It seems ok to me. Remember that Molar mass: M = m * N

_{A}. Insert that and k=R/N

_{A} in v

_{p} and the 2 equations are the same.

The last two equations implies that N*k = n*R which at first glance looks wrong. But N is not Advogadros constant it is the actual number of molecules:

http://en.wikipedia.org/wiki/Ideal_gas_law
So N = n*N

_{A}, and if you insert above:

n*N

_{A}*k = n*R or k = R/N

_{A} , which is the defenition of k.

Edit: Ops I was too slow :)