In this lesson we're going to be looking at a couple of different example problems to test our understanding of the material that we've just presented. The first example problem we're going to take a look at is how do we take a gear which is fabricated from a particular steel, namely a steel that contains two-tenths of a percent carbon? We put that product in a furnace with a particular hydrocarbon atmosphere, and we control that atmosphere so that we can maintain the content of nine-tenths of a percent carbon on the surface. The carbon then diffuses in and what we're now going to look for is what is the amount of carbon that we can expect to see at a depth of 0.0254 centimeters and at a temperature of 1,000 degrees C? And what I have provided here are the information that's necessary to calculate the process of the fusion of the carbon in the steel. Because we're at a temperature of 1,000 degrees C, it means that we're look for the diffusion of carbon in the FCC phase. So in order for me to be able to calculate the diffusivity of carbon at that temperature, I need two pieces of information. I need the D0 and the Q, and those are given at the bottom of the problem. So the D0 and Q will be used to calculate the diffusivity at a temperature of 1,000 degrees C. So let's take a look at all the parameters that we have so far and see how they fit into our equation. First of all we have x, and we're interested in knowing what is the particular carbon content at this position inside of the gear, away from the surface? And what we know is that our initial carbon content is 2 weight percent carbon. And by our particular atmosphere that we've chosen, we've fixed the atmosphere to be nine-tenths of a percent carbon on the surface. And what we're looking at then is what will be the carbon content after a total of ten hours at the surface? And we'll use the diffusivity as calculated using the expression up on the slides. And we're going to wind up plotting the function that I have here which is the solution to this particular problem. Now if I plot this at the value of ten hours, what I would then be able to do is by putting in the appropriate values, I will then be able to calculate what that composition is at that position x, when we are at 1,000 degrees for ten hours. So first of all, let's take a look at this problem. We haven't done these kinds of calculations in a bit, so I put the problem this way so that you need to calculate what the diffusivity is given the two pieces of information, namely the D0 and the Q for the diffusion of carbon in FCC iron. When we do that, we use the erroneous expression to calculate the diffusivity, and what I did was to put this in an Excel spreadsheet, and I went ahead and I calculated the temperature, and I correspondingly changed the temperature from centigrade to kelvin. I took the reciprocal of it because that's the way the expression becomes linear, and then I calculate the diffusivity. And because we're interested in 1,000 degrees, what we see is that diffusivity is on the order of 2.98 times 10 to the minus 11. So this is the value of the diffusivity that we're going to be using in this particular problem. And again, we're looking at units of meters squared per second. So I went ahead and plotted this in our typical way, of the log of diffusivity versus the reciprocal of the temperature, and what you see is the plot over the range of the temperatures of interest. I have the actual temperature in kelvin above the data and along the x-axis the reciprocal of temperature as we normally plot the erroneous behavior. So our function looks something like this when we plot it and that's the governing equation that provides that solution. And what we're looking for is what we have in that blue box, which is what is the composition at that point x after ten hours? And so what do we put in here is the following pieces of information. We have the value of z, because everything on the right-hand side of the equation is known. We know what the value of x is that we're looking for, we know what the value of the diffusivity is, we've calculated that, we know what the time is. So now what that means is we need to calculate what the value of z is. Once we know the value of z then we can determine what the error function of z is. Once we have that then we'll be able to calculate what the value of C(x,t) is. Now I've put up here a dataset that describes the error function, and it's in the tabulated form. And I've done this because sometimes we're interested in more accurate representation of the data rather than the looking at a graph and trying to determine values from the graph. So what we have to do is we have to determine what will be the value of z, and then calculate what its corresponding error function is. So when we do that, and we put all of our numbers into the expression, we find that we are looking at a value of z to be 0.123. And what we need to do then is to interpolate that between those two data points, and what I did was to put up here that set of interpolation. In other words, we're looking for what the value of that error function for the given value of z that we've just calculated, which is 0.123. And when we put all of this into the equation, what we find is that the carbon content that we're going to get at this particular depth is on the order of eight-tenths of a percent by weight of carbon. So this is an example where we know everything on the right-hand side of the equation, and what we're now trying to do is to calculate what the composition is, C(x,t). What we'll take a look at in the next example problem is the reverse. That is what we're going to be doing in this case is we're going to know what the composition is at a particular value of x and t, and the distance, and the initial carbon content. And again, the diffusivity where at the same temperature it's the same overall composition, and therefore, we have the diffusivity that we calculated in the previous example. And we know that once again, we've fixed the carbon to be nine-tenths of a weight percent at the surface, and now we're going to put all of this information into the equation that appears on the slide. And here, what we're going to do is to calculate the time necessary for all of these conditions to be satisfied, that is the value of C(x,t), and given the diffusivity and the position we need to calculate how much time that will take. So when we do that what we now have is the information that is on the left-hand side of the equation. And that is, it gives us the information which is the value of the error function of z. Now then, once we have that error function of z, we can look for the value of z that satisfies that particular error function of z. And so that would then be the solution to this problem. So this gives us an idea of exactly how we can use this solution to Fick's second law for the case of carburizing a surface using a particular carbon atmosphere, thank you.