Mercury is often used in thermometers. The mercury sits in a bulb on the bottom of the thermometer and rises up a thin capillary as the temperature rises. Suppose a mercury thermometer contains 3.380 g of mercury and has a capillary that is 0.200 mm in diameter. How far does the mercury rise in the capillary when the temperature changes from 0.0 °C to 25.0 °C? The density of mercury at these temperatures is 13.596 g/cm³ and 13.534 g/cm³, respectively

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hi everyone for this problem, we're told that ethanol is an alternative used in thermometers. Given an ethanol thermometer with 2.50 g of ethanol and a capillary with a diameter of 0.170 millimeters. We need to determine the height, ethanol rises in the capillary when the temperature rises from 10 degrees Celsius to 20 degrees Celsius. And we're also told that at 10 degrees Celsius, the density is 100.7978 g per cubic centimeter and at 20 degrees Celsius, the density is 200.7892 g per cubic centimeter. So let's go ahead and get started. So we're given density and we're given mass and we have two different temperatures. So let's start off by calculating the volume at each of the temperatures that were given. So we're told that we have 10 degrees Celsius and we have 20 degrees Celsius. And we know that density is equal to mass over volume. So if we wanted to solve for volume at both of these temperatures, we just rearrange this equation and that rearranged equation will be volume is equal to mass over density. and so for 10°C, We know that the mass for both of them is 2.540 grams And the density at 10°C is 0.79 seven eight g per cubic centimeter. And so we see our grams will cancel. And so that will leave us with the volume of 3. cubic centimeters at 10 degrees Celsius. So let's go ahead and do the same thing for 20 degrees Celsius, Our volume is going to equal mass over density. Our mass is the same so our volume is equal to 2.540g. But the density is different at 20°C. We're told that the density is zero two grams per cubic Centimeter. So our g cancel and we're left with the volume of 3.2 sq cm. So now we have the volume at both temperatures. The next thing that we can do is use our area equation to solve for height and so we know that volume is equal to area times height. Okay. And the area of a cylinder is high are squared. So what we can do here is plug in this area here and rearranged the first equation to solve for height. Okay, so let's go ahead and do that at each temperature. And so if we rearrange our volume equals area times height and solve for height, then we get our height is going to equal our volume over area. We know our volume at each temperature because we just solved for that in our area, we're going to substitute it for pi r squared where are is our radius. And the problem they tell us our diameter and to go from diameter to radius, we're going to divide our diameter by two. And so we know that our diameters 20. millimeters. And when we divide that by two we get a radius of 0. millimeters. Okay, But we can't use millimeters because our density is in centimeters. So we need to convert this two centimeters and we can do that by saying one centimeter is equal to millimeters. Okay. And so this gives us a radius of 0. 8,5cm. Okay, so now our height is going to equal our volume over Our area. So our volume at 10°C is 3. 838 cubic centimeters in our area is going to be pi R squared. So pi Times are radius and cm zero centimeters. And that is squared. and so we get a height a final height at 10°C to be 14026. 8541 centimeters. Yes, I know this is a long number. So this is our height at 10 degrees Celsius. And we're going to do the same thing for 20 degrees Celsius. And when we take the difference, this is going to tell us the height that ethanol rises when the temperatures change From 10 to 20. Okay, so let's keep going, we're almost done. So our height Is equal to volume over area and at 20°C we said our volume was 3. cubic centimeters divided by our area, which is pI Times are radius 0. centimeters squared. And so we get a height at 20°C to be 14179 point four 3467 cm. And so our difference in height is going to be so we can say this is height too, and this is height one, so it's going to be height to minus height one. And that gives us a change in height to be 152. cm. This is our final answer. This is going to be the height that ethanol rises in the capillary when the temperature rises from 10°C to 20°C. That's the end of this problem. I hope this was helpful.

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Key Concepts

Density

Thermal Expansion

Capillary Action