Derivation of Exit Velocity <-> Isentropic Relations

[mr-september]

Hi, I have several questions regarding the equations in the docs, under the section "Exit Velocity".

1. Could you please show some in-between steps between the 1st and 2nd equations in that section? This doesn't appear to be a standard isentropic relation, at least in this form.
2. In the 3rd equation, what does the "normal" R represent? Usually it represents Avogadro's constant, but in this case it is Avogadro's divided by Molecular Weight? Is there a name for this variable? This may help me bridge above equations 1 and 2.
3. Still in the 3rd equation, I don't think the units are balanced? Left hand side, velocity, would be m/s, but right hand side resolves to J.g/(mol^2)?