**12. The Simplified Version of Dozy-Chaos Mechanics: Proton-Transfer Reactions. On Symmetry in the Brönsted Relationship**

Grounded on the simplified version of dozy-chaos mechanics [16] (Section 11), in 1990, a theoretical description of the basic experimental patterns in the Brönsted relationship [51] for the reactions of proton transfer (acid-base catalysis) was given [12]. The Brönsted relationship was found by Brönsted and Pedersen in 1924 (see [51]). The theory in [16] is immediately appropriate to the explanation of electron transfers. To explain the reactions of transfers of heavy charged particles (proton transfers), the result of thermic fluctuations of the potential barrier transparence must be considered because of fluctuations in the barrier width. In contrast to the elementary proton transfer, the electron-transfer process is insensitive to small fluctuations in the barrier width due to the large size of the electronic wave function in the initial and final states. The analytical formulas for the proton-transfer rate constants are obtained. In acid catalysis, the empirical Brönsted relationship is

$$
\log \mathsf{K}^{(\text{acid})} = a \,\mathrm{lg} \mathcal{K}\_{\text{eq}}^{\text{emp}} + a \tag{43}
$$

where *K*(acid) is the rate constant, *K*emp eq is the empirical equilibrium constant, and α and *a* are constants. In base catalysis, the empirical Brönsted relationship is

$$\log K^{(\text{base})} = \beta \lg K\_{\text{eq}}^{\text{emp}} + b \tag{44}$$

The theoretically-obtained Brönsted coefficients α and β (the Einstein model of nuclear vibrations ωκ = constant ≡ ω) for direct (acid catalysis) and inverse (base catalysis) reactions [12]

$$\alpha = \frac{1}{2} + \frac{L \, k\_{\rm B} T}{\hbar (2f/m)^{1/2}} - \left[1 - \frac{E \sinh t}{2f \sinh(\hbar \omega / 2k\_{\rm B} T)}\right] \frac{2m(k\_{\rm B} T)^2}{\hbar^2 \gamma\_{\rm b}} \tag{45}$$

and

$$\beta = \frac{1}{2} - \frac{L \, k\_{\rm B} T}{\hbar (2J/m)^{1/2}} + \left[1 - \frac{E \, \sinh t}{2J \sinh(\hbar \omega / 2k\_{\rm B} T)}\right] \frac{2m(k\_{\rm B} T)^2}{\hbar^2 \gamma\_{\rm b}}\tag{46}$$

(*t* is given by Equation (41) and γ<sup>b</sup> is barrier rigidity) are symmetric relative to <sup>1</sup> <sup>2</sup> and meet the generally known empirical equality [53,54]

$$
\alpha + \beta = 1 \tag{47}
$$

(which directly follows from the Brönsted relationships (43) and (44)).
