Transport of liquid mixtures through porous membranes is central to processes such as desalination, chemical separations, and energy harvesting, with ultrathin membranes made from novel 2D nanomaterials showing exceptional promise. Here, we derive, for the first time, general equations for the solution and solute fluxes through a circular pore in an ultrathin planar membrane induced by a solute concentration gradient. We show that the equations accurately capture the fluid fluxes measured in finite-element numerical simulations for weak solute-membrane interactions. We also derive scaling laws for these fluxes as a function of the pore size and the strength and range of solute-membrane interactions. These scaling relationships differ markedly from those for concentration-gradient-driven flow through a long cylindrical pore or for flow induced by a pressure gradient or an electric field through a pore in an ultrathin membrane. These results have broad implications for transport of liquid mixtures through membranes with thickness on the order of the characteristic pore size.