Planck constant in Eyring theory of rate processes and viscosity

Question

In my research I came across an old and probably rather obscure theory, namely Eyring's theory of rate processes [1,2]. This theory is the bedrock of an important model used to explain wetting dynamics at the molecular scale, namely Molecular Kinetic Theory [3].

Eyring equation for chemical kinetics reads: $$ \kappa = \frac{k_BT}{h}\exp{\Big(-\frac{\Delta G}{k_BT}\Big)} , \quad (1) $$ being $\kappa$ the reaction rate, $T$ the temperature, $\Delta G$ the Gibb's free energy of activation, $k_B$ Boltzmann's constant and $h$ Planck's constant.

I work with Molecular Dynamics (MD) simulations which, despite the name, are purely classical simulations, i.e. no quantum effect is considered directly, following the Born-Oppenheimer approximation. People have been using MD for decades and nowadays it is possible to reproduce most of the properties of simple liquids, which leads me to think that liquids are under a first (and maybe second) order approximation 'classical things'. Nonetheless, equation (1) has been candidly employed in modeling liquids. Furthermore, the very partition function of a generic liquid has been postulated to be [4]: $$ f^N = \Big[V_f\Big(\frac{2\pi m k_B T}{h^2}\Big)^{3/2}\Big]^N\exp{\Big(-\frac{E}{k_BT}\Big)} , \quad (2) $$ being $N$ the number of liquid particles, $V_f$ the free volume of the liquid, $m$ the mass of a liquid particle, $J$ the partition function for internal molecular configurations and $E$ the internal energy of the liquid. Provided with a model for the free volume, the combination of (1) and (2) has been used to derive transport coefficients, such as viscosity [5,6]: $$ \eta = \frac{h}{v}\exp{\Big(\frac{\Delta g}{k_BT}\Big)} , \quad (3) $$ being $v$ the volume of a fluid molecule and $\Delta g$ the per-molecule activation energy.

I have a background in mathematics, not in physics, so in my mindset a constant is just a constant. However, as far as I know Planck's constant 'emerges' only when faced with inherently quantum behaviour. So my questions are:

1. Why does the model of a classical transport coefficient such as viscosity involve the Planck's constant?
2. Why would the attempt rate of a rate process (chemical reaction or else) be $k_BT/h$? How general is (1)?

I haven't found much information on the derivation of (1) and I remark that (2) is postulated.

References:
[1] https://en.wikipedia.org/wiki/Eyring_equation
[2] Glasstone, Laidler and Eyring, The Theory of Rate Processes, McGraw Hill (1941)
[3] Blake, The physics of moving wetting lines, J. Colloid Interface Sci. (2006)
[4] Eyring and Hirschfelder, The theory of the liquid state, J. Phys. Chem. (1937)
[5] Hirschfelder, Stevenson and Eyring, A Theory of Liquid Structure, J. Phys. Chem. (1937)
[6] Li and Chang, Self-Diffusion Coefficient and Viscosity in Liquids, J. Phys. Chem. (1955)

Answer 1

The Planck constant arose from a quantum theory that preceded quantum mechanics.

It was found that statistical mechanical computations could reproduce thermodynamic properties on the basis of assigning equal a priori probability to different "states" of the system and counting the ones that had a desired property.

Ideal gasses with N particles have states described by N different positions together with N different momenta. Planck found that the number of different states in a region of this so-called phase-space was proportional to its volume integral over x's and p's.

The product x times p has dimensions of Action, Planck needed a physical scale for action, he called it h. And a good thing, too.

No Schrodinger Equation is needed to bring Planck's constant into statistical models.

This subject is converd nicely in:

How does one quantize the phase-space semiclassically?