text: The derivation starts with Ficks first law using a uniform distance axis y as the coordinate system and having the origin fixed to the location of the markers. It is assumed that the markers move relative to the diffusion of one component and into one of the two initial rods, as was chosen in Kirkendalls experiment. In the following equation, which represents Ficks first law for one of the two components, D is the diffusion coefficient of component one, and C' is the concentration of component one:
  This coordinate system only works for short range from the origin because of the assumption that marker movement is indicative of diffusion alone, which is not true for long distances from the origin as stated before. The coordinate system is transformed using a Galilean transformation, y = x − νt, where x is the new coordinate system that is fixed to the ends of the two rods, ν is the marker velocity measured with respect to the x axis. The variable t, time, is assumed to be constant, so that the partial derivative of C with respect to y is equal to the partial of C with respect to x. This transformation then yields
  The above equation, in terms of the variable x, only takes into account diffusion, so the term for the motion of the markers must also be included, since the frame of reference is no longer moving with the marker particles. In the equation below,  is the velocity of the markers.
  Taking the above equation and then equating it to the accumulation rate in a volume results in the following equation. This result is similar to Fick's second law, but with an additional advection term:
  The same equation can be written for the other component, designated as component two:
  Using the assumption that C, the total concentration, is constant, C and C can be related in the following expression:
  The above equation can then be used to combine the expressions for  and  to yield
  Since C is constant, the above equation can be written as
  The above equation states that  is constant because the derivative of a constant is equal to zero. Therefore, by integrating the above equation it is transforms to , where  is an integration constant.
  At relative infinite distances from the initial interface, the concentration gradients of each of the components and the marker velocity can be assumed to be equal to zero. Based on this condition and the choice for the coordinate axis, where the x axis fixed at the far ends of the rods, I is equal zero. These conditions then allow the equation to be rearranged to give
  Since C is assumed to be constant, . Rewriting this equation in terms of atom fraction  and  yields