**3. Chemical potential and activity revise**

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 recapitulated.

Chemical potential is in classical, equilibrium thermodynamics defined as a partial derivative of Gibbs energy (*G*):

$$
\mu\_{\alpha} = \left(\bigcirc \mathbf{G} / \bigcirc \mathfrak{n}\_{\alpha}\right)\_{\mathbf{T}\_{\cdot}, \mathfrak{p}\_{\prime}, \mathfrak{n}\_{\mathrm{j} \times \mathfrak{a}}} \tag{36}
$$

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 chemical potential is stated defining the activity of a component :

$$a\_{\alpha} = \exp\left(\frac{\mu\_{\alpha} - \mu\_{\alpha}^{\circ}}{RT}\right) \tag{37}$$

which can be transformed to

$$
\mu\_a = \mu\_a^\circ + RT\ln a\_a \tag{38}
$$

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 thermodynamics for the partial molar volume: , / *T nj V p* . In a mixture of ideal gases partial molar volumes are equal to the molar volume of the mixture, *Vm* (Silbey et al., 2005). Because *Vm* = *RT*/*p* we can write:

$$\text{RT } / \text{ p} = \left( \left\| \mu\_{a,\emptyset} / \left\| \mathcal{O} p\_a \right\| \right) \left( \left\| p\_a \right\| / \left\| \mathcal{P} \right\| \right) = \text{x}\_a \left( \left\| \mu\_{a,\emptyset} / \left\| \mathcal{P} p\_a \right\| \right) \tag{39}$$

and

$$\text{RT } / \text{ p}\_a = \left( \partial \mu\_{a,\emptyset} / \partial \text{p}\_a \right) \tag{40}$$

Thermodynamics and Reaction Rates 687

relationship between compositions of equilibrated liquid and gaseous phases final form of ,l dependence on the composition of liquid is obtained. For example, with Raoult's law *p*

> 

mixture with the chemical potential defined, at a given *T* and *p*, as ref

which has, in fact, inspired the definition of an ideal (liquid, solid, or gas) mixture as a

where ref is a function of both *T* and *p*. This definition, as well as the identity in (47), can be simply related to the definition of activity only if the standard state is selected consistently with the reference state, i.e. if the former is a function of both *T* and *p*. If the standard state is selected as dependent on temperature, as it should be, than the pressure factor () should

> ref exp *RT*

Then the activity of a (non-electrolyte) component in real solution is written as *a x*

where is the activity coefficient introduced by the equation ref

Introducing activities in place of concentrations means in this case to know the pressure factor and to transform molar fractions into molar concentrations to be consistent with

The main problems with using activities defined for liquid systems can be summarized as follows. Activity is based on molar fractions whereas kinetic uses concentrations. Although there are formulas for the conversion of these variables they do not allow direct substitution, they introduce other variables (e.g., solution density) and lead to rather complex expression of thermodynamic equilibrium constant in concentrations. Whereas concentrations of all species are independent (variables) this is not true for molar fractions – value of one from them is unambiguously determined by values of remaining ones. Chemical potential in liquid and activity based on it are introduced on the basis of (liquid-gas) equilibrium while kinetics essentially works with reactions out of equilibrium. Applicability of equilibriumbased formulated in fugacities are really rare in nonequilibrium states deserves further study. The problem with molar fractions can be resolved by the use of molar concentration

equations should be formulated with the standard concentration. Sometimes following

all concentrations. In this case, the invertibility for *c* is problematic because it is included in *c*; reaction rates should be then formulated in *c*/*c* instead of concentrations that is quite unusual. Of course, the value of activity is dependent on the selected standard state, anyway. All attempts to relate thermodynamic and kinetic equilibrium constants should pay great attention to the selection of standard state and its consequences to be really rigorous

It is clear from this basic overview that chemical potential, activity and their interrelation are in principle equilibrium quantities which, in kinetic applications, are to be used for

   

 

,l ,

 

*<sup>c</sup> RT c c* ln / (Ederer & Gilles, 2007) where *c* is the sum of

ref

 *RT x p p RT x* ln / 

ln (47)

 

> 

(48)

 *<sup>c</sup> RT c c* ln / , however, rate

 

 *RT x* ln

 

> .

 

 *RT x* ln

= *xp*\* and ideal gas phase we have this equation

be introduced (see, e.g., de Voe, 2001)

thermodynamics.

relationship is used: ref

and correct.

,l ,g

based Henry's law giving for ideal-dilute solution ref

,l ,

 

 

Integration from the standard state to some actual state then yields

$$
\mu\_{a,\emptyset} = \mu\_{a,\emptyset}^\circ + RT\ln\left(p\_a \,/\, p^\circ\right) \tag{41}
$$

Comparing with the definition of activity it follows

$$a\_{\alpha} = p\_{\alpha} / p^{\circ} \text{ (mixture of ideal gases)} \tag{42}$$

Application of this relationship was illustrated in the example given above. Note that (42) was not derived from the definition of activity but comparing the properties of chemical potential in the ideal gas mixture (41) with the definition of activity. Note also that the partial derivative in the original definition of chemical potential is in general a function of molar amounts (contents) of all components but eq. (42) states that the chemical potential of a component is a function only of the content of that component.

In a real gas mixture, non-idealities should be taken into account, usually by substituting fugacity (*f*) for the partial pressure:

$$
\mu\_{a,\emptyset} = \mu\_{a,\emptyset}^\circ + RT\ln\left(f\_a \mid p^\circ\right) \tag{43}
$$

The fugacity can be eliminated in favor of directly measurable quantities using the fugacity coefficient

$$f\_{\alpha} = \phi\_{\alpha} p\_{\alpha} \tag{44}$$

and its relationship to the partial molar volume and the total pressure (de Voe, 2001):

$$
\mu\_{\alpha,\mathfrak{g}} = \mu\_{\alpha,\mathfrak{g}}^{\circ} + RT \ln \left( p\_{\alpha} \,/\, p^{\circ} \right) + \int\_{0}^{p} \left( \overline{V}\_{\alpha} - RT \,/\, p \right) \mathrm{d} \, p \tag{45}
$$

It should be stressed that in derivation of the expression for the fugacity coefficient it was assumed that the Gibbs energy is a function of (only) temperature, pressure, and molar amounts of all components. Comparing with the definition of activity we have

$$a\_a = f\_a \nmid p^\circ \quad \text{(mixture of gases)}\tag{46}$$

If kinetic equations for mixture of real gases are written in partial pressures then thermodynamic and kinetic equilibrium constants are incompatible due to the presence of fugacity coefficient or the integral in eq. (45). Kinetic equations for mixture of real gases could be formulated in terms of fugacities instead of concentrations (or partial pressures) to achieve compatibility between thermodynamic and kinetic equilibrium constants but even than the same problem remains with the presence of the standard pressure in thermodynamic relations. Kinetic equations formulated in fugacities are really rare – some success in this way was demonstrated by Eckert and Boudart (Eckert & Boudart, 1963) while Mason (Mason, 1965) showed, using the same data, that fugacities need not remedy the whole situation.

Similar derivation for liquid state (solutions) has different basis. It stems from the equilibrium between liquid and gaseous phase in which the following identity holds: ,g = ,l. Introducing expression (41) or (43) and using either Raoult's or Henry's law for the

Application of this relationship was illustrated in the example given above. Note that (42) was not derived from the definition of activity but comparing the properties of chemical potential in the ideal gas mixture (41) with the definition of activity. Note also that the partial derivative in the original definition of chemical potential is in general a function of molar amounts (contents) of all components but eq. (42) states that the chemical potential of

In a real gas mixture, non-idealities should be taken into account, usually by substituting

The fugacity can be eliminated in favor of directly measurable quantities using the fugacity

*f p* 

,g ,g <sup>0</sup> ln / / d *<sup>p</sup>*

It should be stressed that in derivation of the expression for the fugacity coefficient it was assumed that the Gibbs energy is a function of (only) temperature, pressure, and molar

If kinetic equations for mixture of real gases are written in partial pressures then thermodynamic and kinetic equilibrium constants are incompatible due to the presence of fugacity coefficient or the integral in eq. (45). Kinetic equations for mixture of real gases could be formulated in terms of fugacities instead of concentrations (or partial pressures) to achieve compatibility between thermodynamic and kinetic equilibrium constants but even than the same problem remains with the presence of the standard pressure in thermodynamic relations. Kinetic equations formulated in fugacities are really rare – some success in this way was demonstrated by Eckert and Boudart (Eckert & Boudart, 1963) while Mason (Mason, 1965) showed, using the same data, that fugacities need not remedy the

Similar derivation for liquid state (solutions) has different basis. It stems from the equilibrium between liquid and gaseous phase in which the following identity holds: ,g = ,l. Introducing expression (41) or (43) and using either Raoult's or Henry's law for the

and its relationship to the partial molar volume and the total pressure (de Voe, 2001):

 

amounts of all components. Comparing with the definition of activity we have

 

 

*RT p p V RT p p*

 (45)

,g ,g *RT p p* ln / (41)

(mixture of ideal gases) (42)

,g ,g *RT f p* ln / (43)

(mixture of gases) (46)

(44)

Integration from the standard state to some actual state then yields

 / *a p p* 

Comparing with the definition of activity it follows

 

is a function only of the content of that component.

 

a component

coefficient

whole situation.

fugacity (*f*) for the partial pressure:

/ *a f p*

 

> 

relationship between compositions of equilibrated liquid and gaseous phases final form of ,l dependence on the composition of liquid is obtained. For example, with Raoult's law *p* = *xp*\* and ideal gas phase we have this equation

$$
\mu\_{\alpha,1} = \mu\_{\alpha,\emptyset}^\circ + RT\ln\left(\mathbf{x}\_{\alpha} p\_{\alpha}^\ast \;/\; p^\circ\right) \equiv \mu\_{\alpha}^{\text{ref}} + RT\ln\mathbf{x}\_{\alpha} \tag{47}
$$

which has, in fact, inspired the definition of an ideal (liquid, solid, or gas) mixture as a mixture with the chemical potential defined, at a given *T* and *p*, as ref *RT x* ln where ref is a function of both *T* and *p*. This definition, as well as the identity in (47), can be simply related to the definition of activity only if the standard state is selected consistently with the reference state, i.e. if the former is a function of both *T* and *p*. If the standard state is selected as dependent on temperature, as it should be, than the pressure factor () should be introduced (see, e.g., de Voe, 2001)

$$\Gamma\_{\alpha} = \exp\left(\frac{\mu\_{\alpha}^{\text{ref}} - \mu\_{\alpha}^{\circ}}{RT}\right) \tag{48}$$

Then the activity of a (non-electrolyte) component in real solution is written as *a x* where is the activity coefficient introduced by the equation ref *RT x* ln . Introducing activities in place of concentrations means in this case to know the pressure factor and to transform molar fractions into molar concentrations to be consistent with thermodynamics.

The main problems with using activities defined for liquid systems can be summarized as follows. Activity is based on molar fractions whereas kinetic uses concentrations. Although there are formulas for the conversion of these variables they do not allow direct substitution, they introduce other variables (e.g., solution density) and lead to rather complex expression of thermodynamic equilibrium constant in concentrations. Whereas concentrations of all species are independent (variables) this is not true for molar fractions – value of one from them is unambiguously determined by values of remaining ones. Chemical potential in liquid and activity based on it are introduced on the basis of (liquid-gas) equilibrium while kinetics essentially works with reactions out of equilibrium. Applicability of equilibriumbased formulated in fugacities are really rare in nonequilibrium states deserves further study. The problem with molar fractions can be resolved by the use of molar concentration based Henry's law giving for ideal-dilute solution ref ,l , *<sup>c</sup> RT c c* ln / , however, rate equations should be formulated with the standard concentration. Sometimes following relationship is used: ref ,l , *<sup>c</sup> RT c c* ln / (Ederer & Gilles, 2007) where *c* is the sum of all concentrations. In this case, the invertibility for *c* is problematic because it is included in *c*; reaction rates should be then formulated in *c*/*c* instead of concentrations that is quite unusual. Of course, the value of activity is dependent on the selected standard state, anyway. All attempts to relate thermodynamic and kinetic equilibrium constants should pay great attention to the selection of standard state and its consequences to be really rigorous and correct.

It is clear from this basic overview that chemical potential, activity and their interrelation are in principle equilibrium quantities which, in kinetic applications, are to be used for

Thermodynamics and Reaction Rates 689

to densities). This invertibility is not self-evident and the best way would be to prove it. Samohýl has proved (Samohýl, 1982, 1987) that if mixture of linear fluids fulfils Gibbs'

ensures the invertibility. This stability is a standard requirement for reasonable behavior of many reacting systems of chemist's interest, consequently the invertibility can be considered

*J J* 1 2 , ,, , ,,,, , ,, , *nn n T J gg gT J* 1 2 1 2 *T*

where the last transformation was made using the following transformation of (specific) chemical potential into the traditional chemical potential (which will be called the molar chemical potential henceforth): = *g M*. Using the definition of activity (37) another transformation, to activities, can be made providing that the standard state is a function of

*J T* 1 2 , ,, , ,,,, *n n J a*1 2 *a a T*

It should be stressed that chemical potential of component as defined by (49) is a function of densities of all components, i.e. of , = 1,..., *n*, therefore also the molar chemical

Although the functions (dependencies) given above were derived for specific case of linear fluids they are still too general. Yet simpler fluid model is the simple mixture of fluids which is defined as mixture of linear fluids constitutive (state) equations of which are independent

/ 0 for ; , 1, ,

a function of density of this component only (and of temperature). Mixture of ideal gases is defined as a simple mixture with additional requirement that partial internal energy and enthalpy are dependent on temperature only. Then it can be proved (Samohýl, 1982, 1987)

> 

that is slightly more general than the common model of ideal gas for which *R* = *R*/*M*. Thus the expression (41) is proved also at nonequilibrium conditions and this is probably only one mixture model for which explicit expression for the dependence of chemical potential on composition out of equilibrium is derived. There is no indication for other cases

check conformity of the traditional ideal mixture model with the definition of simple mixture. For solute in an ideal-dilute solution following concentration-based expression is

 

 

*f*

 

 

 

(50)

(51)

 / 

1 2 , ,, , *<sup>n</sup> T* are invertible (with respect

(, = 1,..., *n*) is regular which

 

*cc cT* 1 2 ,,,, *<sup>n</sup>* . Note that generally

*n* (52)

) is a function of densities, or chemical potentials, or

,*T* , i.e. the chemical potential of any component is

( ) ln / *T RT p p* (53)

,*T* should be just of the logarithmic form like (47). Let us

 

 

on density gradients. Then it can be shown (Samohýl, 1982, 1987) that

 

*g g* 

 

potential, i.e. providing that functions *g g*

 

temperature only:

stability conditions then the matrix with elements *g*

to be guaranteed and we can transform the rate functions as follows:

 

potential is following function of composition:

any rate of formation or destruction (*J*

activities, etc. of all components.

and, consequently, also that *g g*

that chemical potential is given by

while the function *g g*

used:

 

non-equilibrium situations. Let us now trace one relatively simple non-equilibrium approach to description of chemically reacting systems and its results regarding the chemical potential. Samohýl has developed rational thermodynamic approach for chemically reacting fluids with linear transport properties (henceforth called briefly linear fluids) and these fluids seem to include many (non-electrolyte) systems encountered in chemistry (Samohýl & Malijevský, 1976; Samohýl, 1982, 1987). This is a continuum mechanics based approach working with densities of quantities and specific quantities (considered locally, in other words, as fields but this is not crucial for the present text) therefore it primarily uses densities of components (more precisely, the density of component mass) instead of their molar concentrations or fractions that are common in chemistry. This density, in fact, is known in chemistry as a mass concentration with dimension of mass per (unit) volume and can be thus easily recalculated to concentration quantities more common in chemistry. Chemical potential of a reacting component is defined in this theory as follows:

$$\mathcal{g}\_a = \hat{c}\{\rho \overline{f}\} / \hat{c}\rho\_a \tag{49}$$

Here is the density of mixture, i.e. the sum of all component densities , and *f* is the specific free energy of (reacting) mixture as a function of relevant independent variables (the value of this function is denoted by *f*). Inspiration for this definition came from the entropic inequality (the "second law" of thermodynamics) as formulated in rational thermodynamics generally for mixtures and from the fact that this definition enabled to derive classical (equilibrium) thermodynamic relations in the special case that is covered by classical theory. The chemical potential *g* thus has the dimensions of energy per mass. The product *f* essentially transforms the specific quantity to its density and the definition (49) can be viewed as a generalization of the classical definition (36) – partial derivative of mixture free energy (as a function) with respect to an independent variable expressing the amount of a component.

The specific free energy *f* is function of various (mostly kinematic and thermal) variables but here it is sufficient to note that component densities are among them, of course.

In the case of linear fluids it can be proved that free energy is function of densities and temperature only, *f f T* 1 2 , ,, , *<sup>n</sup>* . The same result is proved also for chemical potentials *g* and also for reaction rates expressed as component mass created or destroyed by chemical reactions at a given place and time in unit volume, *rr T* 1 2 , ,, , *<sup>n</sup>* . These rates can be easily transformed to molar basis much more common in chemistry using the molar mass *M*: *J* = *r*/*M*. Component densities are directly related to molar concentration by a similar equation: *c* = /*M*. In this way, the well known kinetic empirical law – the law of mass action – is derived theoretically in the form: *J J cc cT* 1 2 ,,,, *<sup>n</sup>* . Apparently, activities could be introduced into this function as independent variables controlling reaction rates by means of relations as *a cc* / but this is not rigorous because these relations are consequences of chemical potential and its explicit dependence on mixture composition and not definitions per se. Therefore, chemical potentials should be introduced as independent variables at first. This could be done providing that component densities can be expressed as functions of chemical

non-equilibrium situations. Let us now trace one relatively simple non-equilibrium approach to description of chemically reacting systems and its results regarding the chemical potential. Samohýl has developed rational thermodynamic approach for chemically reacting fluids with linear transport properties (henceforth called briefly linear fluids) and these fluids seem to include many (non-electrolyte) systems encountered in chemistry (Samohýl & Malijevský, 1976; Samohýl, 1982, 1987). This is a continuum mechanics based approach working with densities of quantities and specific quantities (considered locally, in other words, as fields but this is not crucial for the present text) therefore it primarily uses densities of components (more precisely, the density of component mass) instead of their molar concentrations or fractions that are common in chemistry. This density, in fact, is known in chemistry as a mass concentration with dimension of mass per (unit) volume and can be thus easily recalculated to concentration quantities more common in chemistry. Chemical potential of a reacting component is

> *g f* /

Here is the density of mixture, i.e. the sum of all component densities , and *f* is the specific free energy of (reacting) mixture as a function of relevant independent variables (the value of this function is denoted by *f*). Inspiration for this definition came from the entropic inequality (the "second law" of thermodynamics) as formulated in rational thermodynamics generally for mixtures and from the fact that this definition enabled to derive classical (equilibrium) thermodynamic relations in the special case that is covered by classical theory. The chemical potential *g* thus has the dimensions of energy per mass. The product

essentially transforms the specific quantity to its density and the definition (49) can be viewed as a generalization of the classical definition (36) – partial derivative of mixture free energy (as a function) with respect to an independent variable expressing the amount of a

The specific free energy *f* is function of various (mostly kinematic and thermal) variables

In the case of linear fluids it can be proved that free energy is function of densities and

potentials *g* and also for reaction rates expressed as component mass created or destroyed by chemical reactions at a given place and time in unit volume, *rr T*

These rates can be easily transformed to molar basis much more common in chemistry using

concentration by a similar equation: *c* = /*M*. In this way, the well known kinetic empirical law – the law of mass action – is derived theoretically in the form:

but this is not rigorous because these relations are consequences of chemical potential and its explicit dependence on mixture composition and not definitions per se. Therefore, chemical potentials should be introduced as independent variables at first. This could be done providing that component densities can be expressed as functions of chemical

 . Apparently, activities could be introduced into this function as independent variables controlling reaction rates by means of relations as *a cc*

1 2 , ,, , *<sup>n</sup>* . The same result is proved also for chemical

= *r*/*M*. Component densities are directly related to molar

 

 /

 1 2 , ,, , *<sup>n</sup>* .

but here it is sufficient to note that component densities are among them, of course.

(49)

*f*

defined in this theory as follows:

temperature only, *f f T*

the molar mass *M*: *J*

*J J cc cT* 1 2 ,,,, *<sup>n</sup>*

   

component.

potential, i.e. providing that functions *g g* 1 2 , ,, , *<sup>n</sup> T* are invertible (with respect to densities). This invertibility is not self-evident and the best way would be to prove it. Samohýl has proved (Samohýl, 1982, 1987) that if mixture of linear fluids fulfils Gibbs' stability conditions then the matrix with elements *g* / (, = 1,..., *n*) is regular which ensures the invertibility. This stability is a standard requirement for reasonable behavior of many reacting systems of chemist's interest, consequently the invertibility can be considered to be guaranteed and we can transform the rate functions as follows:

$$\mathbf{J}^{\alpha} = \overline{\mathbf{J}}^{\alpha} \left( \rho\_1, \rho\_2, \dots, \rho\_{n'} T \right) = \widehat{\mathbf{J}}^{\alpha} \left( \mathbf{g}\_1, \mathbf{g}\_2, \dots, \mathbf{g}\_{n'} T \right) = \overline{\mathbf{J}}^{\alpha} \left( \mu\_1, \mu\_2, \dots, \mu\_{n'} T \right) \tag{50}$$

where the last transformation was made using the following transformation of (specific) chemical potential into the traditional chemical potential (which will be called the molar chemical potential henceforth): = *g M*. Using the definition of activity (37) another transformation, to activities, can be made providing that the standard state is a function of temperature only:

$$\overline{\mathfrak{J}}^{\alpha} \left( \mu\_1, \mu\_2, \dots, \mu\_n, T \right) = \widetilde{\mathfrak{J}}^{\alpha} \left( a\_1, a\_2, \dots, a\_n, T \right) \tag{51}$$

It should be stressed that chemical potential of component as defined by (49) is a function of densities of all components, i.e. of , = 1,..., *n*, therefore also the molar chemical potential is following function of composition: *cc cT* 1 2 ,,,, *<sup>n</sup>* . Note that generally any rate of formation or destruction (*J* ) is a function of densities, or chemical potentials, or activities, etc. of all components.

Although the functions (dependencies) given above were derived for specific case of linear fluids they are still too general. Yet simpler fluid model is the simple mixture of fluids which is defined as mixture of linear fluids constitutive (state) equations of which are independent on density gradients. Then it can be shown (Samohýl, 1982, 1987) that

$$
\partial \overline{\hat{f}}\_{\alpha} \;/ \, \partial \rho\_{\gamma} = 0 \quad \text{for } \alpha \neq \gamma; \; \alpha, \gamma = 1, \dots, n \tag{52}
$$

and, consequently, also that *g g* ,*T* , i.e. the chemical potential of any component is a function of density of this component only (and of temperature). Mixture of ideal gases is defined as a simple mixture with additional requirement that partial internal energy and enthalpy are dependent on temperature only. Then it can be proved (Samohýl, 1982, 1987) that chemical potential is given by

$$\mathbf{g}\_a = \mathbf{g}\_a^\circ(T) + R\_a T \ln\left(p\_a \,/\, p^\circ\right) \tag{53}$$

that is slightly more general than the common model of ideal gas for which *R* = *R*/*M*. Thus the expression (41) is proved also at nonequilibrium conditions and this is probably only one mixture model for which explicit expression for the dependence of chemical potential on composition out of equilibrium is derived. There is no indication for other cases while the function *g g* ,*T* should be just of the logarithmic form like (47). Let us check conformity of the traditional ideal mixture model with the definition of simple mixture. For solute in an ideal-dilute solution following concentration-based expression is used:

Thermodynamics and Reaction Rates 691

also out of it and the final simplified approximating polynomial, called thermodynamic polynomial, follows and represents rate equation of mass action type. More details on this method can be found elsewhere (Samohýl & Malijevský, 1976; Pekař, 2009, 2010). Here it is

First example is the mixture of two isomers discussed in Section 2. 3. Rate of the only one independent reaction, selected as A = B, is approximated by a polynomial of the second

The concentration of B is expressed from the equilibrium constant, (*c*B)eq = *K*(*c*A)eq and substituted into (58) with *J*1 = 0. Following form of the polynomial in equilibrium is

Eq. (59) should be valid for any values of equilibrium concentrations, consequently

considered this is not achieved by simple substitution of *K* for *<sup>j</sup> k*

equilibrium constant defined by (56): ln*K R*

with no need of additional consistency conditions.

extends the scheme (R2) and includes also bimolecular isomerization path: 2A = 2B.

This example illustrated how thermodynamics can be consistently connected to kinetics considering only independent reactions and results of nonequilibrium thermodynamics

Example of simple combination reaction A + B = AB will illustrate the use of molar chemical potential in rate equations. In this mixture of three components composed from two atoms only one independent reaction is possible. Just the given reaction can be selected with

Substituting (60) into (58) the final thermodynamic polynomial (of the second degree) results:

Note, that coefficients *kij* are functions of temperature only and can be interpreted as mass action rate constants (there is no condition on their sign, if some *kij* is negative then traditional rate constant is *kij* with opposite sign). Although only the reaction A = B has been selected as the independent reaction, its rate as given by (61) contains more than just traditional mass action term for this reaction. Remember that component rates are given by (28). Selecting *k*02 = 0 two terms remain in (61) and they correspond to the traditional mass action terms just for the two reactions supposed in (R2). Although only one reaction has been selected to describe kinetics, eq. (61) shows that thermodynamic polynomial does not exclude other (dependent) reactions from kinetic effects and relationship very close to *J*1 = *r*<sup>1</sup> + *r*2, see also (29), naturally follows. No Wegscheider conditions are necessary because there are no reverse rate constants. On contrary, thermodynamic equilibrium constant is directly involved in rate equation; it should be stressed that because no reverse constant are

2 2

22 2 <sup>2</sup>

2 2 1 00 10 A 01 B 20 A 02 B 11 A B *J k k c kc k c k c kcc* (58)

00 10 01 A eq 20 02 11 A eq 0 ( *k k Kk c k K k Kk c* ) ( ) (59)

2 00 10 01 20 02 11 *k k Kk k K k Kk* 0; ; (60)

1 10 A B 02 A B 11 A A B *J k Kc c k K c c k Kc c c* (61)

from (27). Eq. (61) also

A B AB /( ) *T* and equal to

illustrated on two examples relevant for this article.

degree:

obtained:

$$
\mu\_a = \mu\_a^{\text{ref}} + RT \ln \left( c\_a / c^\circ \right) \tag{54}
$$

where ref includes (among other) the gas standard state and concentration-based Henry's constant. Changing to specific quantities and densities we obtain:

$$\mathcal{g}\_a = \mu\_a^{\text{ref}} \;/\; M\_a + \left(\text{RT} \;/\; M\_a\right) \ln\left(\rho\_a \;/\; M\_a \mathbf{c}^\circ\right) \tag{55}$$

which looks like a function of and *T* only, i.e. the simple mixture function *g g* ,*T* . However, the referential state is a function of pressure so this is not such function rigorously. Except ideal gases there is probably no proof of applicability of classical expressions for dependence of chemical potential on composition out of equilibrium and no proof of its logarithmic point. There are probably also no experimental data that could help in resolving this problem.

## **4. Solution offered by rational thermodynamics**

Rational thermodynamics offers certain solution to problems presented so far. It should be stressed that this is by no means totally general theory resolving all possible cases. But it clearly states assumptions and models, i. e. scope of its potential application.

The first assumption, besides standard balances and entropic inequality (see, e.g., Samohýl, 1982, 1987), or model is the mixture of linear fluids in which the functional form of reaction rates was proved: *J J cc cT* 1 2 ,,,, *<sup>n</sup>* (Samohýl & Malijevský, 1976; Samohýl, 1982, 1987). Only independent reaction rates are sufficient that can be easily obtained from component rates, cf. (26) from which further follows that they are function of the same variables. This function, *J Jcc cT ii n* 1 2 ,,,, , is approximated by a polynomial of suitable degree (Samohýl & Malijevský, 1976; Samohýl, 1982, 1987). Equilibrium constant is defined for each independent reaction as follows:

$$-RT\ln K\_p = \sum\_{\alpha=1}^{n} \mu\_{\alpha}^{\circ} P^{\mu \alpha}; \ p = 1, 2, \dots, n-h \tag{56}$$

Activity (37) is supposed to be equal to molar concentrations (divided by unit standard concentration), which is possible for ideal gases, at least (Samohýl, 1982, 1987). Combining this definition of activity with the proved fact that in equilibrium eq 1 () 0 *<sup>n</sup> <sup>p</sup> <sup>P</sup>* (Samohýl, 1982, 1987) it follows

$$K\_p = \prod\_{\alpha=1}^n \left[ \left( c\_{\alpha} \right)\_{\text{eq}} \right]^{p^{\text{pv}}} \tag{57}$$

Some equilibrium concentrations can be thus expressed using the others and (57) and substituted in the approximating polynomial that equals zero in equilibrium. Equilibrium polynomial should vanish for any concentrations what leads to vanishing of some of its coefficients. Because the coefficients are independent of equilibrium these results are valid

ref *g M RT M M c*

which looks like a function of and *T* only, i.e. the simple mixture function

function rigorously. Except ideal gases there is probably no proof of applicability of classical expressions for dependence of chemical potential on composition out of equilibrium and no proof of its logarithmic point. There are probably also no experimental data that could help

Rational thermodynamics offers certain solution to problems presented so far. It should be stressed that this is by no means totally general theory resolving all possible cases. But it

The first assumption, besides standard balances and entropic inequality (see, e.g., Samohýl, 1982, 1987), or model is the mixture of linear fluids in which the functional form of reaction

1987). Only independent reaction rates are sufficient that can be easily obtained from component rates, cf. (26) from which further follows that they are function of the same variables. This function, *J Jcc cT ii n* 1 2 ,,,, , is approximated by a polynomial of suitable degree (Samohýl & Malijevský, 1976; Samohýl, 1982, 1987). Equilibrium constant is defined

> ln ; 1, 2, , *<sup>n</sup> <sup>p</sup> RT K P p n h <sup>p</sup>*

Activity (37) is supposed to be equal to molar concentrations (divided by unit standard concentration), which is possible for ideal gases, at least (Samohýl, 1982, 1987). Combining

eq

Some equilibrium concentrations can be thus expressed using the others and (57) and substituted in the approximating polynomial that equals zero in equilibrium. Equilibrium polynomial should vanish for any concentrations what leads to vanishing of some of its coefficients. Because the coefficients are independent of equilibrium these results are valid

1

this definition of activity with the proved fact that in equilibrium eq

*K c <sup>p</sup>*

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

 

constant. Changing to specific quantities and densities we obtain:

 

clearly states assumptions and models, i. e. scope of its potential application.

**4. Solution offered by rational thermodynamics** 

rates was proved: *J J cc cT* 1 2 ,,,, *<sup>n</sup>* 

for each independent reaction as follows:

(Samohýl, 1982, 1987) it follows

where ref 

*g g* 

in resolving this problem.

ref

 

includes (among other) the gas standard state and concentration-based Henry's

 

,*T* . However, the referential state is a function of pressure so this is not such

 

(Samohýl & Malijevský, 1976; Samohýl, 1982,

(56)

1

(57)

() 0 *<sup>n</sup> <sup>p</sup> <sup>P</sup>*

 

*RT c c* ln / (54)

 

/ / ln / (55)

also out of it and the final simplified approximating polynomial, called thermodynamic polynomial, follows and represents rate equation of mass action type. More details on this method can be found elsewhere (Samohýl & Malijevský, 1976; Pekař, 2009, 2010). Here it is illustrated on two examples relevant for this article.

First example is the mixture of two isomers discussed in Section 2. 3. Rate of the only one independent reaction, selected as A = B, is approximated by a polynomial of the second degree:

$$J\_1 = k\_{00} + k\_{10}c\_A + k\_{01}c\_B + k\_{20}c\_A^2 + k\_{02}c\_B^2 + k\_{11}c\_Ac\_B \tag{58}$$

The concentration of B is expressed from the equilibrium constant, (*c*B)eq = *K*(*c*A)eq and substituted into (58) with *J*1 = 0. Following form of the polynomial in equilibrium is obtained:

$$\text{l.0} = k\_{00} + \left(k\_{10} + K k\_{01}\right) \text{(c}\_{\text{A}}\text{)}\_{\text{eq}} + \left(k\_{20} + K^2 k\_{02} + K k\_{11}\right) \text{(c}\_{\text{A}}\text{)}\_{\text{eq}}^2 \tag{59}$$

Eq. (59) should be valid for any values of equilibrium concentrations, consequently

$$k\_{00} = 0; \quad k\_{10} = -Kk\_{01}; \quad k\_{20} = -K^2k\_{02} - Kk\_{11} \tag{60}$$

Substituting (60) into (58) the final thermodynamic polynomial (of the second degree) results:

$$J\_1 = k\_{10} \left( -Kc\_\text{A} + c\_\text{B} \right) + k\_{02} \left( -K^2 c\_\text{A}^2 + c\_\text{B}^2 \right) + k\_{11} \left( -Kc\_\text{A}^2 + c\_\text{A} c\_\text{B} \right) \tag{61}$$

Note, that coefficients *kij* are functions of temperature only and can be interpreted as mass action rate constants (there is no condition on their sign, if some *kij* is negative then traditional rate constant is *kij* with opposite sign). Although only the reaction A = B has been selected as the independent reaction, its rate as given by (61) contains more than just traditional mass action term for this reaction. Remember that component rates are given by (28). Selecting *k*02 = 0 two terms remain in (61) and they correspond to the traditional mass action terms just for the two reactions supposed in (R2). Although only one reaction has been selected to describe kinetics, eq. (61) shows that thermodynamic polynomial does not exclude other (dependent) reactions from kinetic effects and relationship very close to *J*1 = *r*<sup>1</sup> + *r*2, see also (29), naturally follows. No Wegscheider conditions are necessary because there are no reverse rate constants. On contrary, thermodynamic equilibrium constant is directly involved in rate equation; it should be stressed that because no reverse constant are considered this is not achieved by simple substitution of *K* for *<sup>j</sup> k* from (27). Eq. (61) also extends the scheme (R2) and includes also bimolecular isomerization path: 2A = 2B.

This example illustrated how thermodynamics can be consistently connected to kinetics considering only independent reactions and results of nonequilibrium thermodynamics with no need of additional consistency conditions.

Example of simple combination reaction A + B = AB will illustrate the use of molar chemical potential in rate equations. In this mixture of three components composed from two atoms only one independent reaction is possible. Just the given reaction can be selected with equilibrium constant defined by (56): ln*K R* A B AB /( ) *T* and equal to

Thermodynamics and Reaction Rates 693

Both approaches are closely connected through the question of using activities, that are common in thermodynamics, in place of concentrations in kinetic equations and the

Thermodynamic equilibrium constant and the ratio of forward and reversed rate constants are conceptually different and cannot be identified. Restrictions following from the former

Direct introduction of chemical potential into traditional mass action rate equations is incorrect due to incompatibility of concentrations and activities and is problematic even in

Rational thermodynamic treatment of chemically reacting mixtures of fluids with linear transport properties offers some solution to these problems whenever its clearly stated assumptions are met in real reacting systems of interest. No compatibility conditions, no Wegscheider relations (that have been shown to be results of dependence among reactions) are then necessary, thermodynamic equilibrium constants appear in rate equations, thermodynamics and kinetics are connected quite naturally. The role of ("thermodynamically") independent reactions in formulating rate equations and in kinetics

Future research should focus attention on the applicability of dependences of chemical potential on concentrations known from equilibrium thermodynamics in nonequilibrium states, or on the related problem of consistent use of activities and corresponding standard

Though practical chemical kinetics has been successfully surviving without special incorporation of thermodynamic requirements, except perhaps equilibrium results, tighter connection of kinetics with thermodynamics is desirable not only from the theoretical point of view but may be of practical importance considering increasing interest in analyzing of complex biochemical network or increasing computational capabilities for correct modeling of complex reaction systems. The latter when combined with proper thermodynamic requirements might contribute to more effective practical, industrial exploitation of chemical

The author is with the Centre of Materials Research at the Faculty of Chemistry, Brno University of Technology; the Centre is supported by project No. CZ.1.05/2.1.00/01.0012 from ERDF. The author is indebted to Ivan Samohýl for many valuable discussions on

Blum, L.H. & Luus, R. (1964). Thermodynamic Consistency of Reaction Rate Expressions. *Chemical Engineering Science*, Vol.19, No.4, pp. 322-323, ISSN 0009-2509 Boudart, M. (1968). *Kinetics of Chemical Processes*, Prentice-Hall, Englewood Cliffs, USA Bowen, R.M. (1968). On the Stoichiometry of Chemically Reacting Systems. *Archive for Rational Mechanics and Analysis*, Vol.29, No.2, pp. 114-124, ISSN 0003-9527 Boyd, R.K. (1977). Macroscopic and Microscopic Restrictions on Chemical Kinetics. *Chemical* 

*Reviews*, Vol.77, No.1, pp. 93-119, ISSN 0009-2665

on values of rate constants should be found indirectly as shown in Scheme 1.

problem of expressing activities as function of concentrations.

ideal systems.

in general is clarified.

states in rate equations.

**6. Acknowledgment** 

rational thermodynamics.

**7. References** 

processes.

 AB A B eq *K c cc* / , cf. (57). The second degree thermodynamic polynomial results in this case in following rate equation:

$$J\_1 = k\_{110} (\varepsilon\_\text{A} \varepsilon\_\text{B} - K^{-1} \varepsilon\_\text{AB}) \tag{62}$$

that represents the function 1 1 A B AB *J J Tc c c* (, , , ) . Its transformation to the function 1 1 A B AB *J JT* (, , , ) gives:

$$J\_1 = k\_{110} \exp\left(\frac{-\mu\_\text{A}^\circ - \mu\_\text{B}^\circ}{RT}\right) \left[\exp\left(\frac{\mu\_\text{A} + \mu\_\text{B}}{RT}\right) - \exp\frac{\mu\_\text{AB}}{RT}\right] \tag{63}$$

This is thermodynamically correct expression (for the supposed thermodynamic model) of the function *J* discussed in Section 3 and in contrast to (1). It is clear that proper "thermodynamic driving force" for reaction rate is not simple (stoichiometric) difference in molar chemical potentials of products and reactants. The expression in square brackets can be considered as this driving force. Equation (63) also lucidly shows that high molar chemical potential of reactants in combination with low molar chemical potential of products can naturally lead to high reaction rate as could be expected. On the other hand, this is achieved in other approaches, based on *i i* , due to arbitrary selection of signs of stoichiometric coefficients. In contrast to this straightforward approach illustrated in introduction, also kinetic variable (*k*110) is still present in eq. (63), explaining why some

"thermodynamically highly forced" reactions may not practically occur due to very low reaction rate. Equation (63) includes also explicit dependence of reaction rate on standard state selection (cf. the presence of standard chemical potentials). This is inevitable consequence of using thermodynamic variables in kinetic equations. Because also the molar chemical potential is dependent on standard state selection, it can be perhaps assumed that these dependences are cancelled in the final value of reaction rate.

Rational thermodynamics thus provides efficient connection to reaction kinetics. However, even this is not totally universal theory; on the other hand, presumptions are clearly stated. First, the procedure applies to linear fluids only. Second, as presented here it is restricted to mixtures of ideal gases. This restriction can be easily removed, if activities are used instead of concentrations, i.e. if functions *J* are used in place of functions *J* – all equations remain unchanged except the symbol *a* replacing the symbol *c*. But then still remains the problem how to find explicit relationship between activities and concentrations valid at non equilibrium conditions. Nevertheless, this method seems to be the most carefully elaborated thermodynamic approach to chemical kinetics.
