Deriving Andrade's equation

Question

$$\eta=a\exp\frac{B}{RT}$$I'm a high school student, interested in fluid dynamics. I was studying the relationship between viscosity and temperature, and I understood that when liquid heats up, its molecules become excited and begin to move. The energy of this movement is enough to overcome the intermolecular forces, causing a decrease in viscosity. The picture I posted shows Andrade's equation. I understood that R is a gas constant,a,B is a constant, where B is the activation enthalpy.

However, there are a couple of questions about this.

1. In Andrade's equation, where did the gas constant came from?
2. Where did activation enthalpy come from?
3. Is Andrade's equation used just to linearize the exponential relationship between temperature and viscosity?

These are three questions that I would like to know about. (I'm so sorry for the unclear texts.)

Answer 1

Andrade's model is just one of many attempts to match tractable physical assumptions to the empirical viscosity-temperature relation in liquids. I'll offer a high-level overview of what's going on here.

The logic is that under shear stress, particulate motion is primarily small-scale vibration relative to nearest neighbours, at an energy exponentially suppressed relative to the thermal energy scale $k_BT$ ($k_B$ being $R/N_A$ with $N_A$ Avogadro's constant), because it acts counter to the potential barrier neighbours induce. Indeed, thermodynamics at equilibrium likes $\exp-C/T$ factors (see here for an explanation of why $1/T$ tends to be logarithmic in the odds against high-energy events).

In particular, a particle's vibrational energy (and hence frequency) is proportional to $\exp\frac{-B}{RT}$, where it's convenient to write $C=B/R$ so $B$ has units of energy (as you said, it's enthalpy), and has a natural physical interpretation in terms of the aforementioned barrier. But shear stress shifts this energy a little, which means that particulate shifts in the direction of the stress are slightly more likely than shifts against it, resulting in a gradual flow of the bulk liquid. As a result, the flow rate is exponentially suppressed by an $\exp\frac{-B}{RT}$ factor, i.e. the viscosity scales as $\exp\frac{B}{RT}$.

(I've simplified the maths a bit. See Transport Phenomena 2nd Ed. $\S1.5$ for more detail, if you're good at calculus. It explains the conditions required for an approximately linear relation of shear stress to the velocity gradient.)