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Volume Thermal Expansion

Patrick Ford
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Hey guys in a previous video, we covered linear thermal expansion which had to do with one dimensional objects that were changing temperature. Remember the idea was if you change the temperature of a metal rod or pole or something like that, then the length also changes. And these equations describe the relationship between the changing temperature and the changing length. In this video, we're gonna talk about a very similar concept, something called the volume, thermal thermal expansion or volumetric thermal expansion. The idea here is the exact same except now we're just gonna apply to three dimensional objects like spheres or cubes. So the idea here is that if you increase the temperature of a three D object, you're going to increase their volume. So the volume is going to increase. Alright, so let's take a look here. The idea is that with linear thermal linear thermal expansion, we're talking about one dimensional objects. So what happens is when you change the temperature, the length increases, that's the only dimension that this thing increase. Now we're talking about volumetric, we're talking about three dimensional objects. What happens is if you take a cube or something like that, has some initial volume and now you're going to increase the temperature, then it's going to expand not just along the length, but also the width and the height. It's gonna expand in all three dimensions and it's gonna change a volume delta V. Now the equation that we use for linear thermal expansion was delta L. And for volumetric it's gonna be delta v. So really these equations are going to look very similar. So let's take a look here. The equation for delta V is gonna be beta times V naught times delta T notice the similarities. We have a coefficient. Then we had some initial length here, we have another coefficient called beta and then the initial volume times delta T. Alright, so go ahead and pause the video. What do you think the equation for V final is going to look like? But hopefully you guys realize that these things are also going to look similar as well. The final is just going to be the initial one plus beta times delta T. Right, So it's basically the same exact setup is just some of the letters that are different. Alright, so what you need to know here is that this beta is a new coefficient. This beta has to do with the volumetric expansion coefficient. Whereas alpha had to do with the linear expansion coefficient. Alright, and so what you need to know about this beta is that for the same material like aluminum and aluminum, that beta is actually equal to three times alpha. Because if you have the linear expansion coefficient, beta is just gonna be the same thing in three dimensions. So it's just gonna be three times that I have a couple of examples up here. So for example, we have aluminum is 2.4 times 10 to the -5. And then beta is going to be three times that these are actually the actual values for some of these um for some of these materials here. Alright, so let's go ahead and take a look at our example. That's really all we need to know. So a ball of lead is an initial temperature of 333 and has an initial volume. So we have that tea not is equal to here. We're actually giving it in Calvin 333 and RV not is going to be 50 and this is going to be cm cubed. Now we want to figure out how much the ball shrinks by by how much does the ball shrink when you decrease the temperature. So we're actually looking to find here in part a actually this is the only part here is we're actually looking to find what is this delta v here. Alright, so we're going to decrease the temperature to 303 sisters, Artie final is going to be 303 kelvin. Alright, so we're also told the last thing is that our expansion of coefficient of linear expansion. This is going to be alpha is going to be 2.9 times 10 to the minus five. So these are all our values here. So what's delta v? We're just gonna use the equation, we're looking for delta v not the final and we're just gonna use this equation over here. So delta v is gonna be this is beta times the initial volume, times delta T. So which variables do I have? What I'm looking for? Delta V and I have the initial volume. I don't have the beta. Remember what I was given is the linear coefficient um the the coefficient of linear expansion. And I'm also not sure what the delta T is as well. So let's go ahead and find those outs. So how do we figure out beta? Well, uh for for aluminum, all that all that we know is this coefficient of linear expansion? 2.9. However, what you have to realize is that for the same material we can always relate beta and alpha together. So beta is equal to three alpha. So because we're dealing with volumetric expansion, we're just gonna do three times 2.9 times 10 to the minus fifth. And your beta coefficient is going to be 8.7, 10% of the minus fifth. Alright, so that's the coefficient. Now, what about delta T. Well, how do we get delta t. Remember? We're changing from temperatures were changing from an initial temperature of and their final temperature is going to be 303. So what this means here is that delta T is t final minus t. Initial. What you're gonna get is negative, 30 Kelvin. Alright, so this is actually we're gonna plug into this term right here. So that means delta V is just gonna be this is gonna be uh sorry, this is gonna be 8.7 times 10 to the minus fifth. That's our coefficients. Then we have the initial volume. It's okay. We actually keep it in centimeters cubed because that just means our answer is going to be in centimeters cubed. So we have 50 centimeters cubed and then we have our temperature of negative 30. Alright, if you go out and plug this in, but you're gonna get is negative 0.13 and again, this is going to be centimeters cubed. That's basically the decrease in volume. Once you shrink this, once you decrease the temperature of the ball. All right, So that's it for this one. Guys, let me know if you have any questions.