Lecture 6 - Ideal gas law, rising and sinking air

Up to this point 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 view. We ignore the atmosphere and concentrate on just the air inside a small volume or balloon or parcel* of air. We are going to "derive" an equation that shows how pressure (P) depends on certain properties of the air insidie the balloon.

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

Hot air balloons rise (they also sink), so does the relatively warm air in a thunderstorm updraft (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.

Understanding the ideal gas law is the first step in explaining what actually causes air to rise or sink.

In the second step 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) in Step 3.

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 the balloon inflated.

The pressure produced by the air molecules inside a balloon will first depend on the number of air molecules, N, in the balloon. As you add more and more air to something like a bicycle tire, the pressure increases. If there weren't any air molecules at all there wouldn't be any pressure. 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 the oxygen concentration experiment described in Week 1. 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. We'll demonstrate the effect of temperature on pressure in class on Friday.

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 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).

Air in the atmosphere behaves like air in a balloon. A balloon can grow or shrink in size depending on the pressure of the air inside. When a balloon isn't getting bigger or smaller it means the force inside that is pushing out is balanced by the force outside that is pushing in.

We start in the top figure with air inside a balloon that is exactly the same as the air outside. The air inside and outside have been colored green. The arrows show that the pressure of the air inside pushing outward and the pressure of the air surrounding the balloon pushing inward are all the same strength.

Next we warm the air in the balloon (Fig. 2). The ideal gas law equation tells us that the pressure of the air in the balloon will increase. The increase is momentary though.

Because the pressure inside is now greater (the big yellow arrows) than the pressure outside, the balloon will expand. As volume begins to increase, the pressure of the air inside the balloon will decrease. Eventually the balloon will expand just enough that the pressures inside and outside are again in balance. You end up with a balloon of warm low density air that has the same pressure as the air surrounding it (Fig. 3)

You can use the same reasoning to understand what happens when you cool the air in a balloon.

The air inside and outside are the same in Fig. 1. Cooling the air inside the balloon in Fig. 2 causes a momentary drop in the inside pressure (small yellow colored arrows) and creates a pressure imbalance. The stronger outside air pressure compresses the balloon.

As the balloon volume decreases, pressure inside the balloon increases. It eventually is able to balance the outside air pressure. You end up with a balloon filled with cold high density air.

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.

If you warm a parcel of air the volume will increase and the density will decrease. Pressure inside the parcel remains constant. If you cool the parcel of air it's volume decreases and its density increases. Pressure inside the parcel remains constant.

The balloon shrinks down to practically zero volume when pulled from the liquid nitrogen. It is filled with very cold high density air at that point. As the balloon warms the balloon expands and the density of the air inside the balloon decreases. The volume and temperature kept changing in a way that kept pressure constant. Eventually the balloon ends up back at room temperature (unless it pops).

Now we are in a position to have a quick look at the forces that can cause parcels of air to rise or sink.

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 the parcel (air 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.

We can modify the demonstration that we did earlier to demonstrate Charles' Law. In this case we use balloons filled with helium (or hydrogen). Helium is less dense than air even when the helium has the same temperature as the surrounding air. A helium-filled balloon doesn't need to warmed up in order to rise.

We dunk the helium-filled balloon into some liquid nitrogen to cool it and to cause the density of the helium to increase. When removed from the liquid nitrogen the balloon doesn't rise, the cold helium gas is denser than the surrounding air (the purple and blue balloons in the figure above). As the balloon warms and expands its density of the helium decreases. The balloon at some point has the same density as the air around it (green above) and is neutrally bouyant. Eventually the balloon becomes less dense that the surrounding air (yellow) and floats up to the ceiling.

Something like this happens in the atmosphere.

At (1) sunlight reaching the ground is absorbed and warms the ground. This in turns warms air in contact with the ground (2) Once this air becomes warm and its density is low enough, small "blobs" of air separate from the air layer at the ground and begin to rise. These are called "thermals." (3) Rising air expands and cools (this is something we haven't covered yet). If it cools enough (to the dew point) a cloud will become visible as shown at Point 4. This whole process is called free convection. Many of southern Arizona's summer thunderstorms start this way.

The relative strengths of the downward graviational force and the upward pressure difference force determine whether a parcel of air will rise or sink. Archimedes Law is another way of trying to understand this topic.

A gallon of water weighs about 8 pounds (lbs).

If you submerge a 1 gallon jug of water in a swimming pool, the jug becomes, for all intents and purposes, weightless. Archimedes' Law (see figure below, from p. 53a in the photocopied ClassNotes) explains why this is true.

The upward bouyant force is really just another name for the pressure difference force covered earlier today (higher pressure pushing up on the bottle and low pressure at the top pushing down, resulting in a net upward force). A 1 gallon bottle will displace 1 gallon of pool water. One gallon of pool water weighs 8 pounds. The upward bouyant force will be 8 pounds, the same as the downward force on the jug due to gravity. The two forces are equal and opposite.

Now we imagine pouring out all the water and filling the 1 gallon jug with air. Air is about 1000 times less dense than water;compared to water, the jug will weigh practically nothing.

If you submerge the jug in a pool it will displace 1 gallon of water and experience an 8 pound upward bouyant force again. Since there is no downward force the jug will float.

One gallon of sand (which is about 1.5 times denser than water) jug will weigh 12 pounds.

The jug of sand will sink because the downward force is greater than the upward force.

You can sum all of this up by saying anything that is less dense than water will float in water, anything that is more dense than water will float in water.

The same reasoning applies to air in the atmosphere.

Air that is less dense (warmer) than the air around it will rise. Air that is more dense (colder) than the air around it will sink.

There's a colorful demonstration of how small differences in density can determine whether an object floats or sinks.

Cans of both regular and Diet Pepsi are placed in beakers filled with water (Coke and Diet Coke can also be used).

Both cans are made of aluminum which has a density almost three times higher than water. The drink itself is largely water. The regular Pepsi also has a lot of high-fructose corn syrup, the Diet Pepsi doesn't. The mixture of water and corn syrup has a density greater than plain water. There is also a little air (or perhaps carbon dioxide gas) in each can.

The average density of the can of regular Pepsi (water & corn syrup + aluminum + air) ends up being slightly greater than the density of water. The average density of the can of diet Pepsi (water + aluminum + air) is slightly less than the density of water.

In some respects people in swimming pools are like cans of regular and diet soda. Some people float (they're a little less dense than water), other people sink (slightly more dense than water).

Many people can fill their lungs with air and make themselves float, or they can empty their lungs and make themselves sink. People must have a density that is about the same as water.