Regarding $k_p$:
So, you've got $N_1$ a flux in $mol/ s \cdot m^2 $ and a partial pressures $p_i$ in $Pa$. $k_p$ has got to connect those, and make the units work out.
Exactly what $k_p$ is depends on the flow of the gas. If the gas is stationary, mass transfer will by diffusion alone. Unfortunately, that process doesn't have a steady-state solution in a semi-infinite domain. If you had a thickness for the gas layer ($L_x$), $k_p$ would be:
$$ \frac{D_{AB}}{L_x R T} $$
where $D_{AB}$, $R$, and $T$, are the diffusion coefficient, universal gas constant and temperature. $D_{AB}$ is a property of the species that you're tracking ($A$) and the remaining species that make up the gas ($B$). You can look those up. In a pinch, I assume $2 \cdot 10^{-5}$ for air-like gases.
Now, go with the idea that you've got some kind of flow.
With flow over a surface you'll end up with boundary layer, a thin region where velocity and species concentrations vary between their values at the surface and in the free stream. That flow is generally going to enhance the mass transfer, by actually moving things around a lot faster than diffusion (diffusion is wicked slow).

It is possible to cook up some solutions to the boundary layer equations and actually derive $k_p$. But that's tedious, only works in a few situations and the results are still approximate. What you actually want to do is look up a correlation. Like these:

Those'll give you the Sherwood number based on the Reynolds number and Schmidt number.
$$ Sh = \frac{k_g L}{D_{AB}}, Re = \frac{u L}{\nu}, Sc = \frac{\nu}{D_{AB}} $$
If you aren't familiar, the only thing that might be tricky here is choosing $L$. It's the characteristic length of the flow configurations. Usually, it's just the length of the object in the direction of flow. Except inside pipes, use the diameter.