The Ideal Gas Law

PV = Nk_BT,   where

P = the pressure of the gas
V = the volume of the container the gas occupies
N = the number of molecules of gas in the container
k_B = Boltzmann's constant
T = the temperature of the gas in K°

This equation has four variables in it (P, V, N, T) and that makes it hard to grasp.  Frequently though, we can constrain two of the variables and then we only have to worry about a two-variable equation.

First, we can talk about a particular sample of gas inside a container.  That fixes N, and we don't have to deal with it any further.  We can just set Nk_B = constant, and for our purposes it doesn't matter what the constant is.

Let us consider the rest of the variables, (P, V, T), two at a time.

1)  T = constant

If we fix the temperature, we are just left with PV = constant for the gas law.  So, in this situation, if the volume is doubled, the pressure must go down by one-half.  And vice-versa.  The simplest illustration of this would be a cylinder with a plunger on one end:  if you push the plunger in so that the volume of the cylinder is halved and the temperature remains constant, then the pressure will double.

However, it is important to realize that constant-temperature processes must be pretty slow ones (in other words, you must move the plunger on the gas cylinder very slowly) because there must be enough time for heat to flow between the inside and outside of the cylinder to keep everything at a constant temperature.

Suppose you don't do that.  Suppose you just slam in the plunger very quickly.  Well, you are exerting a force through a distance (as you push the cylinder in, the pressure inside resists, thus you are using force), and as we remember, E = Fd.  So, energy is being expended as you force the plunger inwards.  Where does that energy go?  If you slam the plunger in quickly, it doesn't have time to leave the cylinder, so there is only one place for it to go:  into the gas.  In other words, the gas will heat up.

You can imagine the situation microscopically by visualizing the gas atoms inside the cylinder, bouncing around like ping-pong balls.  Suppose one of the walls starts moving inwards.  What happens to the atoms as they bounce against the wall moving towards them?  Well, what happens to a baseball when it meets a bat moving towards it?  The atoms bounce off the wall with a higher velocity than they had when they hit it.  In other words, their temperature rises.  In order for a cylinder with a plunger being pushed into it to remain at constant temperature, the plunger must be moved in so slowly that the heated gas atoms inside the cylinder have time to lose that heat to the outside world.

Tech Note -- the diesel engine, unlike a gasoline engine, uses no spark plug.  How does it ignite the fuel?  It does it by simply operating at a very high pressure.  When diesel fuel and air are sprayed inside the cylinder in a diesel engine, the cylinder is compressed so highly (and so fast) that the temperature of the fuel-air mixture rises enough to ignite itself.  No other heat input is necessary.

The opposite of the above effect is provided by putting compressed gas into the cylinder, then letting go of the plunger.  The high-pressure gas will drive the plunger out -- and the energy to accelerate the plunger has to come from somewhere.  It can only come from the gas inside the cylinder, so the gas temperature will fall.  (It is left as an exercise to the reader to imagine ping-pong balls hitting a wall and moving it outwards, and thereby losing velocity.)  The expansion can only occur at constant temperature if the expansion is very slow, so that the gas has time to absorb heat from the outside world.

2)  V = constant

In this case, we can write P(const) = (const)T for the Ideal Gas Law, or just P = (const)T

In this case, the pressure will rise or fall directly with the temperature.  Double the temperature, double the pressure.  Constant volume is easy to achieve:  you just need a gas inside a sealed container of some sort.  The only caveat to be kept in mind here is that the temperature must be measured in K°.  "Doubling the temperature" means that you go from 200 K° to 400 K°, not 50 F° to 100 F°.

Tech note -- Ever wonder why manufacturers warn you not to burn empty spray cans in a fire, lest they explode?  You may have wondered (as I did, when I was a little kid), what exactly is it that is going to explode, if the can is empty?  Well, in fact, the can wouldn't explode -- if it were truly "empty".  But, it isn't empty.  It has air inside, and if you heat that air enough, it might reach a high enough pressure to rupture the can, i.e., explode.

Microscopically, the V= constant case is easy to visualize.  You have atoms bouncing around.  You heat them up (or cool them down), i.e., you change their velocity.  They then bang against the walls more (or less) energetically, which is exactly what we call pressure.

3)  P = constant

This case gives us V = (const)T.  Doubling the absolute temperature of a gas also doubles its volume, if the pressure is constant, and vice versa.

This case can sometimes be a bit tricky to visualize.  A constant pressure can only be maintained if there is some force (usually external to the body of gas in question) maintaining a constant force on the gas.  In my lectures, I sometimes use a demonstration device which consists of a cylinder filled with tiny blue pellets to simulate a gas.  An agitator at the bottom whips up the pellets into a little hailstorm, thereby simulating the motion of gas atoms.  A movable plunger setting on top of the cylinder has a constant weight and a constant area, so it represents a constant pressure.  (Since P = force / area).  Temperature is a measure of the average energy of atoms, and by whipping the pellets at different speeds I can show that the volume of the cylinder expands as the speed of the pellets (i.e., their "temperature") is increased, and vice-versa.

Microscopically, we are just saying that gas atoms move faster as they gain heat (increase temperature).  So, they pound the walls harder and create higher pressure.  If the wall is free to move, then it will do so, because unbalanced pressures on opposites sides of a wall mean unbalanced forces, and thus the wall must move unless it is clamped into place.

But why did plunger in my demonstration eventually stop?  If the atoms (blue balls) are at the same temperature, then they are still moving at the same speed.  Why don't they just push out the plunger forever?

To answer this, we must realize that the pressure on the wall is affected not only by how fast the balls (atoms) are hitting it, but also how often.  As the size of the chamber increases, the time it takes for the balls to criss-cross the chamber must also increase.  So, they cannot hit the wall as often.  Thus, as the wall moves further and further back, the pressure on it must decrease.  The wall eventually stops moving, at the point where the internal pressure has become exactly equal to the external (constant) pressure.

Probably the most common constant-pressure phenomena in everyday life are those affected by the Earth's atmosphere, which is way too massive to have its pressure changed by anything humans can do.  (The Sun is more influential -- the weatherman's "barometric pressure" is just a fancy phrase for atmospheric pressure, and this does vary slightly with the weather.)  A party balloon, for example, has constant atmospheric pressure acting on it.  If you blow up a party balloon and place it in the freezer for an hour or so, it will shrink.  If you let it warm up, it will expand again.

Summary

The ideal gas law is very intuitive, if you just remember what the terms in the equation represent physically and keep in mind the picture of small atoms racing about.

For a constant volume, changing the temperature (the speed of the atoms) will change the pressure because the atoms now strike the unmoving walls more (or less) vigorously.

For a constant pressure (like that provided by the weight of a movable plunger) you again change the temperature/speed of the balls, but in this case the walls are now free to move and do so, changing the distance the atoms must travel between hits.  The system will come to equilibrium at a new volume, determined by the point at which the internal pressure once again equals the external pressure.

For a constant temperature, changing the volume will change the pressure (change the number of "hits" on the walls) because now the atoms have a different distance to travel and thus can strike the walls more (or less) frequently, even though their speeds have not changed.

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