Referring back to the derivation for Darken's first equation,  is written as
Inserting this value for  in  gives
As stated before, , which gives
Rewriting this equation in terms of atom fraction  and  yields
By using  and solving to the form , it is found that
Integrating the above gives the final equation:
This equation is only applicable for binary systems that follow the equations of state and the Gibbs–Duhem equation. This equation, as well as Darken's first law, , gives a complete description of an ideal binary diffusion system. This derivation was the approach taken by Darken in his original 1948, though shorter methods can be used to attain the same result.