Once again, the advent calendar is delayed until late at night by a busy day with SteelyKid-- soccer in the morning, playing with a trebuchet after lunch, then Arthur Christmas at the Colonie mall. We're running low on days to honor great milestones in physics, though, so I don't want to skip a day entirely.
I'm also trying to spread this around to cover a fairly representative set of subfields; having done classical mechanics and E&M at some length, I need to rush through a couple of other subfields quickly. One of these is classical thermodynamics, a field with a rich history and wide range of applicability, which nonetheless will be represented here by a single equation:
This is the "Ideal Gas Law," and governs the behavior of a gas composed of idealized molecules that collide elastically with each other but don't interact between collisions. This is, obviously, an imperfect approximation of reality, but it makes a good starting point for thinking about the behavior of gases.
In this equation, P is the pressure of the gas, V the volume of the container, N the number of molecules, and T the temperature. The constant kB is Boltzmann's constant, and serves to get the units right.
So, what does this tell us?
Well, the easiest way to think about it is to imagine keeping one of the variables constant, and looking at how the others behave. So, for example, if you have a system at constant temperature, and increase the pressure, the volume has to decrease (as anybody who has ever squeezed a balloon knows). Or if you've got a system at constant pressure, and increase the temperature, the volume must increase (which makes a hot-air balloon possible). Or if you have a constant volume, and suddenly decrease the pressure, the temperature must decrease (which is what leads to that characteristic little curl of vapor when you open a cold bottle of beer on a humid day). And so on.
These relationships between pressure, volume, and temperature were worked out piecewise over a period of almost 200 years, starting with Robert Boyle in the 1600's. It was all pulled together into this form in the early 1800's, and given a more satisfactory theoretical foundation by Boltzmann and Maxwell in the latter part of the 19th century.
Why is this important? Well, a huge number of things turn out to depend on the behavior of confined gases, chief among them being engines. If you want to analyze the operation of your car in an abstract sense, this is the place you would start.
And it's the analysis of engines (originally steam engines) that leads to all the rest of thermodynamics. In particular, the study of heat engines leads to the idea of entropy, and an understanding of the limits of what can possibly be accomplished in real systems. That, in turn, leads to all sorts of fruitful speculation about the history and future of the universe, and so on.
So, If you're in a place where the temperature and pressure are comfortable, relax, and take a minute to appreciate the hard scientific work that lets us understand and control the behavior of gases. And if you're not in a place where the temperature and pressure are comfortable, what are you doing reading blogs and thinking about physics?
Come back tomorrow, when we will regrettably dispense with all of statistical mechanics in a single equation.
Please remember that the temperature T must be an absolute temperature.
One of the things I like about gas laws written this way is that they explicitly separate volume V and particle number N. By the time you reach senior year or first year of grad school, thinking of a "density" as "amount of a quantity per particle times number of particles per unit volume" is more useful than assuming "density" means "mass per volume".
agm: What is "amount of a quantity"? What is this quantity you are referring to?
The quantity could be mass, so it might be (amount of) mass per particle times the number of particles per unit volume. The "amount of" might be redundant, but aside from that agm wants to be more explicit about density. Other quantities, such as amount of charge could be a way of specifying charge (or whatever) density.
The general idea is *number* density. Multiply by mass per particle, get mass density. Multiply by charge, get charge density. Etc Etc.
In my experience, it's easier for people to go from a general case to particular case than it is to go the other way, so for me it's a matter of pedagogical effectiveness, though admittedly without any support other than my anecdotal experience.