University of North Carolina at Chapel Hill
Equipartition Theorem - Example 4

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Kinetic Theory Of Gases

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Video Transcript

welcome to our fourth example video. Looking at the Echo Partition theorem in this video, we're going to consider a box of gas A with a material be inside. So this is a solid of material. Be And we want to find out if we have temperature a greater than be what would be the change in the temperature of our solid? Okay, Okay. So in order to solve this, we need to think about how is energy being stored inside our solid? Is it being stored like Amman atomic gas, die atomic gas or a poly atomic gas? Well, the model we often use for solids is something known as a crystal lattice, where we have a central atom of interest. And then it has neighboring atoms that are above and below, in front and behind and to the left and right of the central atom. And with each of these adjacent Adams, there is some bond that holds them together. And we can model each of these bonds as functioning like a spring. And so there are three additional sets three sets of springs here that help control the motion of this central atom. Now clear the atom can move in the X, Y and Z, meaning it will have at least three halfs R T. But because it has thes spring like bonds as well, it can also store energy in these directions as well. In the in terms of potential energy construir potential energy and the X in the UAE and in the sea meaning that are actual energy for E. B is going to be equal to three times and be are times temperature. So we're going to treat it like a poly atomic gas. Knowing this, then we know that e b not will be equal to this and e b final well, it equal to B three and B r t b final. Additionally, we know that FB Final is going to be equal to N a or NB apologize over and a plus and be multiplied by E total and again here. E total is going to be the some of the initial energies of the two materials, meaning that we have EB Final is equal to N. B and be over n a plus and be multiplied by three halfs and a r ta. Not plus three times and be are TV? Not so here we have RMB final, we can relate it to our temperature TV final and find Then that T B final is going to be equal to one over n a plus and be multiplied by one half n a t a. Not plus n b t b not you go through and do the algebra there to solve for all that, Then subtracting TB not from this will be able to find our final value for TB final. And for Delta terabytes, so dealt TV is equal to one half and a t a initial plus n b TV initial over n a plus and be all minus t b initial and we have our solution here. Now notice here that this is all true and we're assuming in all of our examples that the number of moles for each material is constant. This is generally true and that we can control the amount of gas and a system or the amount of material in a solid and keep it constant. But there are situations where those change and those Consignia ficken tely complicate these sorts of calculations

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