From One Particle to a Box of Gas
Last timeRopes, Wheels and Forces That Do No Work
Pressure is delivered momentum and temperature is average squared speed, so a gas needs no new physics. What it does need is counting, and counting is where time acquires a direction.
A box of gas has about as many particles in it as there are grains of sand on
Earth, several times over. Nobody is going to track them. What is remarkable is
that the two quantities anyone actually measures about a gas, its pressure and its
temperature, can be obtained from the conservation laws of the earlier lessons
applied to one particle at a time, and then multiplied up.
Start with a single particle bouncing between two walls, straight back and forth.
It hits the right-hand wall travelling right and leaves travelling left at the same
speed, because the wall is heavy and the bounce is elastic. Its momentum has
changed by twice what it was. By the law of the third lesson, the wall received
exactly that much. Then the particle crosses the box, bounces off the far wall, and
comes back, so the delivery repeats at a rate set by its speed and the width of the
box.
- the momentum handed to the wall in one hit, which is twice the particle's momentum because the particle reverses rather than stopping
- the width of the box, which sets how long the particle takes to come back for the next hit
- the average force on that wall from this one particle. Divide by the area of the wall and it is a pressure
- the particle's mass and speed, which are the only properties of it that appear anywhere
Now put in a great many particles, travelling in all directions at a range of
speeds. Two things change and neither is deep. The force adds up, so the number of
particles appears. And a particle moving diagonally delivers less to this pair of
walls than one moving straight at it, which on average costs a factor of three,
because there are three directions to share the motion between.
- the pressure, meaning force per unit area, which is what a gauge reads
- the price of three dimensions. Only a third of a particle's motion is aimed at any one pair of walls on average, and this factor is the whole of that accounting
- how much mass sits in each unit of volume. This is where the number of particles enters, and it enters by simple addition because each particle delivers independently
- the average of the squared speed over all the particles. The angle brackets are the only new notation in the lesson and they mean nothing more than an average
That is the useful half of the lesson, and it is worth seeing how cheap it was. Two
conservation laws already established, applied one bounce at a time, give both
quantities a gas is described by.
The lesson stops here
4 more paragraphs to go
You have read the opening. The rest of the argument, the problems that check whether it landed, and the lines worth keeping at the end all come with a plan.
The first lesson of every course in the library reads the whole way through, free, so you can see exactly what the rest of them are.
See the planThe contentsThis is the reading half
Starting the course gives you your own copy of it. Every idea on every page has problems standing under it, marked with a reason rather than a tick, and any sentence you do not believe can be opened and argued with. None of that can happen on a page nobody owns.
The contents