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Does solubility of the solute affect the value of the mass transfer coefficient?
The book I am following mentions that the mass transfer coefficient is a characteristic of the constituent and its environment, and solubility is a property of the environment.
No.
In all but highly exotic situations, diffusive and convective fluxes of solutes are linear in the solution-phase concentration of the species:
$$ \begin{align} J &= -D {\mathrm dC\over \mathrm dx} & \mathrm{(diffusion)} \\ N_c &= k_c\left(C_\infty - C_\mathrm{surface} \right) \quad & \mathrm{(convection)} \end{align} $$
Whatever the dissolved concentration $C$ (or concentration gradient ${\mathrm dC\over \mathrm dx}$) of the species is, that is the value you use in both of the above equations. Whether that concentration (gradient) arises from a species with low, intermediate, or high solubility doesn't matter at all. I suspect your book is using the term environment to refer to things like the geometry, temperature, and fluid-flow characteristics of the system.
The main way I can think of where the solubility might affect the overall mass transport achievable in the system is in its effect on the maximum possible dissolved solute concentration at the surface. If solubility limitations constrain $C_\mathrm{surface}$ to a low value, then the magnitude of $N_c$ will necessarily be low. The low solubility is typically assumed not to appreciably change the value of $k_c$, however.
