# why drift velocity proportional to driving force

http://www.uio.no/studier/emner/matnat/kjemi/KJM5120/v09/undervisningsmateriale/Defects-and-transport-2009-Ch5-Diffusion.pdf This online pdf file trys to derive Fick's 1st law. In the deriving process(page 5.3), it is says that the drift velocity of a particle $v_i$ is proportional to the driving force acting on particle $i$, $F_i$, that is, $$v_i=B_iF_i$$, where $B_i$ is a constant.

My questions is 1. What does this drift velocity mean here? Is it just the total velocity of particle $i$?

1. According to the Newton's second law, $v_i(t)=v_0(t)+Ft/m$. So why $v_i=B_iF_i$ here? Does it because $\sum{v_0(t)}=0$ as the system is macroscopic ally still so the total initial velocity should be zero. But in this case, $B_i$ should be a value that depends on time $t$? But according to the textbook, $B_i$ seems to be a constant that describes the mobility of particle i?

Can anybody give me some help on it? Thanks!

You are correct in assuming that $v_i$ refers to the total velocity of particle $i$. Note that, since this is an ideal diffusion problem, the direction of the velocity is specified - it points along the concentration gradient.

The issue here is that the author uses the term "driving force" rather colloquially, to refer to something that is not, in fact, a force. He states that the concentration gradient "may be considered to be" (rather than "is") a "driving force." In fact, what you are attempting to justify (i.e. the fact that diffusion velocity is proportional to concentration gradient) is taken as an assumption in Fick's Law. For this reason, I don't think Newtonian mechanics really applies here.

My best advice is to either a.) take the same assumptions as Fick's Law, and proceed, or b.) learn the necessary statistical mechanics to eliminate or justify the assumption made in Fick's Law.

• Thanks, I got your points! Fick's 1st law is just a empirical, true derivation needs statistical mechanics, here $v_i=B_iF_i$ is just the same assumption as the assumption Fick's law used. So rather that derivation of Fick's 1st law, the author just give another type of writting of fick's 1st law
1. The time $t$ in your formula is actually the mean time between collisions. Since the random velocity is very much greater than the drift velocity, the mean time between collisions does not increase as electric field increases, it is approximately constant. Therefore the drift velocity is proportional to the driving force $E$.