One of the questions on the In-class Assignment asked you to identify the forces that would cause a hot air balloon to either sink or rise.  Basically we need to understand why warm air rises and cold air sinks.

Hot air balloons rise (they also sink), so does the relatively warm air in a thunderstorm (its warmer than the air around it).   Conversely cold air sinks.  The surface winds caused by a thunderstorm downdraft (as shown above) can reach speeds of 100 MPH and are a serious weather hazard.

A full understanding of these rising and sinking motions is a 3-step process (the following is from the bottom part of p. 49 in the photocopied ClassNotes)

We will first learn about the ideal gas law.  That is an equation that tells you which/how properties of the air inside a balloon work to determine the air's pressure.  Then we will look at Charles' Law, a special situation involving the ideal gas law (air temperature and density change together in a way that keeps the pressure inside a balloon constant).  Then we'll learn about the vertical forces that act on air (an upward and a downward force).

Students working on Experiment #1 will need to understand the ideal gas law to be able to explain why/how their experiment works.

The figure above makes an important point: the air molecules in a balloon "filled with air" really take up very little space.  A balloon filled with air is really mostly empty space.  It is the collisions of the air molecules with the inside walls of the balloon that keep it inflated.

---

Up to this point in the semester we have been thinking of pressure as being determined by the weight of the air overhead.  Air pressure pushes down against the ground at sea level with 14.7 pounds of force per square inch.  If you imagine the weight of the atmosphere pushing down on a balloon sitting on the ground you realize that the air in the balloon pushes back with the same force.  Air everywhere in the atmosphere pushes upwards, downwards, and sideways. 

The ideal gas law equation is another way of thinking about air pressure, sort of a microscopic scale version.  We ignore the atmosphere and concentrate on just the air inside the balloon.  We are going to "derive" an equation.  Pressure (P) will be on the left hand side.  Relevant properties of the air inside the balloon will be found on the right side of the equation.

In A

the pressure produced by the air molecules inside a balloon will first depend on how many air molecules are there, N.  If there weren't any air molecules at all there wouldn't be any pressure.  As you add more and more add to something like a bicycle tire, the pressure increases.  Pressure is directly proportional to N; an increase in N causes an increase in P.  If N doubles, P also doubles (as long as the other variables in the equation don't change).

In B
air pressure inside a balloon also depends on the size of the balloon.  Pressure is inversely proportional to volume, V .  If V were to double, P would drop to 1/2 its original value.

Note
it is possible to keep pressure constant by changing N and V together in just the right kind of way.  This is what happens in Experiment #1 that some students are working on.  Oxygen in a graduated cylinder reacts with steel wool to form rust.  Oxygen is removed from the air sample which is a decrease in N.  As oxygen is removed, water rises up into the cylinder decreasing the air sample volume.  N and V both decrease in the same relative amounts and the air sample pressure remains constant.  If you were to remove 20% of the air molecules, V would decrease to 20% of its original value and pressure would stay constant.

Part C: Increasing the temperature of the gas in a balloon will cause the gas molecules to move more quickly.  They'll collide with the walls of the balloon more frequently and rebound with greater force.  Both will increase the pressure.  You shouldn't throw a can of spray paint into a fire because the temperature will cause the pressure inside the can to increase and the can could explode. 

Surprisingly, as explained in Part D, the pressure does not depend on the mass of the molecules.  Pressure doesn't depend on the composition of the gas.  Gas molecules with a lot of mass will move slowly, the less massive molecules will move more quickly.  They both will collide with the walls of the container with the same force.

The figure below (which replaces the bottom of p. 51 in the photocopied ClassNotes) shows two forms of the ideal gas law.  The top equation is the one we just derived and the bottom is a second slightly different version.  You can ignore the constants k and R if you are just trying to understand how a change in one of the variables would affect the pressure.  You only need the constants when you are doing a calculation involving numbers (which we won't be doing).

Read through the explanation on p. 52 in the photocopied Classnotes.  In the atmosphere a parcel (balloon) of air will always try to keep its pressure the same as the pressure of the surrounding air.  If they aren't equal the parcel will either expand or shrink until they are again equal.

If you warm air it will expand and density will decrease until the pressure inside and outside the parcel are equal.
If you cool air the parcel will shrink and the density will increase until the pressures balance.

These two associations:

(i)   warm air = low density air
(ii)  cold air = high density air

are important and will come up a lot during the remainder of the semester.

Basically it comes down to this - there are two forces acting on a parcel* of air in the atmosphere:
1. Gravity pulls downward.  The strength of the gravity force depends on the mass of the air inside the parcel.  This force is just the weight of the parcel
2. There is an upward pointing pressure difference force.  This force is caused by the air outside (surrounding) the parcel.  Pressure decreases with increasing altitude.  The pressure of the air at the bottom of a parcel pushing upward is slightly stronger than the pressure of the air at the top of the balloon that is pushing downward.  The overall effect is an upward pointing force.

When the air inside a parcel is exactly the same as the air outside, the two forces are equal in strength and cancel out.  The parcel is neutrally bouyant and doesn't rise or sink.

If you replace the air inside the balloon with warm low density air, it won't weigh as much.  The gravity force is weaker.  The upward pressure difference force doesn't change (because it is determined by the air outside the balloon which hasn't changed) and ends up stronger than the gravity force.  The balloon will rise.

Conversely if the air inside is cold high density air, it weighs more.  Gravity is stronger than the upward pressure difference force and the balloon sinks.

* the word parcel just means a small volume of air.

---