# The Physics Philes, lesson 124: Exploding Potatoes and Equations of State

Last week we finished the chapter on temperature and heat. We learned what heat is, how it moves, and how its measured. It was a good time, but it’s time to move on. Now we’re going to explore the thermal properties of matter.

From just living in the world, I think most of us have a fairly good understanding of heat. We know that we can turn water into steam by heating it, and we can turn water into ice by freezing it. Some of us – Not me. Ok…me. – have accidentally exploded a potato in the microwave by forgetting to poke holes in it. Without the holes, the steam that builds up in the potato can’t escape, and things get messy.

These macroscopic properties – temperature, pressure, volume, and mass – are closely related to the microscopic properties of the substance. So, by investigating masses, speeds, kinetic energies, and momenta of individual atoms and molecules we can determine something about the larger quantities of a substance.

We can describe a substance by using physical quantities. Temperature, pressure, volume, and mass are examples of such physical quantities. In fact, these four specifically describe the state of the material in question and we call them state variables. The volume of something is usually dependent on the pressure, temperature, and amount of the substance (described in the total mass or number of moles). Normally, you can’t change any of these variables without causing a change in one or more of the other variables. Think about that potato that I totally didn’t blow up in the microwave. I changed the temperature, which caused a change in pressure.

The relationship between pressure, temperature, volume, and mass (or number of moles) is called the equation of state. Sometimes these relationships are simple and we can know the equation exactly. Sometimes it isn’t simple at all and we don’t or can’t know an exact equation. We still call these relationships equations of state.

Just as an example, let’s look at a (relatively) simple equation of state. A temperature coefficient of volume expansion is the fractional volume change per unit temperature change. The compressibility constant k is the negative of the fractional volume change per unit of pressure change. If a certain material starts out at a certain volume, temperature, and pressure, then the final volume at a slightly different temperature and pressure can be expressed like this:

This is an equation of state. Notice the negative sign in front of the compressibility constant and pressure term. That indicates that an increase in pressure causes a decrease in volume.

We can look at an even simpler equation of state, this time for an ideal gas. Don’t think too much about what an ideal gas is right now. Let’s just think about gases in general.

Through experimental measurements, we know three things about gases:

- The volume of the gas is proportional to the number of moles. So, if we double the number of moles and hold the pressure and temperature constant, the volume will also double
- The volume has an inverse relationship with absolute pressure. In other words, if we double the pressure and hold the temperature and number of moles constant, the new volume will be one half of the original volume.
- The pressure is proportional to the absolute temperature. You’re probably seeing a pattern at this point. If we double the absolute temperature and hold the volume and number of moles constant, the pressure doubles, as well.

We can combine these relationships into something called the ideal gas equation:

The n stands for the number of moles and the R is the ideal gas constant. It is equal to 8.314 J/mol·K and it is the same for all ideal gases.

Now it’s time to think about what an ideal gas is. An ideal gas is just a gas that obeys the ideal gas equation for all pressures and temperatures. Since it’s an idealized model, it’s not perfect. It works best at very low pressures and very high temperatures. It’s pretty good, like within a few percent – at moderate temperatures, moderate pressures, and at temperatures much higher than temperatures at which a gas liquifies.

We can also write the equation in terms of the total mass of the substance, rather than the number of moles. The total mass of a substance can be found by multiplying the number of moles of a substance you have by that substance’s molar mass M:

All we have to do is rearrange the equation a little and replace n in the ideal gas equation:

Huh. Will you look at that? We can glean an expression for density from this equation. The density of the gas is the total mass of the substance divided by the volume, so we can do a little more rearranging and we get:

One more thing before I skedaddle. For a constant mass or a constant number of moles of an ideal gas, the product of the ideal gas constant and the number of moles is constant, so the product of pressure and volume divided by temperature is also constant. So, for any two states of the same mass of a gas, we get:

The R, remember, is a constant term and ends up getting cancelled out.

And that is the end of The Physics Philes for this week. Stay tuned for more physics next week.

*Featured image credit: Stefano A. via Flickr*

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