underset{j}{\sum }{b}_j\cdot B_j}$ (10)

the chemical equilibrium constant expressed through partial pressures is:

$K_p=\frac{{\displaystyle \underset{j}{\prod }{\left(p_{B_j}\right)}^{b_j}}}{{\displaystyle \underset{i}{\prod }{\left(p_{A_i}\right)}^{a_i}}}.$ (11)

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

${K^{\prime }}_p={\text{e}}^{-\frac{\Delta G}{R_u\cdot T}}=K_p\cdot p_0{}^{-\left({\displaystyle \underset{j}{\sum }{b}_j}-{\displaystyle \underset{i}{\sum }{a}_i}\right)}$ (12)

of which:

$R_u=8.314\frac{\text{kJ}}{\text{kmol}\cdot \text{K}}$—universal gas constant.

Using numerical thermodynamic data for pure components that figure in the reaction ${\text{N}}_2{}_{\left(\text{g}\right)}+3{\text{H}}_2{}_{\left(\text{g}\right)}=2{\text{NH}}_3{}_{\left(\text{g}\right)}$ at 298K and 1.013 × 10⁵ Pa ( Table 1 and Table 2 ) and using expressions from (2) to (12), it is possible to calculate the thermodynamic functions $\Delta H,\Delta S,\Delta G,{K^{\prime }}_p$ of the considered reaction depending on the reaction temperature.

The results of the calculation of the thermodynamic functions $\Delta H,\Delta S,\Delta G,$ ${K^{\prime }}_p,K_p$ are shown in Table 3 and are in agreement with data from the literature [14] [15] . In the temperature interval 298 K to 1000 K, the thermodynamic

Table 1

Table 1. Bond enthalpies, free enthalpies and components with entropy at 1.013 × 10 5 Pa and 298 K [12] .

Substance	Δh (kJ/kmol)	Δg (kJ/kmol)	s (kJ/kmol∙K)
Nitrogen	0	0	191.50
Hydrogen	0	0	130.57
Ammonia	−46,110	−16,480	192.34

Table 2

Table 2. Numerous values of a, b, c coefficients for determining the value of heat capacities of components [13] .

Substance	a	b	c	Temperature range (K)
Nitrogen	27.884	4.270	−	298 - 2500
Hydrogen	27.298	3.266	0.502	298 - 3000
Ammonia	29.768	25.121	−1.549	298 - 1800

Table 3

Table 3. Values of enthalpy, entropy and free enthalpy of the reaction temperature N 2 ( g ) + 3 H 2 ( g ) = 2 NH 3 ( g ) .

T (K)	ΔH (kJ)	ΔS (kJ/K)	ΔG (kJ)	${K^{\prime }}_p$ -	$K_p$ (Pa−2)		$\ln {K^{\prime }}_p$-
298	−92,220	−198.53	−33,058	6.23 × 105	6.07 × 10−5	13.34
300	−92,018	−196.10	−33,187	6.01 × 105	5.85 × 10−5	13.31
400	−95,819	−208.03	−12,604	4.43 × 101	4.31 × 10−9	3.79
500	−99,224	−216.10	8827	1.20 × 10−1	1.16 × 10−11	−2.12
600	−102,246	−221.87	30,874	2.05 × 10−3	2.0 × 10−13	−6.19
700	−104,892	−226.10	53,378	1.04 × 10−4	1.01 × 10−14	−9.17
800	−107,165	−229.24	76,224	1.05 × 10−5	1.03 × 10−15	−11.46
900	−109,066	−231.54	99,325	1.72 × 10−6	1.67 × 10−16	−13.27
1000	−110,597	−233.21	122,612	3.94 × 10−7	3.84 × 10−17	−14.75

Figure 1

Figure 1. Determination of the reaction area for an exothermic reaction N 2 ( g ) + 3 H 2 ( g ) = 2 NH 3 ( g ) depending on the reaction temperature.

functions ΔH and ΔS are negative, so the sign ΔG is determined by the relative ratio of the enthalpy and entropy terms (Equation (9)) ( Figure 1 ). This means that reaction temperature is a decisive factor for the thermodynamic balance of the synthesis reaction NH_3(g). In the temperature interval of 298 K to 464 K, the equilibrium constant of the considered reaction is very large ${K^{\prime }}_p\gg 1$, which means that the reaction is shifted in the direction of product formation NH_3(g). At a reaction temperature above 464 K, the chemical equilibrium constant ${K^{\prime }}_p\ll 1$, so the equilibrium of the reaction shifts in the direction of the formation of reactants of the reaction ( Figure 2 ).These considerations are in line with Le Chatelier's principle. For an exothermic reaction (ΔH < 0), a decrease in temperature shifts the equilibrium of the reaction to the product side.

Figure 2

Figure 2. Dependence of the reaction equilibrium constant N 2 ( g ) + 3 H 2 ( g ) = 2 NH 3 ( g ) from the temperature.

3. Calculation of the Composition of the Equilibrium Mixture of the Synthesis Reaction NH 3(g)

Balance equations of reactants in the reaction:

${\text{N}}_2{}_{\left(\text{g}\right)}+3{\text{H}}_2{}_{\left(\text{g}\right)}=2{\text{NH}}_3{}_{\left(\text{g}\right)},$ (13)

can be written as follows:

nitrogen $n_{{\text{N}}_2{}_{\left(\text{g}\right)}}=a-y,\left(\text{kmol}\right)\text{,}$ (14)

hydrogen $n_{{\text{H}}_2{}_{\left(\text{g}\right)}}=b-3y,\left(\text{kmol}\right)\text{,}$ (15)

ammonia $n_{{\text{NH}}_3{}_{\left(\text{g}\right)}}=2y,\left(\text{kmol}\right),$ (16)

of which:

a—The number of kilomoles of nitrogen entering the reaction (13);

b—The number of kilomoles of hydrogen reacting (13);

2y—The number of kilomoles of ammonia in the mixture after establishing chemical equilibrium.

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

$n_{\sum }=n_{{\text{N}}_{\text{2}}\left(\text{g}\right)}+n_{{\text{H}}_{\text{2}}\left(\text{g}\right)}+n_{{\text{NH}}_{\text{3}}\left(\text{g}\right)},$

$n_{\sum }=\left(a-y\right)+\left(b-3y\right)+2y=a+b-2y.$ (17)

The mole fraction of the components in the mixture after the establishment of chemical equilibrium amounts to:

$y_{{\text{N}}_{\text{2}}\left(\text{g}\right)}=\frac{n_{{\text{N}}_{\text{2}}\left(\text{g}\right)}}{n_{\sum }}=\frac{a-y}{a+b-2y},\left(\text{kmol}/\text{kmol}\right)\text{,}$ (18)

$y_{{\text{H}}_{\text{2}}\left(\text{g}\right)}=\frac{n_{{\text{H}}_{\text{2}}\left(\text{g}\right)}}{n_{\sum }}=\frac{b-3y}{a+b-2y},\left(\text{kmol}/\text{kmol}\right)\text{,}$ (19)

$y_{{\text{NH}}_{\text{3}}\left(\text{g}\right)}=\frac{n_{{\text{NH}}_{\text{3}}\left(\text{g}\right)}}{n_{\sum }}=\frac{2y}{a+b-2y},\left(\text{kmol}/\text{kmol}\right)\text{.}$ (20)

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

$p_{{\text{N}}_{\text{2}}\left(\text{g}\right)}=y_{{\text{N}}_{\text{2}}\left(\text{g}\right)}\cdot p=\frac{a-y}{a+b-2y}\cdot p,\left(\text{Pa}\right)\text{,}$ (21)

$p_{{\text{H}}_{\text{2}}\left(\text{g}\right)}=y_{{\text{H}}_{\text{2}}\left(\text{g}\right)}\cdot p=\frac{a-3y}{a+b-2y}\cdot p,\left(\text{Pa}\right)\text{,}$ (22)

$p_{{\text{NH}}_{\text{3}}\left(\text{g}\right)}=y_{{\text{NH}}_{\text{3}}\left(\text{g}\right)}\cdot p=\frac{2y}{a+b-2y}\cdot p,\left(\text{Pa}\right)\text{,}$ (23)

of which:

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

The chemical equilibrium constant of the reaction (13) is determined using the expression:

$K_p=\frac{p_{{\text{NH}}_{\text{3}}\left(\text{g}\right)}^2}{p_{{\text{N}}_{\text{2}}\left(\text{g}\right)}\cdot p_{{\text{H}}_{\text{2}}\left(\text{g}\right)}^3},\left({\text{Pa}}^{-2}\right).$ (24)

By changing Equations (21) to (23) into Equation (24) and solving for the unknown y, the equation is obtained:

$A{y}^4+B{y}^3+C{y}^2+Dy+E=0,$ (25)

of which:

$A=27p^2{K}_p-16,$

$B=\left(a+b\right)\cdot \left(16-27p^2{K}_p\right),$

$C=9b{p}^2{K}_p\cdot \left(b+3a\right)-4\cdot {\left(a+b\right)}^2,$

$D=-b^2{p}^2{K}_p\cdot \left(9a+b\right),$

$E=a{b}^3{p}^2{K}_p.$

By substituting the solution of Equation (25) into expressions (21) to (23), the numerical values of the mole fractions of the components in the equilibrium mixture are obtained. Degree of responsiveness of reactants N_2(g) and H_2(g) is determined using the expression:

${\eta }_{{\text{N}}_{\text{2}}\left(\text{g}\right)}=\frac{a-\left(a-y\right)}{a}=\frac{y}{a},\left(\text{kmol}/\text{kmol}\right)$ (26)

${\eta }_{{\text{H}}_{\text{2}}\left(\text{g}\right)}=\frac{b-\left(b-3y\right)}{b}=\frac{3y}{b},\left(\text{kmol}/\text{kmol}\right)$ (27)

4. The Results of the Calculation of the Composition of the Equilibrium Mixture of the Reaction N 2 ( g ) + 3 H 2 ( g ) = 2 N H 3 ( g )

The results of the calculation of the composition of the equilibrium mixture at the stoichiometric ratio of reactants, at $T=400\text{K}$ and pressure $p=1.013\times {10}^5\text{Pa}$ are shown in Table 4 and the graphic dependence in Figure 3 .

Table 4

Table 4. Composition of the equilibrium reaction N 2 ( g ) + 3 H 2 ( g ) = 2 NH 3 ( g ) mixture at the stoichiometric number of moles of reactants ( a = 1 kmol , b = 3 kmol ) at pressure 1.013 × 10 5 Pa.

T (K)	$y_{{\text{N}}_{\text{2}}}$ (kmol/kmol)	$y_{{\text{H}}_2}$ (kmol/kmol)	$y_{{\text{NH}}_{\text{3}}}$ (kmol/kmol)	${\eta }_{{\text{N}}_{\text{2}}}$ (kmol/kmol)	${\eta }_{{\text{H}}_{\text{2}}}$ (kmol/kmol)
400	0.1218	0.3654	0.5128	0.6779	0.6779
500	0.2269	0.6808	0.0922	0.1690	0.1690
600	0.2464	0.7393	0.0143	0.0282	0.0282
700	0.2492	0.7475	0.0033	0.0066	0.0066
800	0.2498	0.7493	0.0010	0.0020	0.0020
900	0.2499	0.7496	0.0005	0.0010	0.0010
1000	0.2499	0.7486	0.0005	0.0010	0.0010

Figure 3

Figure 3. Mole fraction of components in the equilibrium reaction mixture N 2 ( g ) + 3 H 2 ( g ) = 2 NH 3 ( g ) depending on the temperature.

It can be seen that ammonia synthesis starts at lower temperatures. At 400 K, the mole fraction of ammonia in the equilibrium mixture is 0.5128 kmol/kmol. At higher temperatures, the mole fraction of ammonia decreases, while at 900 K, the mole fraction of ammonia is equal to 0.0005 kmol/kmol. The degree of conversion (reaction) N_2(g) and H_2(g) depending on the temperature is shown in Figure 4 . Apparently, the increase in the reaction temperature and mole fraction shares N_2(g) and H_2(g) in the equilibrium reaction ${\text{N}}_2{}_{\left(\text{g}\right)}+3{\text{H}}_2{}_{\left(\text{g}\right)}=2{\text{NH}}_3{}_{\left(\text{g}\right)}$ mixture, the degree of conversion of the reactants decreases. At 400 K, the degree of conversion of reactants is as much as 0.6779 kmol/kmol, and at 900 K, the degree of conversion is 0.0010 kmol/kmol. This practically means that the synthesis process NH_3(g) at 900 K is completed because it amounts to only 0.0005 kmol/kmol.

Figure 4

Figure 4. Degree of conversion N 2(g) and H 2(g) in the equilibrium reaction mixture N 2 ( g ) + 3 H 2 ( g ) = 2 NH 3 ( g ) depending on the temperature.

5. Conclusion

The paper presents the thermodynamic analysis of ammonia synthesis and the following conclusions were reached:

This paper can be used like initial basis of phase designing devices and apparatus in various branches process techniques, chemical engineering as well as for a better understanding of the process of combustion and synthesis of the multicomponent systems.

Further research into ammonia synthesis should be directed at determining the speed of the considered reaction. It is also necessary to investigate the influence of different industrial catalysts on the speed of the ammonia synthesis reaction and their application in industrial practice.

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