Gas Law Theory

Fluid dynamics and gases

A lot of what we learned about fluids and liquids works equally well for gases as well as liquids. When it comes to Archimedes principle, we can examine the buoyant force on objects.  This is generally only noticeable when the buoyant force exceeds the force of gravity which you know form playing with Helium balloons.  Generally, you probably do not play with Hydrogen balloons as they tend to be slightly flammable.  The buoyant calculations are simple for airships as they are always totally submerged by the atmosphere.

Interestingly, there is a comeback in interest in hydrogen as a buoyant gas for airships, in particular given the increasing costs of helium.   And only pure hydrogen gas is flammable.  If you dilute it with a small amount of nitrogen, it is not quite as dangerous.

We mentioned air pressure before.  However, instead of having several meters of water over our heads as a diver might, we have several miles of air.  One significant difference is that the density of that air changes significantly with pressure and altitude.

As air pressure varies with altitude, this makes a useful tool for pilots for determining the altitude of an airplane.  In fact, one of our Vernier sensors has a built in air pressure altitude sensor!  This could be a fun experiment to test in the John Hancock Center!  As you get farther from Earth, there is less air over your head, and less air pushing down on you.

However, in order for an airplane to use air pressure to determine the altitude of an airplane, you need to know what the air pressure is on the ground.  That is why there is a network of barometers across the country, and pilots can get barometer/altimeter settings by tuning in to the proper frequencies.

This is particularly important as air pressure changes significantly over fronts (near storms!).  A stable region of High air pressure might be at 31” of Hg, whereas a Low Pressure stormy region might be at 28.”  This is particularly why we care about air pressure. In the days before satellites and the television Weather Man, one could know a storm was coming by the sudden drop in air pressure. [Insert WEATHER MAP PHOTO from Dr. Horn's Room]

Generally, air pressure tends to decrease by 1” of Hg for every increase in altitude of 1000 feet.  So if you fly over a stormy region without realizing it, your  airplane altimeter could be easily thrown off by a few thousand feet!  Many airplane accidents have happened because of pilots with bad altimeter settings.

As mentioned before, Galileo made a crude barometer using a vacuum pump made from a leather bag and water as the fluid.  Torricelli, a student of Galileo’s, instead used mercury, which is thus the convention we will use today to measure air pressure.  Essentially, all a Torricelli Tube is merely a flask where the air has been evacuated connected to an upsidedown tube in a dish full of Mercury.  Air pressure pushes down on the Mercury and then pushes it up into the tube which has been evacuated of air.  The level of the Mercury depends on the air pressure, which is normally around 29” of Hg.

You might be surprise that air pressure could push water 33 feet up a straw, or Mercury 30” up a straw.  But the Mandenberg Spheres give an even more dynamic demonstration of the power of air pressure.  You evacuate the air from the spheres and try to pull them apart!

Robert Boyle first published the relationship between pressure and volume in the 17th century.  The behavior of gasses with changing air pressure was understand long before the underlying physics was, largely because atomic theory and the concepts of energy were not yet developed until the 19th century.  In 1662, Boyle published the results of experiments by of changing volume by increasing the pressure.  The experimental apparatus was actually built by Robert Hooke.

 

Boyle’s law is often expressed as

PV = constant

 

What I like about the development of this law is that it was largely based on observation of what they saw as opposed to the explanation using molecular theory.  They didn’t know what this constant was or what it meant.  Although we can prove it quickly using a Vernier Gas Pressure sensor, one can easily prove Boyle’s Law using a syringe and a stack of textbooks.

 

The second gas law which was empirically studies is Charles’s Law.  Again, this can be easily proved using a closed syringe and a thermometer.  It is important to note that these temperature measurements are in the absolute value scale. This law also said

V/T = constant

Charles developed this rule in the 1780s although is was published by Gay-Lussac and Dalton in 1802.

The third law of importance is known as the Gay-Lussac Law which says that

 

P/T = constant

 

Again, we could prove this easily using a combination of a temperature probe and a gas pressure sensors.

These three laws were combined by Clapeyron in 1834 to produce the ideal gas law.  This is what appears in many physics texts as

PV/T = constant

The modern form of this is

PV = nRT

Where they realized that this depends on the number of molecules in the gas.

n = number of moles of gas

R=8.31 J/K/mol

The above equation is what often appears in chemistry books.

In a physics book you will often see this equation expressed as

PV = NkT

Boltmann k = R/ NA = 1.38 x 10 –23 J/K

NA = 6.022 x 10 -23 per  (g mol)

N = number of molecules or particles

What is important to note here is what we actually mean by temperature, heat, and internal energy, all concepts we will explore in the coming week.

We can think of temperature as proportional to the average kinetic energy per molecule. T = avg KE * 2/(3k)

Heat is actually the flow of energy from one object to another (either by conduction, convection, or radiation).

The internal energy is the sum of the molecular energies of a substance (due to both the kinetic energies and the electrostatic potential interaction energies).

The internal energy is usually expressed as 3/2 NkT