Reaction rate unit conversion: how to convert from molarity to ppm How can I convert a reaction rate $k$ with units of $\mathrm{1\over M~~s}$ to $\mathrm{1\over ppm~~s}$?

For context: I want to make a numerical simulation of a system of reactions that happens in air (and in water droplets suspended in the air), I have already built a part of the model for which I used $\mathrm{ppm}$ for my concentrations and thus, e.g., $\mathrm{1\over ppm~~s}$ as the units for the $k$ values of second-order reactions in the system. (Reactions of different orders have different units, obviously, but those details are irrelevant to the question.) The article I have on hand lists a $k$ value for the reaction I need in $\mathrm{1\over M~~s}$ and the reaction takes place in the liquid phase.I assume that the PPM is weight-based although the article in question does not actually specify this.

I am asking the above question because I am not sure if this conversion is necessarily as straightforward as a simple factor of $10^{-6}$.
We can do it in two ways:

1) For dilute solutions, ppm and mg/L is equivalent. So, for dilute solutions, the straightforward conversion scheme should look like::
$$ppm=\frac{mol}{L}\times MW\times 1000$$  

2) with no assumption::  

first, remember that $ppm=y_i \times 10^{-6}$

$$ppm=\frac{mol}{L} \times \frac{x gm}{mol} \times \frac{1 L}{10^{-3} m^3} \times \frac{m^3}{y~gm} \times 10^{-6}$$  

here, x is the MW of solute and y is the density of the solvent (for dilute solutions) or solution in $\frac{gm}{m^3}$