Mercury is often used in thermometers. The mercury sits in a...

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 $\mathrm{mm}$ in diameter. How far does the mercury rise in the capillary when the temperature changes from $0.0^{\circ} \mathrm{C}$ to $25.0^{\circ} \mathrm{C}$ ? The density of mercury at these temperatures is 13.596 $\mathrm{g} / \mathrm{cm}^{3}$ and $13.534 \mathrm{g} / \mathrm{cm}^{3},$ respectively.

Key Concepts

Conservation of Mass

This principle states that the mass of a closed system remains constant over time, regardless of the physical changes that occur within the system. In thermometers and related applications, while the density and volume of a liquid may change with temperature, the mass remains unchanged. This concept is critical when calculating the change in volume due to temperature variations using the fixed mass of the liquid.

Density and Volume Relationship

Density is defined as the mass of a substance per unit volume. When a substance undergoes a change in temperature, its density can change, which in turn affects its volume, assuming the mass remains constant. By using the relationship volume = mass/density, one can determine how much the volume of a fluid has increased or decreased due to a change in temperature.

Thermal Expansion of Fluids

Thermal expansion is the tendency of matter to change in volume in response to a change in temperature. For fluids, this expansion can be significant and is often observed by changes in density as the temperature increases. Understanding thermal expansion is essential in designing instruments like thermometers, where the change in volume of a liquid is used to measure temperature differences.

Capillary Geometry and Volume Displacement

In instruments such as capillary thermometers, the geometry of the capillary tube, particularly its diameter and cross-sectional area, directly influences how a change in the liquid’s volume translates into a change in the liquid’s level (or height) within the tube. Calculating the displacement involves dividing the volume change by the tube's cross-sectional area, linking geometric principles with fluid statics.

Transcript

00:02 Hi there.

00:03 We are trying to determine how much mercury will move up the thermometer when the temperature changes from 0 to 25.

00:12 So mercury, like most substances, will expand when it warms up.

00:18 Since it expands, its density decreases as it warms up.

00:22 So what i first want to do is determine the volume of mercury we have at each temperature.

00:28 So let's look at zero degrees.

00:33 At zero degrees, we have 3 .380 grams of the mercury, and they tell us that the density at this temperature is 13 .596 grams for every cubic centimeter.

01:00 That tells us that the volume is 024860.

01:10 Cubic centimeters.

01:12 So that is our volume of mercury that we have at zero degrees.

01:15 Let's do the same thing using the density at 25 degrees.

01:21 Again, we still have that same 3 .380 grams of mercury, but now its density has decreased.

01:36 In every centimeter cubed, there's 13 .534 grams.

01:51 Therefore, the at 25 degrees increases to 0 .2494.

02:02 So the mercury expanded a little bit as it warmed up.

02:07 Right, the difference in this is how much the volume would change.

02:14 So i want to see how much the volume increased.

02:19 I'm going to do this just by subtracting those two values.

02:25 So the volume increase is going to be equal to the volume at the warmer temperature minus the volume at the cooler temperature.

02:47 Subtracting these two numbers shows us that the volume has increased by 0 .00114 centimeters cubed.

03:00 All right, so that's how much the volume has increased.

03:05 And we want to see what this volume increase, how this translates, to distance it travels up the capillary tube.

03:15 So if we think about a capillary tube, it is a cylinder, and the mercury is going to move up this cylinder.

03:26 So to figure out how far up this cylinder it's going to move, we're going to have to consider the formula for volume of a cylinder.

03:36 Volume of a cylinder is the area, which is pi r squared times the height.

03:45 So pi r squared h.

03:47 Let's think about what we know...

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