Is there (or was there) a unit of electric current based on Avogadros number or Coulombs constant? This has to do with the SI definition of the Ampere. Why the quantity $2*10^{-7}  $ Newtons in particular? It would make more sense to define 1 Ampere = 1 mole of electron charge per second. Which would be equivalent to 1 Farad/second. The Ampere is not a static unit since it is based on moving electrons. But is there a history as to why they chose that particular amount of force between 2 infinitely long wires 1 meter apart?
 A: Modern SI unit definitions link the value of a unit to a particular mechanical measurement, so that the various base units can be calibrated independently of each other. This is why the definition of the ampere is in terms of an actual mechanical experiment involving two current-carrying wires, long enough that the finite-size corrections don't matter for the precision you want (which is what "infinite" means in practice).
This definition came from an equivalent, easier-to-understand definition of the ampere as "1 coulomb of charge per second." A current of 1 coulomb per second in two wires 1 meter apart will give you a force of $2\times 10^{-7}$ Newtons. The coulomb is not an SI base unit; fully expanded in SI base units, 1 coulomb is 1 ampere-second. Defining the coulomb in terms of the ampere, instead of the other way around, is a choice that is made out of convenience; nowadays, it's far easier to get a steady 1-ampere current in an experimental setting than it is to retain a steady charge of 1 coulomb. Because of this, it's much easier to describe the base electromagnetic unit in terms of a force measurement (which we can do very precisely and easily) rather than a charge measurement (which is harder).
Also, for the record, 1 mole of electrons gives you a charge of $6.022\times 10^{23}*1.6\times 10^{-19}\;\mathrm{C}=96352\;\mathrm{C}$, which, if passed through a wire in one second, would give you a current of $96352$ amperes. The number of electrons that you're looking for is $\frac{1}{1.6\times 10^{-19}}=6.25\times 10^{18}$ electrons, or $1.03\times 10^{-5}$ moles of electrons. The farad, being a unit of capacitance, is irrelevant here.
A: The SI system has just  be revised so that the electron charge is exactly
$$
e= 1.602176634 \times 10^{-19} {\rm C}
$$ and a current of one ampere is 1 C per second. Thus the $2\times 10^{-7}$ definition is obsolete and the new definition is along the lines you suggest.  Similarly a mole is now exactly $6.02214076\times 10^{23}$ elementary entities. 
Here is text from Wkipedia. It agrees with what I read in the physics literature:
"A consequence of this change is that the new definition of the kilogram is dependent on the definitions of the second and the metre.
Ampere:
The definition of the ampere underwent a major revision – the previous definition, which is difficult to realise with high precision in practice, was replaced by a definition that is more intuitive and easier to realise.
Previous definition: The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 m apart in vacuum, would produce between these conductors a force equal to 2$\times 10^{-7}$ newton per metre of length.
2019 definition: The ampere, symbol A, is the SI unit of electric current. It is defined by taking the fixed numerical value of the elementary charge $e$ to be 1.602176634 $\times 10^{−19}$ when expressed in the unit C, which is equal to A⋅s, where the second is defined in terms of $\Delta \nu \,{\rm C}\cdot {\rm s}$."
