Extracting meaningful standard enthalpies and entropies of activation for surface reactions from kinetic rates

Abstract

While analyses based on the Arrhenius kinetic model have been successfully applied since its introduction, the Arrhenius model neglects to describe pre-exponential factor in a scientifically meaningful fashion. Since the 1930s, transition-state theory (TST), has met with success in interpreting the pre-exponential factor’s value, allowing a standard entropy of activation to be estimated. However, analyses based on TST’s assumptions have been applied inconsistently in the literature, particularly in corrosion science, leading to difficulty in comparison of standard entropy of activations from different studies. In this work, the foundational principles of TST and standard states are discussed and a standard method to apply TST in analyzing rates of surface reactions is recommended. When full details are not available, reacting species’ concentrations should be normalized to the concentration of active surface sites. For corrosion reactions, conversion relationships are given to convert from units of corrosion rate to surface reaction rate, consistent with TST. This method is dubbed a surface reactant equi-density approximation. Application of this standard to reported data results in adjustments of the standard entropy of activation between − 65 and +50 J/mol K and brings reported entropies into a narrower range of values.

Appendices

Appendix 1: Examples of the use of standard states in TST

Most textbooks initially teach TST in the context of gas phase reactions, and we follow that approach here as it makes some of the concepts easier to explain. Consider a gas phase reaction of A_(g)$\to$ B_(g) + C_(g), where the activity coefficients are 1, the standard state for each gas is considered an ideal gas at 1 bar at a given temperature. For typographical convenience, a double sided arrow ($\leftrightarrow$) will be used rather than the equilibrium symbol (⇋) in subscripts of the following terms and equations. The standard equilibrium constant, $K_{{{\text{A}} \leftrightarrow {\text{B}} + {\text{C}}}}^\circ$, will be related to the kinetic equilibrium constant, $K_{{{\text{A}} \leftrightarrow {\text{B}} + {\text{C}}}}$,and rate constants $\frac{{{\text{k}}_{\text{f}} }}{{{\text{k}}_{\text{r}} }}$ as follows:

$$K_{{{\text{A}} \leftrightarrow {\text{B}} + {\text{C}}}}^\circ = K_{{{\text{A}} \leftrightarrow {\text{B}} + {\text{C}}}} *\frac{{\left( {{\text{c}}_{\text{f}}^\circ } \right)^{ - 1} }}{{\left( {{\text{c}}_{\text{r}}^\circ } \right)^{ - 1} }} = \frac{{{\text{k}}_{\text{f}} }}{{{\text{k}}_{\text{r}} }}\frac{{\left( {{\text{c}}_{\text{f}}^\circ } \right)^{ - 1} }}{{\left( {{\text{c}}_{\text{r}}^\circ } \right)^{ - 1} }}$$

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In conventional TST, a quasi-equilibrium is established between the reactants in the transition state (the activated state). This is the origin of the $\frac{{\left( {{\text{c}}_{\text{R}}^\circ } \right)^{ - 1} }}{{\left( {{\text{c}}_{\text{TS}}^\circ } \right)^{ - 1} }}$ term in Eq. 3, [22] which has the standard states of the reactants in the transition state. This term is part of the Arrhenius pre-exponential. A correct accounting of units (including the standard states), allows extracting of the standard entropy of activation and standard enthalpy of activation from the rate constant from Eq. 3. In the forward reaction of A_(g)$\to$ B_(g) + C_(g), if we assume activity coefficients of 1 and that the transition state consists of a single gas molecule, then:

$$k_{{{\text{A}} \to {\text{B}} + {\text{C}}}} = \kappa \frac{{k_{B} T}}{h}e^{{ - \frac{{\Delta^{\ddag } G^\circ }}{RT}}} \frac{{\gamma_{\text{TS}} }}{{\gamma_{\text{R}} }}\frac{{(c_{TS}^\circ )^{ - 1} }}{{\left( {c_{R}^\circ } \right)^{ - 1} }} = \kappa \frac{{k_{B} T}}{h}e^{{ - \frac{{\Delta^{\ddag } H^\circ }}{RT} + \frac{{\Delta^{\ddag } S^\circ }}{R}}} \frac{{\left( {{\text{P}}_{\text{TS}}^\circ } \right)^{ - 1} }}{{\left( {{\text{P}}_{\text{A}}^\circ } \right)^{ - 1} }}$$

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For such cases where the transition state and reactant are both a single gas molecule, the standard state terms cancel and this is one reason the term is often omitted in textbooks. The IUPAC recommended value for $P^\circ$ is 1 bar, and the standard states would cancel for Eq. 5: $\frac{{\left( {{\text{P}}_{\text{TS}}^\circ } \right)^{ - 1} }}{{\left( {{\text{P}}_{\text{A}}^\circ } \right)^{ - 1} }} = \frac{P^\circ }{P^\circ }$. However, the $c^\circ$ terms do not always cancel, and the authors recommend including them explicitly in both the numerator and denominator prior to cancellation, to avoid misleading readers.

An example where the terms would not cancel is for the bimolecular reaction of D_(g) + E_(g)$\to$ F_(g), $\frac{{(c_{TS}^\circ )^{ - 1} }}{{(c_{R}^\circ )^{ - 1} }} = \frac{{\left( {{\text{P}}_{\text{TS}}^\circ } \right)^{ - 1} }}{{\left( {{\text{P}}_{\text{E}}^\circ } \right)^{ - 1} \left( {{\text{P}}_{\text{E}}^\circ } \right)^{ - 1} }} = \frac{P^\circ P^\circ }{P^\circ }$. Also consider the case of adsorption to a surface with site designated as S, and a single gas molecule A adsorbing: A + S $\to$ A − S, where A − S is the bound state. If we assume that the transition state is like the adsorbed state (immobile and with geometry similar to A − S), then with activity coefficients of 1 we arrive at right-hand side of the following situation:

$$k_{{{\text{A}} + {\text{S}} \to {\text{A}} - {\text{S}}}} = \kappa \frac{{k_{B} T}}{h}e^{{ - \frac{{\Delta^{\ddag } G^\circ }}{RT}}} \frac{{\gamma_{\text{TS}} }}{{\gamma_{\text{R}} }}\frac{{(c_{TS}^\circ )^{ - 1} }}{{\left( {c_{R}^\circ } \right)^{ - 1} }} = \kappa \frac{{k_{B} T}}{h}e^{{ - \frac{{\Delta^{\ddag } H^\circ }}{RT} + \frac{{\Delta^{\ddag } S^\circ }}{R}}} \frac{{\left( {{{\theta }}_{\text{TS}}^\circ } \right)^{ - 1} }}{{\left( {{\text{P}}_{\text{A}}^\circ } \right)^{ - 1} \left( {{{\theta }}_{{{\text{e}} - {\text{sites}}}}^\circ } \right)^{ - 1} }}$$

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Here relative coverage has been chosen (further explained below) for the concentration units in the standard state of the immobile transition state (${{\theta }}_{\text{TS}}^\circ$), and also for the empty sites $\left( {{{\theta }}_{\text{e-sites}}^\circ } \right)$. Again, the units for the standard states do not cancel, and it becomes more important that the units are carefully accounted for when extracting thermodynamics quantities. In contrast, for an elementary chemical reaction step of molecular desorption from a surface, the terms generally will cancel. For solutes, IUPAC recommends using a value of 1 mol solute/L solvent or 1 mol solute/kg solvent, [33] establishing standard states from which standard entropies can be calculated on a common scale [15, 57].

Appendix 2: Standard states of adsorbates on reacting surfaces

For surface-adsorbed species, it is possible to describe the activity of the adsorbates and of the surface using either an absolute coverage (σ_ads) or a relative coverage (${{\theta }}_{\text{ads}}$). For an absolute coverage, the units are relatively clear, such as 1.39 × 10¹⁶ molecules m⁻² [19]. However, for a relative coverage, defined relative to Langmuir (or lattice) adsorption sites, such as (molecules adsorbed)/(lattice sites), it is common to report them as unitless, such as ${{\theta }}_{\text{ads}}$ = 0.34. This lack of specified units is misleading as the term is not truly unitless: a relative concentration is still a type of concentration, defined on a particular scale. In fact, carrying the units is necessary to directly compare results between studies because there are two types of monolayers (lattice-saturated versus geometric [20]), and also because the denominator is sometimes the saturation coverage, leading to three different possible relative scales even for immobile adsorbates.

The authors thus recommend distinguishing the three types using the following acronyms for the different relative coverage units: relative to lattice (rtL) when using (n molecules adsorbed)/(m total lattice sites of that type), relative to saturate coverage (rtS) when using (n molecules adsorbed)/(n_max saturation coverage), and relative to geometric area (rtG) when using (aggregate area of molecules adsorbed)/(geometric surface area). In this case, the three quantities of ${{\theta }}_{\text{ads}}$ = 0.34 rtL, ${{\theta }}_{\text{ads}}$ = 0.34 rtS, and ${{\theta }}_{\text{ads}}$ = 0.34 rtG each have different meanings, and it is possible to convert between them if one knows the number of sites per unit area. For immobile adsorbates, it has been recommended to set the standard state of the adsorbates to being equal to that of the empty sites ${{\theta }}_{\text{ads}}^\circ = {{\theta }}_{\text{e-sites}}^\circ$, as these terms then cancel [19].

For the 2-D gas adsorbates (including 2D gas transition states), ${{\sigma }}_{\text{ads}}^\circ = 1.39 \times 10^{ - 7} {\text{mol/m}}^{2}$ has been recommended [19], though other recommendations do exist [19, 24, 58, 59]. The above two choices (for ${{\theta }}_{\text{ads}}^\circ = {{\theta }}_{\text{e-sites}}^\circ$ and ${{\sigma }}_{\text{ads}}^\circ = 1.39 \times 10^{ - 7} {\text{mol/m}}^{2}$) are based upon enabling standard entropies to be easily compared between different systems to gain insights about the dynamics of the molecule.

It should be noted that an immobile adsorbate and a 2-D gas have different types of phases and different activity units. Therefore, they cannot share the same kind of standard state, and to discriminate between these two cases thus typically requires doing the calculation twice (once assuming an immobile adsorbate model, once assuming a 2-D gas adsorbate model) [20], resulting in two different values for the standard entropy (one for each model) from a single experiment. The experimentally calculated values from the two different models can then be compared to the theoretical values for each model to see which model has a better match between experiment and theory [9]. For a pure solid or liquid phase reacting with the gas phase in stoichiometric reactions (such as dissolution/sublimation of a sugar crystal) the activity and standard state both take a value of 1. For solutes in solution, the IUPAC recommended standard state is 1 mol/L or 1 mol solute/kg solvent [33].

Appendix 3: Effect of using transition state theory on the estimate of the activation energy

Consider a system investigated at two temperatures, $T_{i}$ and $T_{f}$ with experimental reaction rates $Rate_{i}$ and $Rate_{f}$, respectively, and unit concentrations. Assuming ${{\Delta }}^{\ddag } H^\circ$ and $\Delta^{\ddag } S^\circ$ to be constant across the temperature range and that $P{{\Delta }}^{\ddag } V^\circ = 0$, the activation energy may be empirically expressed simply as Eqs. 25 and 26 for the Arrhenius and TST estimations, respectively.⁹ Thus, a plot of ln(k) vs 1/T is not entirely linear for TST, while it is assumed to be linear in an Arrhenius plot.

$$\frac{{\ln \left( {\frac{{Rate_{i} }}{{Rate_{f} }}} \right)T_{f} T_{i} R}}{{T_{i} - T_{f} }} = E_{{a_{\text{Arr - plot,Emp}} }}$$

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$$\frac{{\ln \left( {\frac{{Rate_{i} T_{f} }}{{Rate_{f} T_{i} }}} \right)T_{f} T_{i} R}}{{T_{i} - T_{f} }} = E_{{a_{TST,Emp} }}$$

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Subtracting Eq. 26 from Eq. 25 yields the upper limit for difference between the empirical activation energy estimate from each type of linearization, Eq. 27:

$${{\Delta }}E_{{a,Emp{ - }Error}} = E_{{a_{{Arr{ - }plot,Emp}} }} - E_{{a_{TST,Emp} }} = \frac{{\ln \left( {\frac{{T_{i} }}{{T_{f} }}} \right)T_{f} T_{i} R}}{{T_{i} - T_{f} }}$$

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This difference term is plotted over a wide range of initial and final temperatures (against the temperature difference of the initial and final data points) in Fig. 2.

Fig. 2

Upper limit for the difference in the estimation of the activation energy between an Arrhenius-plot and TST as function of the temperature range of ${\text{T}}_{\text{i}}$ to ${\text{T}}_{\text{f}}$ using Eq. 27. Arrows and lines represent a constant T_f and T_i, respectively, as labeled on the plot. Changes in the x-axis following one of these lines represents changes in the variable term. For example, the arrow closest to both axes, T_f = 500 K, corresponds to T_i = 0 at the x-axis and T_i = 500 K where T_f − T_i = 0 (y-intercept). (Color figure online)

Several points are notable from this figure. First, the maximum difference between these theories can be substantial, up to ~ 20 kJ/mol. However, this is only valid for extremely high temperature studies going to up to 3500 K and beginning at 1500 K. While some systems, such as those for nuclear space propulsion [61] would certainly find such high temperature ranges relevant, most reaction systems are studied well below these conditions. Even forthcoming Generation IV nuclear power applications are being designed for upper temperatures near 1300 K, for the most aggressive systems [61]. In these systems the maximum deviation is significantly lower, near 10–12 kJ/mol. Depending on the system and data qualty, this difference may be within the experimental error of the data. Second, as inspection of Eq. 27 demonstrates, a lower reaction temperature significantly decreases the difference between the estimates. The effect of temperature range is so impactful, that a study ranging from 100 to 2000 K would report a difference of only 3 kJ/mol in the estimated activation energy. Over ranges of several hundred K or more, other errors may become dominant (for example, the common approximation that the standard enthalpy and standard entropy are independent of temperature may become inaccurate). Thus, while some difference between the theories is inescapable, in most systems experimental errors will be larger than this temperature dependence change. From a practical perspective, even a few kJ/mol can make a difference [62], and Eq. 27 can be used to go from activation energy estimates obtained from the end-points in an Arrhenius-plot to a more accurate TST estimate. The above considerations also suggest that a better choice is Eq. 8 or a modified Arrhenius plot of the form, where in the below equation $u_{T}$ has the units of temperature.

$$\ln \left( {\frac{k}{T}\frac{{u_{T} }}{{u_{k} }}} \right) = \ln \left( {\frac{{k_{B} u_{T} }}{{h u_{k} }}e^{{\frac{{{{\Delta }}^{\ddag } S^\circ }}{R} + 1 - \frac{{P{{\Delta }}^{\ddag } V^\circ }}{RT}}} \frac{{(c_{TS}^\circ )^{ - 1} }}{{\left( {c_{R}^\circ } \right)^{ - 1} }}} \right) - \frac{{E_{a} }}{RT}$$

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