# System with chemical reactions

The change in the chemical composition in a system discussed in the preceding subsection can result from a chemical reaction. During a chemical reaction, some of the chemical bonds binding the atoms into molecules are broken, and new ones are formed. Since the chemical energies associated with the chemical bonds in reactants and products are generally different, the resulting change in chemical energy and its effect on the overall energy balance of the system must be accounted for. For a single-phase reacting system, the change of energy can be due to the change of sensible internal energy associated with temperature and pressure change. It may also reflect changes in chemical energy associated with chemical reactions. The reacting system starts with the mixture of reactants, and the chemical reaction in the system produces new components that will coexist in the mixture. Therefore, a system undergoing chemical reaction can be considered as a mixture that contains both reactants and products. In practical applications, one particular kind of chemical reaction, namely combustion, is particularly important. Combustion is a chemical reaction during which a fuel is oxidized and a large amount of thermal energy is released. Most fuels (such as coal, gasoline, and diesel fuel) consist of hydrogen and carbon, and are called hydrocarbon fuels. The oxidant gas for most combustion processes is air, which can be treated as a mixture of 21% oxygen and 79% nitrogen (each kmol of oxygen in air is accompanied by 0.79/0.21=3.76 kmol of nitrogen). For a given reaction, the chemical equation establishes the relationship between the mole numbers of the reactants consumed and the mole numbers of the products generated. For example, combustion of 1 kmol of methane with air that contains 2 kmol of oxygen can be represented by the following chemical equation:

${\rm{C}}{{\rm{H}}_{\rm{4}}}{\rm{ + 2}}{{\rm{O}}_{\rm{2}}}{\rm{ + 7}}{\rm{.52}}{{\rm{N}}_{\rm{2}}} \to {\rm{C}}{{\rm{O}}_{\rm{2}}}{\rm{ + 2}}{{\rm{H}}_{\rm{2}}}{\rm{O + 7}}{\rm{.52}}{{\rm{N}}_{\rm{2}}}\qquad \qquad(1)$

where nitrogen is present on both sides of the equation and is a non-reacting species that is carried to the products. At high combustion temperature, a small amount of nitrogen N2 is oxidized to nitrogen oxides (NO, NO2). However, this amount is negligibly small as far as the overall combustion reaction is concerned. Combustion is complete if all of the carbon in the fuel burns to carbon dioxide and all of the hydrogen burns to water. The combustion in eq. (1) is a complete combustion. On the other hand, the combustion is incomplete if the product contains any unburned fuel or C, H2, or CO. Common causes of incomplete combustion include an insufficient supply of oxygen and inadequate mixing between fuel and oxygen. The first law of thermodynamics is a general law that applies to any process, including combustion. The first law of thermodynamics for a closed system is expressed as

$\delta Q = dE + pdV\qquad \qquad(2)$

or, for a finite process,

$Q = ({E_2} - {E_1}) + \int_{{V_1}}^{{V_2}} {pdV} \qquad \qquad(3)$

where Q is the heat transfer between the system and its surroundings, $\int_{{V_1}}^{{V_2}} {pdV}$ is the work done by the system on its surroundings, and E1 and E2 are total internal energies, including sensible and latent internal energy and chemical energy, before and after chemical reaction. Equation (3) is a general expression that is applicable to any combustion process. Combustion occurring under two specific conditions is of the greatest interest for practical applications: constant-volume and constant-pressure. For combustion at constant volume, the heat transfer is

$Q = {E_2} - {E_1}\qquad \qquad(4)$

When combustion occurs at constant pressure, eq. (3) becomes

$Q = ({E_2} - {E_1}) + p({V_2} - {V_1}) = {H_2} - {H_1}\qquad \qquad(5)$

Equation (5) is also valid for combustion in an open system provided that H1 and H2 represent enthalpy at the inlet and exit, respectively, of the system.

## Chemical Equilibrium

Before the chemical equilibrium theory was established, it was believed that all chemical reactions would proceed until all reactants were completely converted into products. In fact, chemical reactions proceed only until they reach an equilibrium state, referred to as chemical equilibrium. At the equilibrium state, the chemical reaction then proceeds incrementally in both directions, so that there is no net change in composition. Under these specific conditions, and if the chemical equilibrium is stable, the equilibrium will not change with time. Chemical equilibrium is another cause of incomplete combustion, and cannot be prevented. At chemical equilibrium, the combustion of methane can be represented by

$C{H_4} + 2{O_2} + 7.52{N_2} [] C{O_2} + 2{H_2} + 7.52{N_2} \qquad \qquad(6)$

where the two arrows in eq. (6) indicate that the reaction takes place in both directions at any equilibrium state. Therefore, chemical reaction does not cease at chemical equilibrium, but the reaction rate is the same in both directions. Consequently, there is no notable change of composition for a reacting system at chemical equilibrium. For a system containing Nr reactants and Np products, the generalized chemical equation can be expressed as

${a_{r1}}{A_{r1}} + {a_{r2}}{A_{r2}} + ... + {a_{r{N_r}}}{A_{r{N_r}}} [] {a_{p1}}{A_{p1}} + {a_{p2}}{A_{p2}} + ... + {a_{p{N_p}}} {A_{p{N_p}}}\qquad \qquad(7)$

where ${A_{ri}}{\rm{ }}(i = 1,2 \cdots {N_r})$ are chemical symbols for the reactants and Api $(i = 1,2 \cdots {N_p})$ are chemical symbols for the products, with their corresponding stoichiometric coefficients ar,i and ap,i. In the chemical reaction shown in eq. (6), the total numbers of reactants and products are Nr = 3 and Np = 3, respectively. The chemical symbols of reactants and products are Ar1 = CH4, Ar2 = O2, Ar3 = N2, Ap1 = CO2, Ap2 = H2O, and Ap3 = N2. The coefficients preceding the chemical symbols in eq. (7) are referred to as stoichiometric coefficients. These coefficients describe the proportion of the mole numbers of reactants disappearing and mole numbers of products appearing during the reaction process. The stoichiometric coefficients for reactants CH4,O2 and N2 are 1, 2, and 7.52, respectively. Equation (7) can also be written as a more compact form (Bejan, 1997):

$0 [] \sum\limits_{i=1}^{{N_r}+{N_p}} {{a_i}{A_i}} \qquad \qquad(8)$

Where

${a_i} = \left\{ {\begin{array}{*{20}{c}} { - {a_{ri}}} & {i = 1,2, \cdots {N_r}} \\ {{a_{p(i - {N_r})}}} & {i = {N_r} + 1,{N_r} + 2, \cdots {N_r} + {N_p}} \\ \end{array}} \right.\qquad \qquad(9)$

and

${A_i} = \left\{ {\begin{array}{*{20}{c}} {{A_{ri}}} & {i = 1,2, \cdots {N_r}} \\ {{A_{p(i - {N_r})}}} & {i = {N_r} + 1,{N_r} + 2, \cdots {N_r} + {N_p}} \\ \end{array}} \right.\qquad \qquad(10)$

During a chemical reaction process, the decreasing mole number of reactants and increasing mole number of products must be proportional to the corresponding stoichiometric coefficients. If a chemically-reacting system initially contains Nr reactants and the mole number of the ith reactant is $n_i^0$, when the chemical reaction reaches chemical equilibrium, the mole number of the ith reactant becomes ${n_{ri}} = n_{ri}^0 - {a_{ri}}\zeta {\rm{ (}}i = 1,2, \cdots ,{N_r})$ and the mole number of the ith product is ${n_{pi}} = {a_{pi}}\zeta {\rm{ }}(i = 1,2, \cdots ,{N_p}).$ In the above notation, ζ indicates the degree of advancement of the chemical reaction (i.e., ζ = 0 means no reaction and a very large ζ represents a large mole number of products). The maximum value ζmax, is reached when at least one of the reactants is exhausted. When the system is at chemical equilibrium with a degree of advancement ζ, and the mole number of each of the reactants and products is represented by ${n_i}{\rm{ }}(i = 1,2, \cdots {N_r},{N_r} + 1, \cdots {N_r} + {N_p})$, a slight advancement of the chemical reaction will bring the system to a new equilibrium state represented by ζ + dζ, in which state the new mole numbers of each of the components become ${n_i} + d{n_i}{\rm{ }}(i = 1,2, \cdots {N_r},$ ${N_r} + 1, \cdots {N_r} + {N_p}).$ The change of mole number of each component is then

$d{n_i} = {a_i}d\zeta \begin{array}{*{20}{c}} {} & {i = 1,2, \cdots {N_r},{N_r} + 1, \cdots {N_r} + {N_p}} \\ \end{array}\qquad \qquad(11)$

The change in internal energy for the chemically-reacting system can be obtained by substituting eq. (11) into eq. $dE = TdS - pdV + \sum\limits_{i = 1}^N {{\mu _i}d{n_i}}$ from Closed Systems with Compositional Change, i.e.,

$dE = TdS - pdV + \left( {\sum\limits_{i = 1}^{{N_r} + {N_p}} {{\mu _i}{a_i}} } \right)d\zeta \qquad \qquad(12)$

Introducing De Donder’s affinity function (Bejan, 1997),

$Y = - \sum\limits_{i = 1}^{{N_r} + {N_p}} {{\mu _i}{a_i}} \qquad \qquad(13)$

eq. (12) becomes

$dE = TdS - pdV - Yd\zeta \qquad \qquad(14)$

The affinity function is a linear combination of the chemical potentials of reactants and products; therefore, the affinity function itself is a property of the chemically reacting system. Equation (14) suggests that the internal energy of a chemically reactive system is a function of entropy, volume, and degree of affinity, i.e.,

$E = E(S,V,\zeta )\qquad \qquad(15)$

Expanding eq. (15) in terms of each independent variable while holding all other properties constant produces the following:

$dE = {\left( {\frac{{\partial E}}{{\partial S}}} \right)_{V,\zeta }}dS + {\left( {\frac{{\partial E}}{{\partial V}}} \right)_{S,\zeta }}dV + {\left( {\frac{{\partial E}}{{\partial \zeta }}} \right)_{S,V}}d\zeta \qquad \qquad(16)$

Comparing the third terms on the right hand side of eqs. (14) and (16) yields

$Y = - {\left( {\frac{{\partial E}}{{\partial \zeta }}} \right)_{S,V}}\qquad \qquad(17)$

Other representations of the fundamental relation for chemically-reactive systems can be directly obtained from eq. (14) by using the definitions of enthalpy, Helmholtz free energy, and Gibbs free energy, i.e.,

$dH = Vdp + TdS - Yd\zeta \qquad \qquad(18)$

$dF = - SdT - pdV - Yd\zeta \qquad \qquad(19)$

$dG = Vdp - SdT - Yd\zeta \qquad \qquad(20)$

It can be readily determined from eqs. (18) – (20) that other expressions of chemical equilibrium are

$Y = - {\left( {\frac{{\partial H}}{{\partial \zeta }}} \right)_{S,p}} = - {\left( {\frac{{\partial F}}{{\partial \zeta }}} \right)_{T,V}} = - {\left( {\frac{{\partial G}}{{\partial \zeta }}} \right)_{T,p}}\qquad \qquad(21)$

For a typical process wherein pressure and temperature are constant, the equilibrium condition requires that [see eq. dGT,p = 0 from Equilibrium Criteria for Pure Substances]

$d{G_{T,p}} = 0\qquad \qquad(22)$

Comparison of eqs. (22) and (20) reveals that for a chemical reaction occurring at constant pressure and temperature, the degree of affinity at equilibrium is zero.

$Y = - \sum\limits_{i = 1}^{{N_r} + {N_p}} {{\mu _i}{a_i}} = 0\qquad \qquad(23)$

In order for the chemical equilibrium of the reacting system with constant pressure and temperature to be stable, it is necessary for the Gibbs free energy to satisfy eq. $\Delta {G_{T,p}} \le 0$ from Equilibrium Criteria for Pure Substances as well, i.e., the Gibbs free energy must be at its minimum as shown in Fig. 1. Mathematically, the condition for stability can be expressed as

${\left( {\frac{{{\partial ^2}G}}{{\partial {\zeta ^2}}}} \right)_{T,p}} > 0 \qquad \qquad (24)$

Figure 1 Chemical equilibrium at constant temperature and pressure.

Substituting eq. (21) into eq. (24), the condition for stability becomes

${\left( {\frac{{\partial Y}}{{\partial \zeta }}} \right)_{T,p}} < 0\qquad \qquad(25)$

$\begin{array}{l} Y = - \sum\limits_{i = 1}^{3 + 3} {{\mu _i}{a_i}} \\ {\rm{ }} = - ({\mu _{{\rm{C}}{{\rm{H}}_{\rm{4}}}}}{a_{{\rm{C}}{{\rm{H}}_{\rm{4}}}}} + {\mu _{{{\rm{O}}_{\rm{2}}}}}{a_{{{\rm{O}}_{\rm{2}}}}} + {\mu _{{{\rm{N}}_{\rm{2}}}}}{a_{{{\rm{N}}_{\rm{2}}}}} + {\mu _{{\rm{C}}{{\rm{O}}_{\rm{2}}}}}{a_{{\rm{C}}{{\rm{O}}_{\rm{2}}}}} + {\mu _{{{\rm{H}}_{\rm{2}}}{\rm{O}}}}{a_{{{\rm{H}}_{\rm{2}}}{\rm{O}}}} + {\mu _{{{\rm{N}}_{\rm{2}}}}}{a_{{{\rm{N}}_{\rm{2}}}}}) \\ {\rm{ = }}{\mu _{{\rm{C}}{{\rm{H}}_{\rm{4}}}}} + 2{\mu _{{{\rm{O}}_{\rm{2}}}}} + 7.52{\mu _{{{\rm{N}}_{\rm{2}}}}} - {\mu _{{\rm{C}}{{\rm{O}}_{\rm{2}}}}} - 2{\mu _{{{\rm{H}}_{\rm{2}}}{\rm{O}}}} - 7.52{\mu _{{{\rm{N}}_{\rm{2}}}}} = 0 \\ \end{array}$

Thus the chemical potentials of the reactants and products at equilibrium satisfy

${\mu _{{\rm{C}}{{\rm{H}}_{\rm{4}}}}} + 2{\mu _{{{\rm{O}}_{\rm{2}}}}} + 7.52{\mu _{{{\rm{N}}_{\rm{2}}}}} = {\mu _{{\rm{C}}{{\rm{O}}_{\rm{2}}}}} + 2{\mu _{{{\rm{H}}_{\rm{2}}}{\rm{O}}}} + 7.52{\mu _{{{\rm{N}}_{\rm{2}}}}}$

It is possible to define an equilibrium constant for chemical reactions from knowledge of the standard-state Gibbs function change. The equilibrium constant can be used to determine the equilibrium composition of chemical reactions (Cengel and Boles, 2008). Considering a generic chemical reaction represented by eq. (7), the change in Gibbs free energy is equal to the difference between the chemical potentials of the products and the reactants. To satisfy chemical equilibrium, the sum of the Gibbs energies of the reactants must be equal to the sum of the Gibbs energies of the products:

${a_{r1}}{\mu _{r1}} + {a_{r2}}{\mu _{r2}} + ... + {a_{r{N_r}}}{\mu _{r{N_r}}} = {a_{p1}}{\mu _{p1}} + {a_{p2}}{\mu _{p2}} + ... + {a_{p{N_p}}}{\mu _{p{N_p}}}\qquad \qquad(26)$

When the system with constant temperature and pressure slightly departs from equilibrium, the change of Gibbs free energy is:

$d{G_{T,p}} = \sum\limits_{i = 1}^{{N_r} + {N_p}} {{\mu _i}{a_i}} \qquad \qquad(27)$

For a 2-reactant and 2-product reaction, eq. (27) gives:

$d{G_{T,p}} = {a_{r1}}{\mu _{r1}} + {a_{r2}}{\mu _{r2}} - {a_{p1}}{\mu _{p1}} - {a_{p2}}{\mu _{p2}}\qquad \qquad(28)$

Defining chemical potential in terms of activity:

$\mu = {\mu ^*} + {R_u}T\ln \left[ A \right]\qquad \qquad(29)$

and substituting eq. (29) into eq. (28) yield:

$\begin{array}{l} d{G_{T,p}} = \left( {{a_{r1}}\mu _{r1}^* + {a_{r2}}\mu _{r2}^*} \right) - \left( {{a_{p1}}\mu _{p1}^* + {a_{p2}}\mu _{p2}^*} \right) \\ + \left( {{a_{r1}}{R_u}T\ln \left[ {{A_{r1}}} \right] + {a_{r2}}{R_u}T\ln \left[ {{A_{r2}}} \right] - {a_{p1}}{R_u}T\ln \left[ {{A_{p1}}} \right] - {a_{p2}}{R_u}T\ln \left[ {{A_{p2}}} \right]} \right) \\ \end{array}$

which simplifies to:

$d{G_{T,p}} = \sum\limits_{i = 1}^{{N_r} + {N_p}} {\mu _i^*{a_i}} + {R_u}T\ln \frac{{{{\left[ {{A_{r1}}} \right]}^{{a_{r1}}}}{{\left[ {{A_{r2}}} \right]}^{{a_{r2}}}}}}{{{{\left[ {{A_{p1}}} \right]}^{{a_{p1}}}}{{\left[ {{A_{p2}}} \right]}^{{a_{p2}}}}}}$

The required condition for chemical equilibrium at constant pressure and temperature is dGT,p = 0, i.e.

$\sum\limits_{i = 1}^{{N_r} + {N_p}} {\mu _i^*{a_i}} + {R_u}T\ln \frac{{{{\left[ {{A_{r1}}} \right]}^{{a_{r1}}}}{{\left[ {{A_{r2}}} \right]}^{{a_{r2}}}}}}{{{{\left[ {{A_{p1}}} \right]}^{{a_{p1}}}}{{\left[ {{A_{p2}}} \right]}^{{a_{p2}}}}}} = 0$

which is rewritten as:

$\Delta {G^*} = - {R_u}T\ln \frac{{{{\left[ {{A_{r1}}} \right]}^{{a_{r1}}}}{{\left[ {{A_{r2}}} \right]}^{{a_{r2}}}}}}{{{{\left[ {{A_{p1}}} \right]}^{{a_{p1}}}}{{\left[ {{A_{p2}}} \right]}^{{a_{p2}}}}}} = - {R_u}T\ln \left[ {K_P^o} \right]$

where ΔG * is the standard state Gibbs function change and the equilibrium constant $K_P^o$ is defined as:

$K_P^o = \frac{{[{A_{r1}}]}^{{a_{r1}}}{[{A_{r2}}]}^{{a_{r2}}}}{{[{A_{p1}}]}^{{a_{p1}}}{[{A_{p2}}]}^{{a_{p2}}}} = e^{- \Delta {G^*}/{R_u}T} \qquad \qquad (30)$

## References

Bejan, A., 1997, Advanced Engineering Thermodynamics, 2nd ed., John Wiley & Sons, New York, NY.

Cengel, Y.A., and Boles, M.A., 2008, Thermodynamics – An Engineering Approach, 6th ed., McGraw-Hill, New York, NY.

Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Elsevier, Burlington, MA.