And now we'll see how the Arrhenius relationship applies to the increasing number of vacant sites in a crystal as we go to higher temperatures. And so now we tie together an important comment and concept from the previous lesson on defects in solids with our current interest in how atoms migrate through a solid material. This shows that the Arrhenius equation applies equally well to the case of creating point defects, one of the early topics in our lesson on the defects in crystalline materials. The fraction of the atomic sites in a material above absolute zero is gonna be given again by an Arrhenius expression, pre-exponential constant times an exponential term. And in this case, we're dealing with a value of activation energy corresponding to the energy necessary to create a single vacancy in the crystal structure. See a little bit of difference here in the Arrhenius expression compared to the earlier example. Again, remember, that Arrhenius gave us originally that any chemical reaction rate is equal to again that pre-exponential constant times e to the minus Q over RT. Now we see that instead of R we have this k. Of course I'm sure you've seen an constant perhaps a previous course on chemistry or physics. And the relationship is simply that R is equal to Avogadro's number times k. So k is the corresponding fundamental unit that appears for individual atomic mechanisms and R is the so-called gas constant that corresponds to that expression for macroscopic amount of materials, Avogadro's number of atoms. And of course we can this the gas a constant, but we're talking in this course by and large about solid state materials. So we again, have this unfortunate label because of the ideal gas law. Again you'll recall from any introductory course on chemistry and while this is a very useful fundamental constant for describing the behavior of gases relationship between the pressure and volume at a given temperature, and the number of moles of that gas. Again, that gave us the label. But this is a universal constant that applies across the broad range of material states I guess gases, liquids, and solids. So it's perfectly valid here. So because we're talking about the energetics necessary to create an individual, Vacancy One of those point defects we discussed early on in the previous lesson on crystal defects. So we're using the Boltzmann constant in that regard. >> Then if we again take the logarithm of that, we see that, again, we have the intercept defined. And then the slope of our logarithmic Arrhenius plot that's going to give us the energy necessary to create an individual vacancy and divide it by the [INAUDIBLE] constant.