20. Heat and Temperature

Volume Thermal Expansion

# Overflowing Mercury

Patrick Ford

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Hey guys, we've got an interesting problem here for you. So we've got a glass flask that is completely filled with mercury and then we're going to increase the temperature of both the flask and the mercury and we're gonna try to figure out how much mercury is sort of overflowing and spilling out of the flask. Let me just draw this out for you just to make this really clear for you. So, I have this sort of glass flask like this, right, it's filled with mercury at 0°C. Then what happens is you increase the temperature to 100°C, you have an increase of temperature and things start expanding. Now we have a three dimensional expansion because we have a three dimensional object, like a glass that holds some liquid in it. So what happens here is that the glass and the mercury both expands? So the question is, do they expand the same amount? And the answer is no, because remember the expansion depends on these beta coefficients, the coefficient of volumetric expansion. You'll notice that the one for mercury is actually bigger than the one for the glass. So, here's what's going on here, you're increasing from 0 to 100. And what happens is the glass changes in volume by some amount, you want to call that delta v glass now, for the same change 0-100, you also have the mercury that starts to change in volume. What happens is this delta v is going to be greater than the delta V for the glass. So what happens is if this thing is already completely filled with mercury, if it expands more volume, basically, you're gonna have mercury that's just to leak out of this container and it starts to spill out over the edges. This is really what we want to find here. So, I'm gonna call this V spill, that's really what we're looking at here. And what happens is if the the volume for the mercury changes more than the volume for the glass, then v spill is just gonna be the subtraction of those two. It's going to be the delta V for the mercury minus the delta V for the glass. So really simply here, if the mercury increases by 15, but the glass increases by 10, then the amount that spills over, It's just the difference between them. It's five, Right? That's what spills out of the container. So, that's really what we've got going on here. And because we're looking at these delta V equations, these delta V variables, we're gonna be using our delta V equation for volumetric thermal expansion. So basically what I have to do is just replace these equations here with um the correct substance. Right? So I've got delta V for the mercury. So this is going to be the beta for the mercury times the initial volume of the mercury. I'm just going to use H G. That's the chemical symbol for mercury, times the change in temperature of the mercury minus the beta for the glass? Mine times the the initial for the glass times delta T. For the glass. Right, so we just got the uh the coefficients there. Alright, so basically what I've got here is I've got the two coefficients like this. Now what happens is I need to figure out the initial volumes of the glass and also the change in the temperatures. Now, if you think about what's going on here, is that both of these were initially at zero degrees and then they both end up at 100 degrees. So these two delta teas are going to be the same. This delta T that we're working with here is just gonna be 100 Now, what about the initial volume? Well the initial volume here is just gonna be 250 for both. If the glass holds 250 centimeters cubed then it's completely fooled. The mercury. The mercury also has 250 centimeters cubed. So basically what happens is these two variables on both of the terms here are actually going to be the same. And because of that actually just makes the equation a little bit simpler here. So V spill, which is what I'm looking for here is actually just going to be the initial times delta T remember, it's going to be the same for both times. This is gonna be beta for H G minus beta for the glass. All I've done here is a sort of just grouped together these two variables which are the same and then these to sort of get combined into a parentheses. Alright, so this v spill here. Alright, so now it's time to go ahead and start plugging in now. Normally what I would do is I would convert this to meters cubed but it actually asks us to keep it in cm cube so I don't have to do any converting. So the volume initial is going to be 250 cm cubed Times The Delta T, which is just 100. And now the beta coefficients, the one for mercury is 1.8 times 10 to the - -1.2 times 10 to the -5. All right, so you go ahead and work this out and what you're going to get here is that the V spill is just equal to 4.2 cm. So we got a positive number, which just means that we were right selling out of mercury is gonna spill out of the container. And the amount that spills out is just this amounts 4.2 cm cubed. So hopefully that makes sense. Guys, let me know if you any questions

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