# The Physics Philes, lesson 119: Turn Up the Heat

Wow. It’s been three weeks since I last wrote about physics. Three whole weeks. Life, man. You know how it is. But I have not forgotten. Let me pick up where I left off.

Let’s see…*flips through notes*…aha! It’s time to talk about thermal expansion.

Thermal expansion is pretty much what it sounds like. When most materials are heated, they expand. We can see this in everyday life. Have you ever used hot water to loosen a metal lid? If you have, you’ve had experience with thermal expansion.

As you might expect, the relationship between the temperature and the expansion of the material can be expressed with a few simple formulas. First, let’s think about the expansion of a rod.

OK, so you’ve got this rod made of some material, any material. It doesn’t matter. It has some initial length and initial temperature. Experiment has shown that if we change that temperature a little, the change in length of the rod is directly proportional to the temperature change. In addition, the change in length is also proportional to the the rod’s initial length. That is, if you have two rods made of the same material, but one rod is twice as long as the other, the change in its length will be twice as great. Each material has its own proportionality constant α, so we can express the relationship like this:

where ΔL is the change in length, L0 is the initial length, and ΔT is the change in temperature. The proportionality constant α is called the coefficient of linear expansion. It’s different for every material and the unit is inverse Kelvin or inverse degree Celsius. It’s weird that we can use Kelvin *or* Celsius, right? Actually, we can use either because we’re dealing with a *change* in temperature. The difference between degrees is the same in Kelvin and Celsius. That is, if the temperature in a room goes up one degree Celsius, it also goes up one Kelvin. Since this this is a relationship involving a change in temperature, we can use either Celsius or Kelvin. (Fahrenheit, however, is off limits. The interval between degrees are smaller those of Celsius or Kelvin.)

For a lot of materials, every linear dimension changes in accordance with this equation. Whether it’s the rod’s thickness, or the side of a square sheet, or the diameter of a hole, many materials will expand consistent with the above equation.

What a minute. The diameter of a hole? Wouldn’t the material just expand into the hole?

It turns out, no. As the object with the hole in it expands, the hole will expand, too. Remember that the material will expand in every linear dimension. Think about the atoms of material that outline the hole. As the material is heated and the material expands, the atoms spread apart, which increases the size of the hole. Neat!

In reality, the direct proportionality of the linear thermal expansion relationship is not exact. It’s only an approximation for when the change in temperature is not too large. However, for practical purposes this relationship works just fine.

We’re not done though, because, just has heat causes an expansion in linear dimensions, heat causes an increase in volume, as well. Just like with linear thermal expansion, we have determined through experiment that the change in volume is proportional to the change in temperature (provided that the temperature change isn’t too great, of course). The form of the relationship is much the same as the linear thermal expansion:

where ΔV is the change in length, V0 is the initial length, and ΔT is the change in temperature. The proportionality constant β is called the coefficient of volume expansion. It uses the same units as the coefficient of linear expansion, and for the same reasons.

There is an interesting relationship between α and β. Think about a cube made of some material. It has a side length L and a volume V that is equal to L cubed. When the temperature increases by a very small amount, the side lengths will also increase by a very small amount, as will the volume. In math, we can represent this by

where dV is that tiny increase in volume and dL is that tiny increase in length. We can now replace L and V with their initial values. In that case, dL is

where L0 is the initial length and dT is the tiny increase in temperature. Now, since the initial volume is equal to the initial length cubed, we can write the tiny increase in volume as

Notice that this equation looks a lot like the equation for volume thermal expansion. Except that, instead of β, we have 3α. So, this equation is consistent with the volume thermal expansion equation if β = 3α.

Neat, right? We’re not done with thermal expansion, though. Next week (I promise!) we’ll talk about the thermal expansion of water – which is pretty interesting in itself – and thermal stress.

*Featured image credit: Mauro Cateb via Flickr*

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