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In computational chemistry when using transition state theory we often use standard states for all involved structures. However, sometimes people argue that it should be corrected for concentration (and/or pressure) since that can influence the free energy.

Let's assume we got a reaction A + B $\rightarrow$ C. A is 5 molar, B is 2 molar. What concentration should be used for each structure A, B, C and the transition state TS?

If I use starting concentrations the concentration of C would be 0, which is problematic, but if I use final concentrations B would be 0. And in any case in my understanding of transition state theory the concentration of the transition state is considered to be 0.

And what is best to do in cases with several intermediates, different equilibrium and things like that?

So how does one actually correct for concentration?

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  • $\begingroup$ Nvm my deleted comment, I think I see your question now. I will ponder it and come back. But this reinforces why I don't take transition state calculations seriously :) $\endgroup$ – B. Kelly Sep 4 '20 at 16:58
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    $\begingroup$ I believe this is something handled by micro-kinetic modeling normally. Maybe someone can give a related answer $\endgroup$ – Tristan Maxson Sep 5 '20 at 17:53
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TLDR: The rate constant calculated by transition state theory depends on the standard-state (gas-phase: ideal gas at a partial pressure of 1 atm or any state concentration (1 cm$^3$ molecule$^{-1}$ or 1 mol L$^{-1}$), liquid-phase solutes as in an ideal solution of 1 mol L$^{-1}$) and all species (reactants and transition state) should be in that specific reference state. The introduction of concentration effects can be made with microkinetic models.$^1$


Now let me try to convince you of that. The rate law for a bimolecular elementary reaction $\mathrm{A} + \mathrm{B} \to \mathrm{C} + \mathrm{D}$ is $$ \frac{d[\mathrm{C}]}{dt} = k [\mathrm{A}] [\mathrm{B}] $$ and according to transition state theory the rate constant, $k$, at temperature $T$ can be calculated as $$ k(T) = \frac{k_\mathrm{B}T}{h}\frac{Q^\ddagger(T)}{Q^\mathrm{A}(T)Q^\mathrm{B}(T)} \exp[-V^\ddagger_\mathrm{MEP}/k_\mathrm{B}T]$$ in which $k_\mathrm{B}$ is the Boltzmann constant, $h$ is the Planck constant, $Q^\ddagger(T)$, $Q^\mathrm{A}(T)$, and $Q^\mathrm{B}(T)$ are the partition functions of the transition state, reactant A and reactant B, respectively, and $V^\ddagger_\mathrm{MEP}$ is the classical barrier along the minimum energy path. $^2$

Each partition function is a product of the translational partition function per unit volume, $\Phi_\mathrm{tr}(T)$, and the internal partition function. $$ Q(T) = \Phi_\mathrm{tr}(T) Q_\mathrm{int}(T)$$ with $$ \Phi_\mathrm{tr}(T) = \left( \frac{2\pi m k_\mathrm{B}T}{h^2} \right)^{3/2}$$

Note that the ratio between the translational partition functions per unit volume of transition state and reactants can be written as the partition function for the relative motion of the collision: $$ \frac{\Phi_\mathrm{tr}^\ddagger(T)}{\Phi_\mathrm{tr}^\mathrm{A}(T)\Phi_\mathrm{tr}^\mathrm{B}(T)} = \left( \frac{h^2}{2\pi\mu k_\mathrm{B}T}\right)^{3/2}$$ with $\mu = \frac{m_\mathrm{A}m_\mathrm{B}}{m_\mathrm{A} + m_\mathrm{B}}$. This only makes sense if the partition functions are calculated at the same reference state, implicitly here the volume.

So, when using the Eyring equation $$ k(T) = \frac{k_\mathrm{B}T}{hc^\circ} \exp[-\Delta^\ddagger G^\circ/RT]$$ the Gibbs free energies for the transition state and reactants must be calculated at the standard-state indicated in $c^\circ$.

To include concentration effects of has to integrate the rate laws for all reactions of interest. There are several options of codes that do that given a set of chemical equations, rate constants and initial conditions (and other techinical stuff to solve the equations): Acuchem$^3$, Copasi$^4$, Pilgrim$^5$, and probably others.

  1. M. Besora, F. Maseras, Microkinetic modeling in homogeneous catalysis. Wiley Interdiscip. Rev. Comput. Mol. Sci. 8, 1–13 (2018).

  2. A. Fernández-Ramos, J. A. Miller, S. J. Klippenstein, D. G. Truhlar, Modeling the Kinetics of Bimolecular Reactions. Chem. Rev. 106, 4518–4584 (2006).

  3. W. Braun, J. T. Herron, D. K. Kahaner, Acuchem: A computer program for modeling complex chemical reaction systems. Int. J. Chem. Kinet. 20, 51–62 (1988).

  4. S. Hoops, S. Sahle, R. Gauges, C. Lee, J. Pahle, N. Simus, M. Singhal, L. Xu, P. Mendes, U. Kummer, COPASI--a COmplex PAthway SImulator. Bioinformatics. 22, 3067–74 (2006).

  5. D. Ferro-Costas, D. G. Truhlar, A. Fernández-Ramos, Pilgrim: A thermal rate constant calculator and a chemical kinetics simulator. Comput. Phys. Commun. 256, 107457 (2020).

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  • $\begingroup$ Interesting, thank you. Just a question to clarify, the way I read your answer TST is only valid for standard states? Or can I use any concentration/pressure and temperature as long as the same is used for all involved species? $\endgroup$ – DSVA Nov 6 '20 at 11:03
  • $\begingroup$ @DSVA TST can be used for any reference state you want and the same reference state has to be the same for all involved species ( reactants and transition state). Otherwise, the quasiequilibrium that TST assumes is not valid and it is not applicable. $\endgroup$ – Antonio de Oliveira-Filho Nov 7 '20 at 20:57

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