In the previous lesson, we looked at the solution of the problem where we can look for times and compositions during the carburization process. In this lesson, what we're going to do is to develop a very useful approximation. And you'll hear, very often, people in the area of material science and engineering talk about a particular phenomenon that follows the kinetics of something we call the square root of Dt. We begin by looking at the relationship between z and the arrow function of z. And what we can see is if we use a value of z, the arrow function of z, that's equal to 0.5, what we find is that it's not too far in error to assume that for that error function of z, the value of z is also equal to about 0.5. Now, there's variation in this, but it's a fairly decent approximation. So we're going to use this approximation to our advantage. The way we do that is the following. We're going to describe the solution to this equation, which is given in the upper right-hand corner of the box. And what we want to do is we want to follow the average composition between what is at the surface composition and what is the average composition that we began with in terms of this part. So that composition is nothing more than the representation of Cx plus C0 divided by 2. So that gives us our average composition. And we'll put this, then, into our equation that describes the particular composition, x of t. And we'll go through, put those values in, and what we do is to rearrange the equation. And what we find is that we have the value of 0.5 is equal to the error function times what is in the parentheses. And that's what we want to take a look at. Well, as it turns out, what we have seen regarding our approximation that when we have an error function whose number is 0.5, we can estimate the value of z as 0.5, as well. So, that's the approximation we're using from the previous figure. And, of course, when we do that, what we get is the effective diffusion distance is equal to the square root of Dt. Now, let's look at this expression. D represents the diffusivity. And the diffusivity has to do with how many atoms we have passing a particular area of a plane per unit time. We multiply that by time, and what we get is the units of linear dimension, meters or centimeters or the like. So this is going to give us the idea of the effective diffusion distance as measured by the average composition between what was at the surface and what is the C0, or the beginning composition. And what we can see from this is that the effective diffusion distance, that is, the distance over which diffusion is important, is determined by these two parameters. So we would expect to see the longer period of time that the material is being exposed to a particular temperature, the effective diffusion distance gets larger. On the other hand, when we look at at the diffusivity, as we increase the temperature, once again, we expect to see the effective diffusion distance to be increased. And it's this proportionality of the square root of Dt. So when you hear people talking about, in the materials community, that we're looking at a process that's controlled, the kinetics are controlled, by the square root of Dt, we're saying effectively that the process is controlled by diffusion. Thank you.