Welcome back. In this lesson, we're going to describe actually what the barrier to the homogeneous nucleation process is. So, as a result, what we'll do is we will go back to the equation that describes the two contributions, one that is providing free energy and the other one is taking the free energy away. And those are described in terms of our volume term and our surface area term. Now, what I want to stress here is that we are going to choose a particular material. And as a result of determining a particular material, we're going to have a delta g that's going to be associated with that material and we're going to have an interfacial energy that is associated with that material. We also will have the characteristic, namely the melting temperature of the material, so we're fixing the particular material. So, now what we want to do is to use the equation that is up on the screen to actually predict the critical point. That is, the point where the derivative of the r, the radius, with respect to the free energy is going to be equal to zero. So we look at the undercooling and we are looking at the amount of undercooling as witnessed by the red delta T. And at that particular red delta T you can see the value of the free energy as we go from the liquid phase down to the solid phase. And as a consequence of that, because we're looking at these two behaviors, one increasing and one decreasing, we're going to come up with a maximum. And so, when we take the first derivative of this, we find that that first derivative is going to be equal to 0. That determines for us what the critical size of the process of nucleation is. Everything that lies to the left for this particular temperature, this particular amount of undercooling, we're going to have a circle that represents the point where the r star term can be calculated from the equation. We put that value back into the delta G term and what we're then able to do is to connect that with the barrier. So, it's like looking at the coordinates of the x and y for a maximum and a minimum. Now, what we're going to do is we're going to look at the same equation, this time for the equation, we're going to be looking at a larger delta T. When we look at that larger amount of undercooling, we put it into this equation. And what we're going to see is the location of that maximum has shifted to smaller values of r star and smaller energy barriers dealta G star. We'll try one more and when we go further down, again it's the same material and this time we have a much larger undercooling. And what we're going to do is we're going to calculate this turning point and we will be examining what the actual meaning of that turning point is later. But what you will see is the size of the particle, in terms of r star has gotten significantly smaller than in the two previous diagrams, as well as the barrier term. We'll see more about this as we go through the description of homogenous nucleation. Thank you.