Fick's Laws

Continuous Approximations

So far, we have been using a discrete model of diffusion in which individual atoms random walk and then we count up the number of atoms at each $x$-position to construct the concentration profile. Each $x$-position in our models is about the width of an atom. This is a good model of the real world since atoms are indeed discrete entities. However, in most real systems, there are on the order of $10^{23}$ atoms present! There are so many atoms that we can approximate the discrete reality with a continuous model. Figure 7.6.1 below shows a continuous approximation for three different distributions with different numbers of columns. By the time we get to 101 columns, it is hard to tell the difference. On this page we will zoom out to this continuous approximation and discover how to describe concentration profiles with continuous mathematical models instead of discrete computational ones.

Three distributions (or concentration profiles) with discrete values and continuous approximation for three different numbers of columns.

Figure 7.6.1 Three distributions (or concentration profiles) with discrete values and continuous approximation for three different numbers of columns.

Exercise 7.6.1: Fick's 1st Law
Not Currently Assigned

  1. Fick's 1st Law is an equation which tells you the net flux, denoted $J$, at any point on a concentration profile based on the profile itself. Assuming you have a continuous function, $C(x)$, describing the concentration profile, what would $J$ equal? Take 2-3 minutes to try to come up with something reasonable before looking at the answer.

    Hint: Think about what would determine the flux be between each pair of columns in Figure 7.6.1 and what aspect of the continuous approximation would relate to that.

    If you know how to use $\LaTeX$, you can use that to input math surrounded by dollar signs (e.g. $f(x)=x^2$ will produce $f(x)=x^2$) . Otherwise, just type in the equation the way you would type computer code. If you want to learn $\LaTeX$, this online editor is helpful.

  2. (optional)

    Sketch a concentration profile that has a region with zero net flux and a region with negative net flux. Label which region is which.

Exercise 7.6.2: Fick's 2nd Law
Not Currently Assigned

  1. Fick's 2nd Law is an equation that describes how the instantaneous rate of change of the concentration, $\frac{\partial C}{\partial t}$, at each $x$ position relates to the current concentration profile, $C(x)$. In the discrete case which we looked at on the previous page (Exercise 7.5.2) this would be how much the concentration profile will go up or down at each $x$ position the next time step. Now we are using a continuous approximation, so it becomes an instantaneous rate of change, i.e., a derivative with respect to time.

    Again take 3-5 minutes to try to come up with what $\frac{\partial C}{\partial t}$ will equal as a function of $C(x)$ before looking at the answer.