Hey guys. In a previous video, we covered linear thermal expansion, which had to do with 1-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 going to talk about a very similar concept, something called volumetric thermal expansion. The idea here is the exact same, except now we're just going to apply it to three-dimensional objects like spheres or cubes. So the idea here is that if you increase the temperature of a 3D object, you're going to increase their volume, so their volume is going to increase. Alright, so let's take a look here. The idea is that with linear thermal expansion, we're talking about 1-dimensional objects. So what happens is when you change the temperature, the length increases. That's the only dimension that this thing increased. Now when we're talking about volumetric, we're talking about three-dimensional objects. What happens is if you take a cube or something like that, it 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 going to expand in all three dimensions and it's going to change a volume, ∆v. Now the equation that we use for linear thermal expansion was ∆l, and for volumetric, it's going to be ∆v. So really these equations are going to look very similar.
So let's take a look here. The equation for ∆v is going to be:
∆v
=
β
V
₀
∆T
Notice the similarities. We had a coefficient, then we had some initial length. Here we have another coefficient called β and then the initial volume times ∆T. Alright? So go ahead and pause the video. What do you think the equation for V final is going to look like? Well, hopefully you guys realize that these things are also going to look similar as well. V final is just going to be V initial times (1 plus β times ∆T). Right? So it's basically the same exact setup, just some of the letters are different.
Alright, so what you need to know here is that this β is a new coefficient. This β has to do with the volumetric expansion coefficient, whereas α had to do with the linear expansion coefficient. This β is actually equal to 3 times α because if you have the linear expansion coefficient β is going to be the same thing in three dimensions, so it's just going to be 3 times that. I have a couple of examples here. So for example, we have aluminum as α = 2.4 × 10¯5 and β is going to be 3 times that. These are actually the actual values for some of these, 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 has an initial temperature of 333 K and has an initial volume. So we have that T₀ is equal to here we're actually given it in Kelvin 333, and our V₀ is going to be 50 cm³. Now we want to figure out how much the ball shrinks 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 ∆V here. Alright. So we're going to decrease the temperature to 303, so this is our T final. This is going to be 303 K. Alright? So we're also told the last thing is that our coefficient of linear expansion, this is going to be α, is going to be 2.9 × 10¯5. So these are all our values here. So what's ∆v? We're just going to use the equation if we're looking for ∆v not V final, then we're just going to use this equation over here. So ∆v is going to be this is β times the initial volume times ∆T. So which variables do I have? Well, I'm looking for ∆v and I have the initial volume. I don't have the β. Remember what I was given is the coefficient of linear expansion, the coefficient of linear expansion, and I'm also not sure what the ∆T is as well. So let's go ahead and find those out. So how do I figure out β? Well, for aluminum, all we know is this coefficient of linear expansion to 2.9. However, what you have to realize is that for the same material, we can always relate β and α together. So β is equal to 3α. So because we're dealing with volumetric expansion, we're just going to do 3 times 2.9 × 10¯5, and your β coefficient is going to be 8.7 × 10¯5.
Alright. So that's the coefficient. Now what about ∆T? Well, how do we get ∆T? Remember we're changing from temperatures; we're changing from an initial temperature of 333, and then our final temperature is going to be 303. What this means here is that ∆T is T final - T initial, which is going to be −30 K. Alright? So this is actually what we're going to plug into this term right here. So that means ∆v is just going to be this is going to be 8.7 × 10¯5. That's our coefficient. 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 cm³ and then we have our temperature of −30.
If you go ahead and plug this in, what you're going to get is —0.13 cm³. That's basically the decrease in volume once you shrink this, once you decrease the temperature of the ball.
Alright, so that's it for this one, guys. Let me know if you have any questions.
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Volume Thermal Expansion
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