To derive Darkens second equation the gradient in Gibbs chemical potential is analyzed. The gradient in potential energy, denoted by F, is the force which causes atoms to diffuse. To begin, the flux J is equated to the product of the differential of the gradient and the mobility B, which is defined as the diffusing atoms velocity per unit of applied force. In addition, N is the Avogadro constant, and C' is the concentration of diffusing component two. This yields
which can be equated to the expression for Fick's first law:
so that the expression can be written as
After some rearrangement of variables the expression can be written for D, the diffusivity of component two:
Assuming that atomic volume is constant, so C = C + C,
Using a definition activity, , where R is the gas constant, and T is the temperature, to rewrite the equation in terms of activity gives
The above equation can be rewritten in terms of the activity coefficient γ, which is defined in terms of activity by the equation . This yields
The same equation can also be written for the diffusivity of component one, , and combining the equations for D and D gives the final equation: