Video: Thermal Expansion in Three Dimensions

A glass beaker has a capacity of 397 mL. The beaker is filled to the brim with water that is at a temperature of 15.0°C. Determine the volume of water that will overflow the beaker when the water’s temperature increases to 25.0°C. Use a value of 207 × 10⁻⁶°C⁻¹ for the coefficient of volumetric expansion of water.

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Video Transcript

A glass beaker has a capacity of 397 millilitres. The beaker is filled to the brim with water that is at a temperature of 15.0 degrees Celsius. Determine the volume of water that will overflow the beaker when the water’s temperature increases to 25.0 degrees Celsius. Use a value of 207 times 10 to the negative sixth inverse degrees Celsius for the coefficient of volumetric expansion of water.

Let’s start off our solution with a sketch of this glass beaker. We start out with this beaker of known volume. And we’re told that it’s then filled to the brim with water. So the water completely fills this beaker. And it’s just about to spill over the edge. And initially, this volume of water filling the beaker is at a temperature of 15.0 degrees Celsius. Then, say that we heat the water, perhaps by a flame underneath, somehow adding heat to it to raise its temperature. The water goes from 15.0 to 25.0 degrees Celsius. Based on this temperature change and something called the coefficient of volumetric expansion of water, we want to figure out how much water will overflow the beaker.

Now here is what this number, the coefficient of volumetric expansion, means. It tells us that, as the temperature of the water increases, so does the volume of the water. In other words, the hotter the water, the more space it takes up. Now, if we rewind to the time when the water temperature in the beaker was 15.0 degrees Celsius, remember we’re told that, at that temperature, the water completely fills the beaker. It’s absolutely brimming full. So if water expands when it’s heated, that means when the temperature of the water goes up, the volume will go up as well. And some of this water will spill over the edges of the beaker. It’s that volume of spilled water that we want to solve for. And we can do it using a mathematical equation that ties together the change in the volume of a substance with its coefficient of volumetric expansion and its temperature change.

Here it is. It’s a somewhat simple mathematical relationship that tells us that the ratio between the change in an object’s volume and its original volume is equal to 𝛼 sub 𝑣, its coefficient of volumetric expansion, multiplied by its change in temperature Δ𝑇. Since we want to solve for the amount of water that overflows, we know that that’s Δ𝑉, the change in volume, because the initial volume 𝑉 completely filled the beaker. Here’s what we can write then. We can say that the change in volume of the water, that is, the amount that overflows the beaker, is equal to its initial volume multiplied by 𝛼 sub 𝑣 multiplied by the change in temperature. Now capital 𝑉, the original volume, is given as 397 millilitres. The coefficient of volumetric expansion is given as 207 times 10 to the negative sixth inverse degrees Celsius. Now what about Δ𝑇? Well, that’s equal to the final temperature of the water minus its initial temperature, 25.0 minus 15.0 degrees Celsius.

When we plug all this in, recognising that 25.0 minus 15.0 degrees Celsius is 10.0 degrees Celsius, notice how the units of degrees Celsius cancel out with inverse degrees Celsius, leaving us with the units of volumes in millilitres. That’s a good sign, because we’re calculating a change in volume, Δ𝑉. Multiplying these numbers together to three significant figures, we find the result of 0.822 millilitres. So this tells us that when we heat up our original volume of water, 397 millilitres, by 10.0 degrees Celsius, then the volume of that water expands by a little bit less than one millilitre. That’s how much overflows the beaker.

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