Miloslav Pekař

*Brno University of Technology Czech Republic* 

## **1. Introduction**

672 Thermodynamics – Interaction Studies – Solids, Liquids and Gases

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Thermodynamics has established in chemistry principally as a science determining possibility and direction of chemical transformations and giving conditions for their final, equilibrium state. Thermodynamics is usually thought to tell nothing about rates of these processes, their velocity of approaching equilibrium. Rates of chemical reactions belong to the domain of chemical kinetics. However, as thermodynamics gives some restriction on the course of chemical reactions, similar restrictions on their rates are continuously looked for. Similarly, because thermodynamic potentials are often formulated as driving forces for various processes, a thermodynamic driving force for reactions rates is searched for.

Two such approaches will be discussed in this article. The first one are restrictions put by thermodynamics on values of rate constants in mass action rate equations. The second one is the use of the chemical potential as a general driving force for chemical reactions and also "directly" in rate equations. These two problems are in fact connected and are related to expressing reaction rate as a function of pertinent independent variables.

Relationships between chemical thermodynamics and kinetics traditionally emerge from the ways that both disciplines use to describe equilibrium state of chemical reactions (chemically reacting systems or mixtures in general). Equilibrium is the main domain of classical, equilibrium, thermodynamics that has elaborated elegant criteria (or, perhaps, definitions) of equilibria and has shown how they naturally lead to the well known equilibrium constant. On the other hand, kinetics describes the way to equilibrium, i.e. the nonequilibrium state of chemical reactions, but also gives a clear idea on reaction equilibrium. Combining these two views various results on compatibility between thermodynamics and kinetics, on thermodynamic restrictions to kinetics etc. were published. The main idea can be illustrated on the trivial example of decomposition reaction AB = A + B with rate (kinetic) equation AB A B *r kc kc c* where *r* is the reaction rate, , *k k* are the forward and reverse rate constants, and *c* are the concentrations. In equilibrium, the reaction rate is zero, consequently A B AB eq *k k cc c* / / . Because the right hand side corresponds to the thermodynamic equilibrium constant (*K*) it is concluded that / *K kk* . However, this is simplified approach not taking into account conceptual differences between the true thermodynamic equilibrium constant and the ratio of rate constants that is called here the kinetic equilibrium constant. This discrepancy is sometimes to be removed by restricting this approach to ideal systems of elementary reactions but even then some questions remain.

Thermodynamics and Reaction Rates 675

Scheme 1. Connecting thermodynamics and kinetics correctly (Eckert et al., 1986)

2 eq CO eq

(3)

Equilibrium composition (*a*eq, *c*eq)

Substitution into kinetic equation *r*(*c*eq) = 0

Kinetic equilibrium constant

*k c c k c* 

It is clear that thermodynamic and kinetic equilibrium constants need not be equivalent even in ideal systems. For example, the former does not contain concentration of carbon and though this could be remedied by stating that carbon amount does not affect reaction rate and its concentration is included in the reverse rate constant, even then the kinetic equilibrium constant could depend on carbon amount in contrast to the thermodynamic equilibrium constant. Some discrepancies could not be remedied by restricting on elementary reactions only – in this example the presence of *p*rel and of the total molar amount, generally, the presence of quantities transforming composition variables into standard state-related (activity-related) variables, and, of course, discrepancy in

Let us use the same example to illustrate the procedure suggested by Eckert et al. (1986). At 1300 K and 202 kPa the molar standard Gibbs energies are (Novák et al., 1999): *G . <sup>m</sup>*(CO) 395 3 kJ/mol , 2 *Gm*(CO ) 712 .7 kJ/mol , (C) 20 97 kJ/mol *G . <sup>m</sup>* and from them the value of thermodynamic equilibrium constant is calculated: *K* = 0.00515. Equilibrium molar balance gives CO eq C eq <sup>2</sup> ( ) () *n nx* , CO eq ( ) 12 *n x* , *n x* 1 . Then from (2) follows *x* = 0.0107 (Novák et al., 1999). Equilibrium composition is substituted

On contrary, the ratio of rate constants is given by

Thermodynamic data (e.g. Δ*Gf* )

dimensionalities of the two equilibrium constants.

into (3):

CO C <sup>2</sup>

Kinetic data (rate coefficients/constants)

Chemical potential () is introduced into chemical kinetics by similar straightforward way (Qian & Beard, 2005). If it is expressed by *RT c* ln , multiplied by stoichiometric coefficients, summed and compared with rate equation it is obtained for the given example that:

$$
\Delta\mu \equiv -\mu\_{\rm AB} + \mu\_{\rm A} + \mu\_{\rm B} = RT \ln \frac{\mathcal{C}\_{\rm A} \mathcal{C}\_{\rm B}}{\mathcal{K} \mathcal{c}\_{\rm AB}} = RT \ln \left( \bar{r} / \bar{r} \right) \tag{1}
$$

(note that the equivalence of thermodynamic and kinetic equilibrium constants is supposed again; *r r*, are the forward and reverse rates). Equation (1) used to be interpreted as determining the (stoichiometric) sum of chemical potentials () to be some (thermodynamic) "driving force" for reaction rates. In fact, there is "no kinetics", no kinetic variables in the final expression *RT r r* ln / and reaction rates are directly determined by chemical potentials what is questionable and calls for experimental verification.

## **2. Restrictions put by thermodynamics on values of rate constants**

#### **2.1 Basic thermodynamic restrictions on rate constants coming from equilibrium**

Perhaps the only one work which clearly distinguishes kinetic and thermodynamic equilibrium constant is the kinetic textbook by Eckert and coworkers (Eckert et al., 1986); the former is in it called the empirical equilibrium constant. This book stresses different approaches of thermodynamics and kinetics to equilibrium. In thermodynamics, equilibrium is defined as a state of minimum free energy (Gibbs energy) and its description is based on stoichiometric equation and thermodynamic equilibrium constant containing activities. Different stoichiometric equations of the same chemical equation can give different values of thermodynamic equilibrium constant, however, equilibrium composition is independent on selected stoichiometric equation. Kinetic description of equilibrium is based on zero overall reaction rate, on supposed reaction mechanism or network (reaction scheme) and corresponding kinetic (rate) equation. Kinetic equilibrium constant usually contains concentrations. According to that book, thermodynamic equilibrium data should be introduced into kinetic equations indirectly as shown in the Scheme 1.

Simple example reveals basic problems. Decomposition of carbon monoxide occurs (at the pressure *p*) according to the following stoichiometric equation:

$$\text{'2 CO} = \text{CO}\_2 + \text{C} \tag{\text{R1}}$$

Standard state of gaseous components is selected as the ideal gas at 101 kPa and for solid component as the pure component at the actual pressure (due to negligible effects of pressure on behavior of solid components, the dependence of the standard state on pressure can be neglected here). Ideal behavior is supposed. Then *a* = *p*/*p*° = *p*rel *n*/*n* for = CO, CO2, where *p*rel = *p*/*p*°, and *a*C = 1; *a* is the activity, *p* is the partial pressure, *p*° the standard pressure, *n* is the number of moles, and *n* the total number of moles. Thermodynamic equilibrium constant is then given by

$$K = \left(\frac{n\_{\Sigma}n\_{\text{CO}\_2}}{p\_{\text{rel}}n\_{\text{CO}}^2}\right)\_{\text{eq}} = \left[\frac{(c\_{\text{CO}} + c\_{\text{CO}\_2})c\_{\text{CO}\_2}}{p\_{\text{rel}}c\_{\text{CO}}^2}\right]\_{\text{eq}}\tag{2}$$

Chemical potential () is introduced into chemical kinetics by similar straightforward way

coefficients, summed and compared with rate equation it is obtained for the given example

(note that the equivalence of thermodynamic and kinetic equilibrium constants is supposed

determining the (stoichiometric) sum of chemical potentials () to be some (thermodynamic) "driving force" for reaction rates. In fact, there is "no kinetics", no kinetic

by chemical potentials what is questionable and calls for experimental verification.

**2.1 Basic thermodynamic restrictions on rate constants coming from equilibrium**  Perhaps the only one work which clearly distinguishes kinetic and thermodynamic equilibrium constant is the kinetic textbook by Eckert and coworkers (Eckert et al., 1986); the former is in it called the empirical equilibrium constant. This book stresses different approaches of thermodynamics and kinetics to equilibrium. In thermodynamics, equilibrium is defined as a state of minimum free energy (Gibbs energy) and its description is based on stoichiometric equation and thermodynamic equilibrium constant containing activities. Different stoichiometric equations of the same chemical equation can give different values of thermodynamic equilibrium constant, however, equilibrium composition is independent on selected stoichiometric equation. Kinetic description of equilibrium is based on zero overall reaction rate, on supposed reaction mechanism or network (reaction scheme) and corresponding kinetic (rate) equation. Kinetic equilibrium constant usually contains concentrations. According to that book, thermodynamic equilibrium data should

**2. Restrictions put by thermodynamics on values of rate constants** 

be introduced into kinetic equations indirectly as shown in the Scheme 1.

pressure *p*) according to the following stoichiometric equation:

Simple example reveals basic problems. Decomposition of carbon monoxide occurs (at the

Standard state of gaseous components is selected as the ideal gas at 101 kPa and for solid component as the pure component at the actual pressure (due to negligible effects of pressure on behavior of solid components, the dependence of the standard state on pressure can be neglected here). Ideal behavior is supposed. Then *a* = *p*/*p*° = *p*rel *n*/*n* for = CO, CO2, where *p*rel = *p*/*p*°, and *a*C = 1; *a* is the activity, *p* is the partial pressure, *p*° the standard pressure, *n* is the number of moles, and *n* the total number of moles. Thermodynamic

> CO2 2 CO CO CO2 2 2 rel CO eq rel CO eq

*nn c c c* ( ) *<sup>K</sup> p n p c* 

 

A B

*RT r r* ln / and reaction rates are directly determined

2 CO = CO2 + C (R1)

(2)

(1)

AB ln ln / *c c RT RT r r*

*Kc*

are the forward and reverse rates). Equation (1) used to be interpreted as

*RT c* ln , multiplied by stoichiometric

 

AB A B

 

(Qian & Beard, 2005). If it is expressed by

variables in the final expression

equilibrium constant is then given by

 

that:

again; *r r*,

Scheme 1. Connecting thermodynamics and kinetics correctly (Eckert et al., 1986) On contrary, the ratio of rate constants is given by

$$
\left(\frac{\vec{k}}{\vec{k}}\right)\_{\text{eq}} = \left(\frac{c\_{\text{CO}\_2}c\_{\text{C}}}{c\_{\text{CO}}^2}\right)\_{\text{eq}}\tag{3}
$$

It is clear that thermodynamic and kinetic equilibrium constants need not be equivalent even in ideal systems. For example, the former does not contain concentration of carbon and though this could be remedied by stating that carbon amount does not affect reaction rate and its concentration is included in the reverse rate constant, even then the kinetic equilibrium constant could depend on carbon amount in contrast to the thermodynamic equilibrium constant. Some discrepancies could not be remedied by restricting on elementary reactions only – in this example the presence of *p*rel and of the total molar amount, generally, the presence of quantities transforming composition variables into standard state-related (activity-related) variables, and, of course, discrepancy in dimensionalities of the two equilibrium constants.

Let us use the same example to illustrate the procedure suggested by Eckert et al. (1986). At 1300 K and 202 kPa the molar standard Gibbs energies are (Novák et al., 1999): *G . <sup>m</sup>*(CO) 395 3 kJ/mol , 2 *Gm*(CO ) 712 .7 kJ/mol , (C) 20 97 kJ/mol *G . <sup>m</sup>* and from them the value of thermodynamic equilibrium constant is calculated: *K* = 0.00515. Equilibrium molar balance gives CO eq C eq <sup>2</sup> ( ) () *n nx* , CO eq ( ) 12 *n x* , *n x* 1 . Then from (2) follows *x* = 0.0107 (Novák et al., 1999). Equilibrium composition is substituted into (3):

Thermodynamics and Reaction Rates 677

 *f* – 1 = (*Qr/K* – 1) (*c*, *T, uj*) (8) in the neighbourhood of *Qr*/*K* = 1 (i.e., of equilibrium); *uj* stands for a set of nonthermodynamic variables. Example of practical application of Hollingsworth's approach in

Blum (Blum & Luus, 1964) considered a general mass action rate law formulated as follows:

where is some function of activities, *a*, of reacting species,

which seems to be improbable (rather strong inhibition by reactant).

**2.3 Independence of reactions, Wegscheider conditions** 

1 1

 

which may differ from the stoichiometric coefficients (), in fact, reaction orders. Supposing that both the equilibrium constant and the ratio of the rate constants are dependent only on

/ *<sup>z</sup> kk K* (10)

*z n* ( ) / ; 1, ,

General law (9) is rarely used in chemical kinetics, in reactions of ions it probably does not work (Laidler, 1965; Boudart, 1968). It can be transformed, particularly simply in ideal systems, to concentrations. Samohýl (personal communication) pointed out that criteria (11) may be problematic, especially for practically irreversible reactions. For example, reaction orders for reaction 4 NH3 + 6 NO = 5 N2 + 6 H2O were determined as follows: NH3

of practically irreversible nature of the reaction. Natural selection could be, e.g.,

(reaction is not inhibited by reactant), then *z* = 1/12 and from this follows NH3

Wegscheider conditions belong also among "thermodynamic restrictions" on rate constants and have been introduced more than one hundred years ago (Wegscheider, 1902). In fact, they are also based on equivalence between thermodynamic and kinetic equilibrium constants disputed in previous sections. Recently, matrix algebra approaches to find these conditions were described (Vlad & Ross, 2009). Essential part of them is to find (in)dependent chemical reactions. Problem of independent and dependent reactions is an interesting issue sometimes found also in studies on kinetics and thermodynamics of reacting mixtures. As a rule, a reaction scheme, i.e. a set of stoichiometric equations (whether elementary or nonelementary), is proposed, stoichiometric coefficients are arranged into stoichiometric matrix and linear (matrix) algebra is applied to find its rank which determines the number of linearly (stoichiometrically) independent reactions; all other reactions can be obtained as linear combinations of independent ones. This procedure can be viewed as an a posteriori analysis of the proposed reaction mechanism or network. Bowen has shown (Bowen, 1968) that using not only matrix but also vector algebra interesting results can be obtained on the basis of knowing only components of reacting

 

   

 

(9)

 and 

(11)

are coefficients

1 ,

NO 0

2/3

0 . Orders for reversed direction are unknown, probably because

*m m rk a k a* 

an ideal system is given by Boyd (Boyd, 1977).

temperature, they proved that

NO 0.5 , N HO 2 2 

where

$$
\left(\frac{\vec{k}}{k}\right)\_{\text{eq}} = \frac{0.0107 \times 0.0107}{0.09786} = 0.012\tag{4}
$$

and this is real and true result of thermodynamic restriction on values of rate constants valid at given temperature. More precisely, this is a restriction put on the ratio of rate constants, values of which are supposed to be independent on equilibrium, in other words, dependent on temperature (and perhaps on pressure) only and therefore this restriction is valid also out of equilibrium at given temperature. The numerical value of this restriction is dependent on temperature and should be recalculated at every temperature using the value of equilibrium constant at that temperature.

Thus, simple and safe way how to relate thermodynamics and kinetics, thermodynamic and kinetic equilibrium constants, and rate constants is that shown in Scheme 1. However, it gives no general equations and should be applied specifically for each specific reaction (reacting system) and reaction conditions (temperature, at least). There are also works that try to resolve relationship between the two types of equilibrium constant more generally and, in the same time, correctly and consistently. They were reviewed previously and only main results are presented here, in the next section. But before doing so, let us note that kinetic equilibrium constant can be used as a useful indicator of the distance of actual state of reacting mixture from equilibrium and to follow its approach to equilibrium. In the previous example, actual value of the fraction 2 <sup>2</sup> *ccc* CO C CO / can be compared with the value of the ratio /*k k* and relative distance from equilibrium calculated, for more details and other examples see our previous work (Pekař & Koubek, 1997, 1999, 2000).

#### **2.2 General thermodynamic restrictions on rate constants**

As noted in the preceding section there are several works that do not rely on simple identification of thermodynamic and kinetic equilibrium constants. Hollingsworth (1952a, 1952b) generalized restriction on the ratio of forward and reverse reaction rates (*f*) defined by

$$f(\mathbf{c}\_{a'},T) = \overline{f}(\mathbf{c}\_{a'},T) / \langle \overline{f}(\mathbf{c}\_{a'},T) \equiv \overline{r} / \overline{r} \tag{5}$$

Hollingsworth showed that sufficient condition for consistent kinetic and thermodynamic description of equilibrium is

$$F(Q\_\prime, T) = \Phi(Q\_\prime \%) \text{ and } \Phi(1) = 1 \tag{6}$$

where *F* is the function *f* with transformed variables, ( ,) ( ,) *FQ T f c T <sup>r</sup>* , and *Qr* is the well known reaction quotient. The first equality in (6) says that function *F* should be expressible as a function of *Qr/K*. This is too general condition saying explicitly nothing about rate constants. Identifying kinetic equilibrium constant with thermodynamic one, condition (6) is specialized to

$$\Phi(\mathbb{Q}\_{\delta}/\mathbb{K}) = (\mathbb{Q}\_{\delta}/\mathbb{K})^{\cdot \cdot z} \tag{7}$$

where *z* is a positive constant. Equation (7) is a generalization of simple identity / *K kk* from introduction. Hollingsworth also derived the necessary consistency condition:

$$f - 1 = (Q\_\vartheta \mathcal{K} - 1) \text{ } \Psi(\mathbf{c}\_w \ T, u\_\flat) \tag{8}$$

in the neighbourhood of *Qr*/*K* = 1 (i.e., of equilibrium); *uj* stands for a set of nonthermodynamic variables. Example of practical application of Hollingsworth's approach in an ideal system is given by Boyd (Boyd, 1977).

Blum (Blum & Luus, 1964) considered a general mass action rate law formulated as follows:

$$r = \bar{k}\phi \prod\_{a=1}^{m} a\_{\alpha}^{\alpha\_a} - \bar{k}\phi \prod\_{a=1}^{m} a\_{\alpha}^{\alpha\_a'} \tag{9}$$

where is some function of activities, *a*, of reacting species, and are coefficients which may differ from the stoichiometric coefficients (), in fact, reaction orders. Supposing that both the equilibrium constant and the ratio of the rate constants are dependent only on temperature, they proved that

$$
\vec{k} \,/\, \vec{k} = \mathbf{K}^z \tag{10}
$$

where

676 Thermodynamics – Interaction Studies – Solids, Liquids and Gases

0.09786

and this is real and true result of thermodynamic restriction on values of rate constants valid at given temperature. More precisely, this is a restriction put on the ratio of rate constants, values of which are supposed to be independent on equilibrium, in other words, dependent on temperature (and perhaps on pressure) only and therefore this restriction is valid also out of equilibrium at given temperature. The numerical value of this restriction is dependent on temperature and should be recalculated at every temperature using the value of

Thus, simple and safe way how to relate thermodynamics and kinetics, thermodynamic and kinetic equilibrium constants, and rate constants is that shown in Scheme 1. However, it gives no general equations and should be applied specifically for each specific reaction (reacting system) and reaction conditions (temperature, at least). There are also works that try to resolve relationship between the two types of equilibrium constant more generally and, in the same time, correctly and consistently. They were reviewed previously and only main results are presented here, in the next section. But before doing so, let us note that kinetic equilibrium constant can be used as a useful indicator of the distance of actual state of reacting mixture from equilibrium and to follow its approach to equilibrium. In the

As noted in the preceding section there are several works that do not rely on simple identification of thermodynamic and kinetic equilibrium constants. Hollingsworth (1952a, 1952b) generalized restriction on the ratio of forward and reverse reaction rates (*f*) defined

Hollingsworth showed that sufficient condition for consistent kinetic and thermodynamic

 *F*(*Qr*, *T*) = (*Qr/K*) and (1) = 1 (6)

known reaction quotient. The first equality in (6) says that function *F* should be expressible as a function of *Qr/K*. This is too general condition saying explicitly nothing about rate constants. Identifying kinetic equilibrium constant with thermodynamic one, condition (6) is

where *z* is a positive constant. Equation (7) is a generalization of simple identity / *K kk*

from introduction. Hollingsworth also derived the necessary consistency condition:

and relative distance from equilibrium calculated, for more details and

0.0107 0.0107 0.012

(4)

<sup>2</sup> *ccc* CO C CO / can be compared with the value

(5)

(*Qr/K*) = (*Qr/K*)–*z* (7)

, and *Qr* is the well

2

equilibrium constant at that temperature.

previous example, actual value of the fraction 2

other examples see our previous work (Pekař & Koubek, 1997, 1999, 2000).

**2.2 General thermodynamic restrictions on rate constants** 

 ( , ) ( , )/ ( , ) / *f c T f c T f cT rr* 

where *F* is the function *f* with transformed variables, ( ,) ( ,) *FQ T f c T <sup>r</sup>*

of the ratio /*k k*

description of equilibrium is

specialized to

by

eq

*k k*

$$z = (\alpha\_a' - \alpha\_a) / \,\text{v}\_a; \quad \alpha = 1, \dots, n \tag{11}$$

General law (9) is rarely used in chemical kinetics, in reactions of ions it probably does not work (Laidler, 1965; Boudart, 1968). It can be transformed, particularly simply in ideal systems, to concentrations. Samohýl (personal communication) pointed out that criteria (11) may be problematic, especially for practically irreversible reactions. For example, reaction orders for reaction 4 NH3 + 6 NO = 5 N2 + 6 H2O were determined as follows: NH3 1 , NO 0.5 , N HO 2 2 0 . Orders for reversed direction are unknown, probably because of practically irreversible nature of the reaction. Natural selection could be, e.g., NO 0 (reaction is not inhibited by reactant), then *z* = 1/12 and from this follows NH3 2/3 which seems to be improbable (rather strong inhibition by reactant).

#### **2.3 Independence of reactions, Wegscheider conditions**

Wegscheider conditions belong also among "thermodynamic restrictions" on rate constants and have been introduced more than one hundred years ago (Wegscheider, 1902). In fact, they are also based on equivalence between thermodynamic and kinetic equilibrium constants disputed in previous sections. Recently, matrix algebra approaches to find these conditions were described (Vlad & Ross, 2009). Essential part of them is to find (in)dependent chemical reactions. Problem of independent and dependent reactions is an interesting issue sometimes found also in studies on kinetics and thermodynamics of reacting mixtures. As a rule, a reaction scheme, i.e. a set of stoichiometric equations (whether elementary or nonelementary), is proposed, stoichiometric coefficients are arranged into stoichiometric matrix and linear (matrix) algebra is applied to find its rank which determines the number of linearly (stoichiometrically) independent reactions; all other reactions can be obtained as linear combinations of independent ones. This procedure can be viewed as an a posteriori analysis of the proposed reaction mechanism or network. Bowen has shown (Bowen, 1968) that using not only matrix but also vector algebra interesting results can be obtained on the basis of knowing only components of reacting

Thermodynamics and Reaction Rates 679

that component molar masses and rates should form two perpendicular vectors, i.e. vectors with vanishing scalar product. Let us introduce *n*-dimensional vector space, called the component space and denoted by U, with base vectors **e** and reciprocal base vectors **e**

1, 2,..., *n*). Then the vector of molar masses **M** and the vector of reaction rates **J** are defined in

1 1 , *n n M J* 

To proceed further we use relations (14) and (15) because in contrast to relations (12) and (13) the matrix ║*S*║ is of "full rank" (does not contain linearly dependent rows). The

> 1 1 1 1 1 1 . . . 0 *n h n h n n MS J M S J*

where the latter equality follows using (14). Because the matrix ║*S*║ has rank *h*, the

 

e e

product of the two vectors can be then expressed in the following form:

1

lies in the complementary orthogonal subspace V, **J** V.

 

   

 

*α α*

**MJ e e e e** (17)

; 1,2, , *<sup>n</sup> <sup>α</sup> S h*

e 1

that appear in (17) are linearly independent and thus form a basis of a *h*-dimensional subspace W of the space U (remember that *h* < *n*). This subspace unambiguously determines complementary orthogonal subspace V (of dimension *n*–*h*), i.e. U = V W, V W. From (17)

> *h M*

which shows that **M** can be expressed in the basis of the subspace W or **M** W. From (14)

0; 1,2, ,*h* **J.f**

which means that **J** is perpendicular to all basis vectors of the subspace W, consequently, **J**

Let us now select basis vectors in the subspace V and denote them **d***p*, *p* = 1, 2,..., *n*–*h*. Of course, these vectors lie also in the (original) space U and can be expressed using its basis

> 1 *<sup>n</sup> p p <sup>P</sup>*

> > 1

*<sup>n</sup> p p S P*

Because of orthogonality of subspaces V and W, their bases conform to equation

 0

 

**M eJ e** (16)

**f e** (18)

**M f** (19)

(20)

**d e** (21)

**f .d** (22)

 

 this space as follows:

vectors

follows:

and (16)2 follows:

vectors analogically to (16):

( =

mixture, i.e. with no reaction scheme. This is a priori type of analysis and is used in continuum nonequilibrium (rational) thermodynamics. Because Bowen's results are important for this article they are briefly reviewed now for reader's convenience.

Let a reacting mixture be composed from *n* components (compounds) which are formed by *z* different atoms. Atomic composition of each component is described by numbers *T* that indicate the number of atoms (= 1, 2,..., *z*) in component (= 1, 2,..., *n*). Atomic masses *M*<sup>a</sup> in combination with these numbers determine the molar masses *M*:

$$M\_{\alpha} = \sum\_{\sigma=1}^{x} M\_{\text{a}}^{\sigma} T\_{\sigma \alpha} \tag{12}$$

Although compounds are destroyed or created in chemical reactions the atoms are preserved. If *J* denotes the number of moles of the component formed or reacted per unit time in unit volume, i.e. the reaction rate for the component (component rate in short), then the persistence of atoms can be formulated in the form

$$\sum\_{\alpha=1}^{n} T\_{\sigma\alpha} f^{\alpha} = 0; \qquad \sigma = 1, 2, \dots, z \tag{13}$$

This result expresses, in other words, the mass conservation.

Atomic numbers can be arranged in matrix ║*T*║ of dimension *z n*. Chemical reactions are possible if its rank (*h*) is smaller than the number of components (*n*), otherwise the system (13) has only trivial solution, i.e. is valid only for zero component rates. If *h* < *z* then a new *h n* matrix ║*S*║ with rank *h* can be constructed from the original matrix ║*T*║ and used instead of it:

$$\sum\_{\alpha=1}^{n} S\_{\alpha \alpha} f^{\alpha} = 0; \qquad \sigma = 1, 2, \dots, h \tag{14}$$

In this way only linearly independent relations from (13) are retained and from the chemical point of view it means that instead of (some) atoms with masses *M*<sup>a</sup> only some their linear combinations with masses *M*<sup>e</sup> should be considered as elementary building units of components:

$$M\_{\alpha} = \sum\_{\sigma=1}^{h} M\_{\text{e}}^{\sigma} S\_{\sigma \alpha} \tag{15}$$

Example. Mixture of NO2 and N2O4 has the matrix ║*T*║ of dimension 2 2 and rank 1; the matrix ║*S*║ is of dimension 1 2 and can be selected as 1 2 which means that the elementary building unit is NO2 and 11 2N O ea aa a *M M MM M* 2 2 .

Multiplying each of the *z* relations (13) by corresponding *M*<sup>a</sup> and summing the results for all it follows that 1 0 *n M J* . This fact can be much more effectively formulated in vector form because further important implications than follow. The last equality indicates

mixture, i.e. with no reaction scheme. This is a priori type of analysis and is used in continuum nonequilibrium (rational) thermodynamics. Because Bowen's results are

Let a reacting mixture be composed from *n* components (compounds) which are formed by *z* different atoms. Atomic composition of each component is described by numbers *T* that indicate the number of atoms (= 1, 2,..., *z*) in component (= 1, 2,..., *n*). Atomic masses

z

Although compounds are destroyed or created in chemical reactions the atoms are

time in unit volume, i.e. the reaction rate for the component (component rate in short),

*T J z*

Atomic numbers can be arranged in matrix ║*T*║ of dimension *z n*. Chemical reactions are possible if its rank (*h*) is smaller than the number of components (*n*), otherwise the system (13) has only trivial solution, i.e. is valid only for zero component rates. If *h* < *z* then a new *h n* matrix ║*S*║ with rank *h* can be constructed from the original matrix ║*T*║

a 1 *M M T*

0; 1,2, ,

0; 1,2, ,

*S J h*

In this way only linearly independent relations from (13) are retained and from the chemical

1

Example. Mixture of NO2 and N2O4 has the matrix ║*T*║ of dimension 2 2 and rank 1; the matrix ║*S*║ is of dimension 1 2 and can be selected as 1 2 which means that the

vector form because further important implications than follow. The last equality indicates

ea aa a *M M MM M* 2 2 .

*h M M S*

 denotes the number of moles of the component formed or reacted per unit

(12)

(13)

(14)

should be considered as elementary building units of

. This fact can be much more effectively formulated in

(15)

only some their linear

and summing the results for

important for this article they are briefly reviewed now for reader's convenience.

in combination with these numbers determine the molar masses *M*:

then the persistence of atoms can be formulated in the form

1

1

point of view it means that instead of (some) atoms with masses *M*<sup>a</sup>

*n*

elementary building unit is NO2 and 11 2N O

0

Multiplying each of the *z* relations (13) by corresponding *M*<sup>a</sup>

e

1

*n M J*

 *n*

This result expresses, in other words, the mass conservation.

*M*<sup>a</sup> 

preserved. If *J*

and used instead of it:

components:

all it follows that

combinations with masses *M*<sup>e</sup>

that component molar masses and rates should form two perpendicular vectors, i.e. vectors with vanishing scalar product. Let us introduce *n*-dimensional vector space, called the component space and denoted by U, with base vectors **e** and reciprocal base vectors **e** ( = 1, 2,..., *n*). Then the vector of molar masses **M** and the vector of reaction rates **J** are defined in this space as follows:

$$\mathbf{M} = \sum\_{\alpha=1}^{n} M\_{\alpha} \mathbf{e}^{\alpha}, \quad \mathbf{J} = \sum\_{\alpha=1}^{n} f^{\alpha} \mathbf{e}\_{\alpha} \tag{16}$$

To proceed further we use relations (14) and (15) because in contrast to relations (12) and (13) the matrix ║*S*║ is of "full rank" (does not contain linearly dependent rows). The product of the two vectors can be then expressed in the following form:

$$\mathbf{M}\mathbf{J} = \left(\sum\_{\alpha=1}^{n} \sum\_{\sigma=1}^{h} M\_{\mathbf{e}}^{\sigma} \mathbf{S}\_{\sigma\alpha} \mathbf{e}^{a}\right) \cdot \left(\sum\_{\alpha=1}^{n} f^{\alpha} \mathbf{e}\_{\alpha}\right) = \left(\sum\_{\sigma=1}^{h} M\_{\mathbf{e}}^{\sigma} \sum\_{\alpha=1}^{n} \mathbf{S}\_{\sigma\alpha} \mathbf{e}^{a}\right) \cdot \left(\sum\_{\alpha=1}^{n} f^{\alpha} \mathbf{e}\_{\alpha}\right) = \mathbf{0} \tag{17}$$

where the latter equality follows using (14). Because the matrix ║*S*║ has rank *h*, the vectors

$$\mathbf{f}\_{\sigma} = \sum\_{a=1}^{n} \mathbf{S}\_{\sigma a} \mathbf{e}^{a}; \ \sigma = 1, 2, \dots, h \tag{18}$$

that appear in (17) are linearly independent and thus form a basis of a *h*-dimensional subspace W of the space U (remember that *h* < *n*). This subspace unambiguously determines complementary orthogonal subspace V (of dimension *n*–*h*), i.e. U = V W, V W. From (17) follows:

$$\mathbf{M} = \sum\_{\sigma=1}^{h} M\_{\mathbf{e}}^{\sigma} \mathbf{f}\_{\sigma} \tag{19}$$

which shows that **M** can be expressed in the basis of the subspace W or **M** W. From (14) and (16)2 follows:

$$\mathbf{J}.\mathbf{f}\_{\sigma} = 0; \ \sigma = 1, 2, \ldots, h \tag{20}$$

which means that **J** is perpendicular to all basis vectors of the subspace W, consequently, **J** lies in the complementary orthogonal subspace V, **J** V.

Let us now select basis vectors in the subspace V and denote them **d***p*, *p* = 1, 2,..., *n*–*h*. Of course, these vectors lie also in the (original) space U and can be expressed using its basis vectors analogically to (16):

$$\mathbf{d}^p = \sum\_{\alpha=1}^n P^{p\alpha} \mathbf{e}\_{\alpha} \tag{21}$$

Because of orthogonality of subspaces V and W, their bases conform to equation

$$\mathbf{f}\_{\sigma}, \mathbf{d}^{p} = \sum\_{\alpha=1}^{n} S\_{\sigma\alpha} P^{p\alpha} = 0 \tag{22}$$

Thermodynamics and Reaction Rates 681

Of course, so far we have seen only relationships between reaction rates and no explicit equations for them like, e.g., the kinetic mass action law. Analysis based only on permanence of atoms cannot give such equations – they belong to the domain of chemical

Simple example on Wegscheider conditions was presented by Vlad and Ross (Vlad & Ross,

A = B, 2A = A + B (R2)

Construct the matrix ║*T*║ and determine its rank

Construct the matrix ║*S*║

Select the stoichiometric matrix ║*Ppα*║ fulfilling (23)

> Find component rates from (24)

> > *Kcc* / and

kinetics although they can also be devised by thermodynamics, see Section 4.

2009) – isomerization taking place in two ways:

Find out components of reacting mixture

Scheme 2. Alternative procedure to find reaction rates

*n*–*h* independent reactions and their rates to describe chemical transformations

Use the method of Section 4

Vlad and Ross note that if the (thermodynamic) equilibrium constant is B A eq

if kinetic equations are expressed e.g. 1 1A 1B *r kc kc* then the consistency between

which can be alternatively written in matrix form as

$$\left\| \left\| P^{\mu} \right\| \times \left\| S\_{\sigma \alpha} \right\| \mathbb{T} = \left\| \left\| O \right\| \right\| \tag{23}$$

Meaning of the matrix ║*Pp* ║ can be deduced from two consequences. First, because the reaction vector **J** lies in the subspace V, it can be expressed also using its basis vectors, *n h p p J* **<sup>J</sup> <sup>d</sup>** . Substituting for **J** from (16)2 and for **d***p* from (21), it follows:

$$J^{\alpha} = \sum\_{p=1}^{n-h} J\_p P^{p\alpha}; \ \alpha = 1, 2, \dots, n \tag{24}$$

Second, because the vector of molar masses **M** is in the subspace W, it is perpendicular to all vectors **d***p* and thus

$$0 = \mathbf{d}^p.\mathbf{M} = \sum\_{\alpha=1}^n P^{p\alpha} M\_\alpha; p = 1, 2, \dots, n-h \tag{25}$$

as follows after substitution from (19), (21), (220. Eq. (25) shows that matrix ║*Pp* ║ enables to express component rates in *n*–*h* quantities *Jp* which are, in fact, rates of *n*–*h* independent reactions shown by (25) if instead of molar masses *M* the corresponding chemical symbols are used. In other words ║*Pp* ║ is the matrix of stoichiometric coefficients of component in (independent) reaction *p*.

Vector algebra thus shows that chemical transformations fulfilling persistence of atoms (mass conservation) can be equivalently described either by component reaction rates or by rates of independent reactions. The number of the former is equal to the number of components (*n*) whereas the number of the latter is lower (*n*–*h*) which could decrease the dimensionality of the problem of description of reaction rates. In kinetic practice, however, changes in component concentrations (amounts) are measured, i.e. data on component rates and not on rates of individual reactions are collected. Reactions, in the form of reaction schemes, are suggested a posteriori on the basis of detected components, their concentrations changing in time and chemical insight. Then dependencies between reactions can be searched. Vector analysis offers rather different procedure outlined in Scheme 2. Dependencies are revealed at the beginning and then only independent reactions are included in the (kinetic) analysis. Vector analysis also shows how to transform (measured) component rates into (suggested, selected) rates of independent reactions. This transformation is made by standard procedure for interchange between vector bases or between vector coordinates in different bases. First, the contravariant metric tensor with components *drp* = **d***r*.**d***p* is constructed and then its inversion (covariant metric tensor) with components *drp* is found. From 1 *n h p p p J* **<sup>J</sup> <sup>d</sup>** it follows that 1 *n h p r p r r p J J* **J.d d .d** . Using in

the latter equation the well known relationship between metric tensors and corresponding base vectors and the definition of base vectors (21) it finally follows:

$$J\_p = \sum\_{\alpha=1}^{n} \left( J^{\alpha} \sum\_{r=1}^{n-h} P^{r\alpha} d\_{rp} \right); p = 1, 2, \dots, n-h \tag{26}$$

1

*p*

reaction vector **J** lies in the subspace V, it can be expressed also using its basis vectors,

; 1, 2, ,

║║*S*║T = ║0║ (23)

║ can be deduced from two consequences. First, because the

(24)

**d .M** (25)

║ is the matrix of stoichiometric coefficients of component in

1

*p r p r r*

*J J*

**J.d d .d** . Using in

*n h*

*p*

║ enables to

║*Pp*

**<sup>J</sup> <sup>d</sup>** . Substituting for **J** from (16)2 and for **d***p* from (21), it follows:

1

1 0 ; 1, 2, ,

as follows after substitution from (19), (21), (220. Eq. (25) shows that matrix ║*Pp*

1

1 1

*n nh <sup>r</sup> p rp r*

base vectors and the definition of base vectors (21) it finally follows:

*p p p J* 

**<sup>J</sup> <sup>d</sup>** it follows that

the latter equation the well known relationship between metric tensors and corresponding

*J J Pd p n h*

  ; 1, 2, ,

(26)

*n h*

*p p p*

 

*J JP n*

Second, because the vector of molar masses **M** is in the subspace W, it is perpendicular to all

*<sup>n</sup> p p PM p nh* 

express component rates in *n*–*h* quantities *Jp* which are, in fact, rates of *n*–*h* independent reactions shown by (25) if instead of molar masses *M* the corresponding chemical symbols

Vector algebra thus shows that chemical transformations fulfilling persistence of atoms (mass conservation) can be equivalently described either by component reaction rates or by rates of independent reactions. The number of the former is equal to the number of components (*n*) whereas the number of the latter is lower (*n*–*h*) which could decrease the dimensionality of the problem of description of reaction rates. In kinetic practice, however, changes in component concentrations (amounts) are measured, i.e. data on component rates and not on rates of individual reactions are collected. Reactions, in the form of reaction schemes, are suggested a posteriori on the basis of detected components, their concentrations changing in time and chemical insight. Then dependencies between reactions can be searched. Vector analysis offers rather different procedure outlined in Scheme 2. Dependencies are revealed at the beginning and then only independent reactions are included in the (kinetic) analysis. Vector analysis also shows how to transform (measured) component rates into (suggested, selected) rates of independent reactions. This transformation is made by standard procedure for interchange between vector bases or between vector coordinates in different bases. First, the contravariant metric tensor with components *drp* = **d***r*.**d***p* is constructed and then its inversion (covariant metric tensor) with

*n h*

which can be alternatively written in matrix form as

Meaning of the matrix ║*Pp*

1

*p p p J* 

vectors **d***p* and thus

are used. In other words ║*Pp*

components *drp* is found. From

(independent) reaction *p*.

*n h*

Of course, so far we have seen only relationships between reaction rates and no explicit equations for them like, e.g., the kinetic mass action law. Analysis based only on permanence of atoms cannot give such equations – they belong to the domain of chemical kinetics although they can also be devised by thermodynamics, see Section 4.

Simple example on Wegscheider conditions was presented by Vlad and Ross (Vlad & Ross, 2009) – isomerization taking place in two ways:

$$\mathbf{A} = \mathbf{B}\_t \ \mathbf{\mathcal{Z}} \mathbf{A} = \mathbf{A} + \mathbf{B} \tag{\mathbb{R}}$$

Scheme 2. Alternative procedure to find reaction rates

Vlad and Ross note that if the (thermodynamic) equilibrium constant is B A eq *Kcc* / and if kinetic equations are expressed e.g. 1 1A 1B *r kc kc* then the consistency between

Thermodynamics and Reaction Rates 683

There is a thermodynamic method giving kinetic description in terms of independent

More complex reaction mixture and scheme was discussed by Ederer and Gilles (Ederer & Gilles, 2007). Their mixture was composed from six formal components (A, B, C, AB, BC, ABC) formed by three atoms (A, B, C). Three independent reactions are possible in this mixture while four reactions were considered by Ederer and Gilles (Ederer & Gilles, 2007)

1 1 A B 1 AB 2 2 AB C 2 ABC 3 3 B C 3 BC 4 4 A BC 4 ABC *r kc c kc r kc c kc r kc c kc r kc c kc* , ,, (32)

Let us suppose that the fourth reaction rate can be expressed through the other three rates: *b*1*r*1 + *b*2*r*2 + *b*3*r*3. By similar procedure as in the preceding example we arrive at conditions *b kk* 2 42 / , *b kc k* 3 4A 3 / , and *b bkc k bkc kc* 1 2 2C 1 3 3C 1A / / from which it follows that

> 1234 1234 <sup>1</sup> *kkkk kkkk*

i.e., Wegscheider condition derived in (Ederer & Gilles, 2007) from equilibrium considerations. Thus also here Wegscheider condition seems to be a result of mutual dependence of reaction rates and not a necessary consistency condition between

If reactions A + B = AB, AB + C = ABC, and B + C = BC are selected as independent ones

 *J*A = –*J*1, *J*B = –*J*1 – *J*3, *J*C = –*J*2 – *J*3, *J*AB = *J*1 – *J*2, *J*BC = *J*3, *J*ABC = *J*2 (34) Remember that, e.g., *J*<sup>1</sup> *r*1 but that the relationships between rates of independent reactions

 *J*1 = *r*1 + *r*4, *J*2 = *r*2 + *r*4, *J*3 = *r*3 – *r*4 (35) Eq. (26) gives more complex expressions for independent rates, e.g. *J*1 = – *J*A/2 – *J*B/4 + *J*AB/4 – *J*BC/4 + *J*ABC/4, whereas from (24), i.e. (34), simply follow: *J*1 = –*J*A, *J*2 = *J*ABC, *J*3 = *J*BC.

dimensional space are transformed to components *Ji* in three dimensional subspace. Consequently, in practical applications (24) should be preferred in favor of (26) also to

Message from the analysis of independence of reactions in this example is that it is sufficient to measure three component rates only (*J*A, *J*ABC, *J*BC); the remaining three component rates are determined by them. Although concentrations, i.e. component rates, are measured in kinetic experiments, results are finally expressed in reaction rates, rates of reactions occurring in suggested reaction scheme. Component rates are simply not sufficient in kinetic analysis and they are (perhaps always) translated into rates of reaction steps. However, from the three independent rates there cannot be unambiguously determined rates of four reactions in suggested reaction schemes as (35) demonstrates (three equations for four unknown *ri*). One equation more is needed and this is the above equation relating *r*4 to the remaining three rates. Equations containing *ri* are too general and in practice are replaced by mass action expressions shown in (32) – eight parameters (rate constants) are thus

This is because the rates are considered as vector components – components *J*

(33)

 of six

reactions as noted in Scheme 2, see Section 4.

thermodynamics and kinetics.

and mass action rates (32) follow from (34):

 .

then (24) gives

express *Ji* in terms of *J*

*r*4 = *b*1*r*1 + *b*2*r*2 + *b*3*r*3 with following mass action rate equations:

thermodynamic and kinetic description of equilibrium is achieved only if the following (Wegscheider) condition holds:

$$
\vec{k}\_1 \, / \, \vec{k}\_1 = \vec{k}\_2 \, / \, \vec{k}\_2 = \text{K} \tag{27}
$$

It can be easily checked that in this mixture of one kind of atom and two components the rank of the matrix ║*T*║ (dimension 1 2) is 1 and there is only one independent reaction. The matrix ║*S*║ can be selected as equal to the matrix ║*T*║ and then the stoichiometric matrix can be selected as 1 1 which corresponds to the first reaction (A = B) selected as the independent reaction. There is one base vector 1 **d ee** 1 2 giving one component contravariant tensor *d*11 = 2 and corresponding component of covariant tensor *d*11 = 1/2. Consequently, the rate of the independent reaction is related to component reaction rates by:

$$\mathbf{J}^{\mathsf{A}} = \mathbf{J}\_1 \mathbf{P}^{\mathsf{11}} = \mathbf{\!}\_1 \mathbf{I}\_1 \mathbf{J}^{\mathsf{B}} = \mathbf{J}\_1 \mathbf{P}^{\mathsf{12}} = \mathbf{J}\_1 \tag{28}$$

and *J*A = –*J*B which follows also from (14). Kinetics of transformations in a mixture of two isomers can be thus fully described by one reaction rate only – either from the two component rates can be measured and used for this purpose, the other component rate is then determined by it, can be calculated from it. At this stage of analysis there is no indication that two reactions should be considered and this should be viewed as some kind of "external" information coming perhaps from experiments. At the same time this analysis does not provide any explicit expression for reaction rate and its dependence on concentration – this is another type of external information coming usually from kinetics. Let us therefore suppose the two isomerization processes given above and their rates formulated in the form of kinetic mass action law:

$$r\_1 = \vec{k}\_1 \mathbf{c}\_A - \vec{k}\_1 \mathbf{c}\_B \quad r\_2 = \vec{k}\_2 \mathbf{c}\_A^2 - \vec{k}\_2 \mathbf{c}\_A \mathbf{c}\_B \tag{29}$$

Then the only one independent reaction rate is in the form *J*1 = *r*1 + *r*2. Note, that although the first reaction has been selected as the independent reaction, the rate of independent reaction is not equal to (its mass action rate) *r*1. This interesting finding has probably no specific practical implication. However, individual traditional rates (*ri*) should not be independent. Let us suppose that *r2* is dependent on *r1*, i.e. can be expressed through it: *r*2 = *br*1; then

$$\left(b\vec{k}\_1 - \vec{k}\_2 c\_\mathcal{A}\right) c\_\mathcal{A} - \left(b\vec{k}\_1 - \vec{k}\_2 c\_\mathcal{A}\right) c\_\mathcal{B} = 0\tag{30}$$

should be valid for any concentrations. Sufficient conditions for this are *b kc k kc k* 2A 1 2A 1 / / and from them follows:

$$
\vec{k}\_1 \vec{k}\_2 = \vec{k}\_1 \vec{k}\_2 \tag{31}
$$

i.e. "kinetic part" of Wegscheider condition (27). Substituting derived expressions for *b* into *br*1 it can be easily checked that *r*2 really results. Although the derivation is rather straightforward and is not based on linear dependency with constant coefficients it points to assumption that Wegscheider conditions are not conditions for consistency of kinetics with thermodynamics but results of dependencies among reaction rates. Moreover, this derivation need not suppose equality of thermodynamic and kinetic equilibrium constant.

thermodynamic and kinetic description of equilibrium is achieved only if the following

It can be easily checked that in this mixture of one kind of atom and two components the rank of the matrix ║*T*║ (dimension 1 2) is 1 and there is only one independent reaction. The matrix ║*S*║ can be selected as equal to the matrix ║*T*║ and then the stoichiometric matrix can be selected as 1 1 which corresponds to the first reaction (A = B) selected as the independent reaction. There is one base vector 1 **d ee** 1 2 giving one component contravariant tensor *d*11 = 2 and corresponding component of covariant tensor *d*11 = 1/2. Consequently, the rate of the independent reaction is related to component reaction rates by:

 *J*A = *J*1*P*11 = –*J*1, *J*<sup>B</sup> = *J*1*P*12 = *J*1 (28) and *J*A = –*J*B which follows also from (14). Kinetics of transformations in a mixture of two isomers can be thus fully described by one reaction rate only – either from the two component rates can be measured and used for this purpose, the other component rate is then determined by it, can be calculated from it. At this stage of analysis there is no indication that two reactions should be considered and this should be viewed as some kind of "external" information coming perhaps from experiments. At the same time this analysis does not provide any explicit expression for reaction rate and its dependence on concentration – this is another type of external information coming usually from kinetics. Let us therefore suppose the two isomerization processes given above and their rates

2

Then the only one independent reaction rate is in the form *J*1 = *r*1 + *r*2. Note, that although the first reaction has been selected as the independent reaction, the rate of independent reaction is not equal to (its mass action rate) *r*1. This interesting finding has probably no specific practical implication. However, individual traditional rates (*ri*) should not be independent. Let us suppose that *r2* is dependent on *r1*, i.e. can be expressed through it: *r*2 =

should be valid for any concentrations. Sufficient conditions for this are

i.e. "kinetic part" of Wegscheider condition (27). Substituting derived expressions for *b* into *br*1 it can be easily checked that *r*2 really results. Although the derivation is rather straightforward and is not based on linear dependency with constant coefficients it points to assumption that Wegscheider conditions are not conditions for consistency of kinetics with thermodynamics but results of dependencies among reaction rates. Moreover, this derivation need not suppose equality of thermodynamic and kinetic equilibrium constant.

1 1A 1B 2 2A 2AB *r kc kc r kc kc c* , (29)

*bk k c c bk k c c* 1 2A A 1 2A B <sup>0</sup> (30)

12 12 *kk kk* (31)

*kk kk K* 11 22 / / (27)

(Wegscheider) condition holds:

formulated in the form of kinetic mass action law:

*b kc k kc k* 2A 1 2A 1 / / and from them follows:

*br*1; then

There is a thermodynamic method giving kinetic description in terms of independent reactions as noted in Scheme 2, see Section 4.

More complex reaction mixture and scheme was discussed by Ederer and Gilles (Ederer & Gilles, 2007). Their mixture was composed from six formal components (A, B, C, AB, BC, ABC) formed by three atoms (A, B, C). Three independent reactions are possible in this mixture while four reactions were considered by Ederer and Gilles (Ederer & Gilles, 2007) *r*4 = *b*1*r*1 + *b*2*r*2 + *b*3*r*3 with following mass action rate equations:

$$r\_1 = \vec{k}\_1 \mathbf{c}\_A \mathbf{c}\_B - \vec{k}\_1 \mathbf{c}\_{AB'} \quad r\_2 = \vec{k}\_2 \mathbf{c}\_{AB} \mathbf{c}\_C - \vec{k}\_2 \mathbf{c}\_{ABC'} \quad r\_3 = \vec{k}\_3 \mathbf{c}\_B \mathbf{c}\_C - \vec{k}\_3 \mathbf{c}\_{BC'} \quad r\_4 = \vec{k}\_4 \mathbf{c}\_A \mathbf{c}\_{BC} - \vec{k}\_4 \mathbf{c}\_{ABC} \tag{32}$$

Let us suppose that the fourth reaction rate can be expressed through the other three rates: *b*1*r*1 + *b*2*r*2 + *b*3*r*3. By similar procedure as in the preceding example we arrive at conditions *b kk* 2 42 / , *b kc k* 3 4A 3 / , and *b bkc k bkc kc* 1 2 2C 1 3 3C 1A / / from which it follows that

$$\frac{\vec{k}\_1 \vec{k}\_2 \vec{k}\_3 \vec{k}\_4}{\vec{k}\_1 \vec{k}\_2 \vec{k}\_3 \vec{k}\_4} = 1\tag{33}$$

i.e., Wegscheider condition derived in (Ederer & Gilles, 2007) from equilibrium considerations. Thus also here Wegscheider condition seems to be a result of mutual dependence of reaction rates and not a necessary consistency condition between thermodynamics and kinetics.

If reactions A + B = AB, AB + C = ABC, and B + C = BC are selected as independent ones then (24) gives

$$\mathbf{l}\_{\text{V}} = -\mathbf{l}\_{\text{V}}\mathbf{l}\_{\text{B}} = -\mathbf{l}\_{\text{I}} - \mathbf{l}\_{\text{J}}\mathbf{l}\_{\text{J}} = -\mathbf{l}\_{\text{I}}\mathbf{\hspace{1cm}}\mathbf{l}\_{\text{J}}\mathbf{l}\_{\text{AB}} = \mathbf{l}\_{\text{I}}\mathbf{\hspace{1cm}}\mathbf{l}\_{\text{J}}\mathbf{l}\_{\text{BC}} = \mathbf{l}\_{\text{I}}\mathbf{\hspace{1cm}}\mathbf{l}\_{\text{J}}\mathbf{l}\_{\text{BC}} = \mathbf{l}\_{\text{I}}\mathbf{\hspace{1cm}}\mathbf{\hspace{1cm}}\mathbf{\hspace{1cm}}\mathbf{\hspace{1cm}}\mathbf{\hspace{1cm}}\mathbf{\hspace{1cm}}\mathbf{\hspace{1cm}}\mathbf{\hspace{1cm}}\mathbf{\hspace{1cm}}\mathbf{\hspace{1cm}}\mathbf{\hspace{1cm}}\mathbf{\hspace{1cm}}\mathbf{\hspace{1cm}}\mathbf{\hspace{1cm}}\mathbf{\hspace{1cm}}\mathbf{\hspace{1cm}}\mathbf{\hspace{1cm}}\mathbf{\hspace{1cm}}\mathbf{\hspace{1cm}}\mathbf{\hspace{1cm}}\mathbf{\hspace{1cm}}\mathbf{\hspace{1cm}}\mathbf{\hspace{1cm}}\mathbf{\hspace{1cm}}\mathbf{\hspace{1cm}}\mathbf{\hspace{1cm}}\mathbf{\hspace{1cm}}\mathbf{\hspace{1cm}}\mathbf{\hspace{1cm}}\mathbf{\hspace{1cm}}\mathbf{\hspace{1cm}}\mathbf{\hspace{1cm}}\mathbf{\hspace{1cm}}\mathbf{\hspace{1cm}}\mathbf{\hspace{1cm}}\mathbf{\hspace{1cm}}\mathbf{\hspace{1cm}}\mathbf{\hspace{1cm}}\mathbf{\hspace{1cm}}\mathbf{\hspace{1cm}}\mathbf{\hspace{1cm}}\mathbf{\hspace{1cm}}\mathbf{\hspace{1cm}}\mathbf{\hspace{$$

Remember that, e.g., *J*<sup>1</sup> *r*1 but that the relationships between rates of independent reactions and mass action rates (32) follow from (34):

$$I\_1 = r\_1 + r\_4, \quad I\_2 = r\_2 + r\_4, \quad I\_3 = r\_3 - r\_4 \tag{35}$$

Eq. (26) gives more complex expressions for independent rates, e.g. *J*1 = – *J*A/2 – *J*B/4 + *J*AB/4 – *J*BC/4 + *J*ABC/4, whereas from (24), i.e. (34), simply follow: *J*1 = –*J*A, *J*2 = *J*ABC, *J*3 = *J*BC. This is because the rates are considered as vector components – components *J* of six dimensional space are transformed to components *Ji* in three dimensional subspace. Consequently, in practical applications (24) should be preferred in favor of (26) also to express *Ji* in terms of *J* .

Message from the analysis of independence of reactions in this example is that it is sufficient to measure three component rates only (*J*A, *J*ABC, *J*BC); the remaining three component rates are determined by them. Although concentrations, i.e. component rates, are measured in kinetic experiments, results are finally expressed in reaction rates, rates of reactions occurring in suggested reaction scheme. Component rates are simply not sufficient in kinetic analysis and they are (perhaps always) translated into rates of reaction steps. However, from the three independent rates there cannot be unambiguously determined rates of four reactions in suggested reaction schemes as (35) demonstrates (three equations for four unknown *ri*). One equation more is needed and this is the above equation relating *r*4 to the remaining three rates. Equations containing *ri* are too general and in practice are replaced by mass action expressions shown in (32) – eight parameters (rate constants) are thus

Thermodynamics and Reaction Rates 685

Chemical potential is used in discussions on thermodynamic implications on reaction rates, particularly in the form of (stoichiometric) difference between chemical potentials of reaction products and reactants and through its explicit relationship to concentrations (activities, in general). Before going into this type of analysis basic information is

Chemical potential is in classical, equilibrium thermodynamics defined as a partial

 , , / *Tpnj G n*

Although another definitions through another thermodynamic quantities are possible (and equivalent with this one), the definition using the Gibbs energy is the most useful for chemical thermodynamics. Chemical potential expresses the effect of composition and this effect is also essential in chemical kinetics. To make the mathematical definition of the chemical potential applicable in practice its relationship to composition (concentration) should be stated explicitly. Practical chemical thermodynamics suggests that this is an easy task but we must be very careful and bear all (tacit) presumptions in mind to arrive at proper conclusions. Generally the explicit relationship between chemical composition and

> *<sup>a</sup>* exp *RT*

 

> 

*V p*

 

 

 

> 

,g *p p px p*

gases partial molar volumes are equal to the molar volume of the mixture, *Vm* (Silbey et al.,

 

*RT p* / // /

*RT* / / *p p* 

 

 

 

but this still lacks direct interconnection/linkage to measurable concentrations. Just this is the main problem of applying chemical potential (and activities) in rate equations which systematically use molar concentrations. Even when reaction rates would be expressed using activities in place of concentrations the activities should be properly calculated from the measured concentrations, in other words, the concentrations should be correctly transformed to the activities. Activity is very easily related to measurable composition variable in the case of mixture of ideal gases. Providing that Gibbs energy is a function of temperature, pressure and molar amounts, following relation is well known from

 

chemical potential is stated defining the activity of a component :

thermodynamics for the partial molar volume: , / *T nj*

ln *RT a*

(36)

(37)

(38)

 ,g (39)

,g (40)

. In a mixture of ideal

**3. Chemical potential and activity revise** 

recapitulated.

derivative of Gibbs energy (*G*):

which can be transformed to

2005). Because *Vm* = *RT*/*p* we can write:

and

introduced in this example. They can be in principle determined from three equations (35) with the three measured independent reactions, four equations relating equilibrium composition (or thermodynamic equilibrium constant) and kinetic equilibrium constant and one Wegscheider condition (33), i.e. eight equations in total. Alternative thermodynamic method is described in Section 4.

Algebraically more rigorous is this analysis in the case of first order reactions as was illustrated on a mixture of three isomers and their triangular reaction scheme which is traditional example used to discuss consistency between thermodynamics and kinetics. Here, Wegscheider relations are consequences of linear dependence of traditional mass action reaction rates (Pekař, 2007).

### **2.4 Note on standard states**

Preceding sections demonstrated that one of the main problems to be solved when relating thermodynamics and kinetics is the transformation between activities and concentration variables. This is closely related to the selection of standard state (important and often overlooked aspect of relating thermodynamic and kinetic equilibrium constants) and to chemical potential. Standard states are therefore briefly reviewed in this section and chemical potential is subject of the following section.

Rates of chemical reactions are mostly expressed in terms of concentrations. Among standard states introduced and commonly used in thermodynamics there is only one based on concentration – the standard state of nonelectrolyte solute on concentration basis. Only this standard state can be directly used in kinetic equations. Standard state in gaseous phase or mixture is defined through (partial) pressure or fugacity. As shown above even in mixture of ideal gases it is impossible to simply use this standard state in concentration based kinetic equations. Although kinetic equations could be reformulated into partial pressures there still remains problem with the fact that standard pressure is fixed (at 1 atm or, nowadays, at 105 Pa) and its recalculation to actual pressure in reacting mixture may cause incompatibility of thermodynamic and kinetic equilibrium constants (see the factor *p*rel in the example above in Section 2.1). This opens another problem – the very selection of standard state, particularly in relation to activity discussed in subsequent section. In principle, it can be selected arbitrarily, as dependent only on temperature or on temperature and pressure. Standard states strictly based on the (fixed) standard pressure are of the former type and only such will be considered in this article. All other states, including states dependent also on pressure, will be called the reference state; the same approach is used, e.g. by de Voe (de Voe, 2001).

The value of thermodynamic equilibrium constant and its dependence or independence on pressure is thus dependent on the selected standard (or reference) state. This is quite uncommon in chemical kinetics where the dependence of rate constants is not a matter of selection of standard states but result of experimental evidence or some theory of reaction rates. As a rule, rate constant is always function of temperature. Sometimes also the dependence on pressure is considered but this is usually the case of nonelementary reactions. Consequently, attempts to relate thermodynamic and kinetic equilibrium constants should select standard state consistently with functional dependence of rate constants. On the other hand, the method of Scheme 1 is self-consistent in this aspect because equilibrium composition is independent of the selection of standard state.

introduced in this example. They can be in principle determined from three equations (35) with the three measured independent reactions, four equations relating equilibrium composition (or thermodynamic equilibrium constant) and kinetic equilibrium constant and one Wegscheider condition (33), i.e. eight equations in total. Alternative thermodynamic

Algebraically more rigorous is this analysis in the case of first order reactions as was illustrated on a mixture of three isomers and their triangular reaction scheme which is traditional example used to discuss consistency between thermodynamics and kinetics. Here, Wegscheider relations are consequences of linear dependence of traditional mass

Preceding sections demonstrated that one of the main problems to be solved when relating thermodynamics and kinetics is the transformation between activities and concentration variables. This is closely related to the selection of standard state (important and often overlooked aspect of relating thermodynamic and kinetic equilibrium constants) and to chemical potential. Standard states are therefore briefly reviewed in this section and

Rates of chemical reactions are mostly expressed in terms of concentrations. Among standard states introduced and commonly used in thermodynamics there is only one based on concentration – the standard state of nonelectrolyte solute on concentration basis. Only this standard state can be directly used in kinetic equations. Standard state in gaseous phase or mixture is defined through (partial) pressure or fugacity. As shown above even in mixture of ideal gases it is impossible to simply use this standard state in concentration based kinetic equations. Although kinetic equations could be reformulated into partial pressures there still remains problem with the fact that standard pressure is fixed (at 1 atm or, nowadays, at 105 Pa) and its recalculation to actual pressure in reacting mixture may cause incompatibility of thermodynamic and kinetic equilibrium constants (see the factor *p*rel in the example above in Section 2.1). This opens another problem – the very selection of standard state, particularly in relation to activity discussed in subsequent section. In principle, it can be selected arbitrarily, as dependent only on temperature or on temperature and pressure. Standard states strictly based on the (fixed) standard pressure are of the former type and only such will be considered in this article. All other states, including states dependent also on pressure, will be called the reference state; the same approach is used,

The value of thermodynamic equilibrium constant and its dependence or independence on pressure is thus dependent on the selected standard (or reference) state. This is quite uncommon in chemical kinetics where the dependence of rate constants is not a matter of selection of standard states but result of experimental evidence or some theory of reaction rates. As a rule, rate constant is always function of temperature. Sometimes also the dependence on pressure is considered but this is usually the case of nonelementary reactions. Consequently, attempts to relate thermodynamic and kinetic equilibrium constants should select standard state consistently with functional dependence of rate constants. On the other hand, the method of Scheme 1 is self-consistent in this aspect

because equilibrium composition is independent of the selection of standard state.

method is described in Section 4.

action reaction rates (Pekař, 2007).

chemical potential is subject of the following section.

**2.4 Note on standard states** 

e.g. by de Voe (de Voe, 2001).
