Wednesday, November 25, 2009

GAS PRESSURE

Introduction

Gases

The first step to understanding gases is to spell out what exactly a gas is. Gases have two properties that set them apart from solids and liquids. First, gases spontaneously expand to fill the container they occupy, no matter its size. In other words, a gas has no fixed volume or shape. Secondly, gases are easily compressible.

You can imagine a gas as a busy swarm of molecules. Each molecule moves randomly and travels great distances before bouncing off another molecule. This occurs because the individual molecules comprising a gas are generally far apart. In fact, for a gas at low pressure, we can approximate that aside from a few random collisions, individual gas molecules do not interact. This approximation is what separates gases from solids and liquids, whose molecules always interact. The series of SparkNotes on Gases SparkNote seek to use this approximation about gases to establish the ideal gas law and the kinetic molecular theory. The ideal gas law macroscopically describes how gases behave under nearly all conditions. The kinetic molecular theory describes how sub-microscopic gas molecules interact with each other.

Pressure

Of the three general terms used to describe gases (volume, temperature, pressure), pressure is the least familiar. Before we can delve into the gas theories, we need a firm understanding of it. Pressure is defined as force divided by the area over which the force acts:

pressure

P =

Ice skates are familiar examples of the effects of pressure. The area of the blades of a skate are much smaller than, say, the soles of your feet. So if you strap on ice skates, your weight will act on an area much smaller than it would if you were wearing normal shoes. Since A decreases while F stays the same, by @@Equation@@, the pressure you exert on the ice will be much greater if you're wearing skates. This pressure is often enough to melt a layer of ice, which allows your skate to glide smoothly across an ice rink. If you try the same maneuver with normal shoes, you will not generate enough pressure to melt the ice and won't get anywhere fast.

So how does pressure relate to gases? If you will remember, a gas will fill any container that holds it. It is easy to see why with our swarm analogy. If a compact swarm of molecules is placed into a large container, the individual molecules will move about randomly and eventually stray from their original dimensions. Eventually, some intrepid molecules will reach the walls of the container. When they do, they will impact the walls of the container. These impacts generate a force, and hence a pressure on the walls of the container.

Terms

Atmosphere - A unit of measurement defined as 101,325 Pascals. The typical pressure at sea level varies around one standard atmosphere (atm). Atmospheric pressure P can be calculated via the following equation:

P = ghρ

Bar - A unit of measurement equivalent to 1×105 Pascals.
Barometer - A device used to measure atmospheric pressure.
Barometers.
mm Hg - A unit of pressure commonly used with the barometer. It corresponds to 1 torr and 1/760 atm at 0 o Celsius only.
Pascal - The SI unit of pressure. 1 Pascal = 1 N/m2 .
Pounds per Square Inch - A unit of pressure commonly used in the United States. 1 psi = 1 lb in-2 . 14.6960 psi corresponds to one atmosphere.
Pressure - Pressure = Force/Area. The standard atmospheric pressure is 101,325 Pascals.
Torr - A unit of pressure closely related to mm Hg, but more convenient and absolute. 1 Torr = 1 mm Hg at 0 o Celsius. 1 Torr always equals 1/760 atm, irrespective of the temperature.

Pressure and the Barometer

Pressure

As a student, you are familiar with pressure. Work needs to be done, and there is always a limited time to do it. The less time there is, the more pressure you feel. Gaseous pressure works in much the same way. A force acts on a limited area to give pressure. If the area shrinks (you have less time), something has got to give: either the force shrinks (you take on less work) or the pressure rises.

Pressure is defined mathematically as force divided by the area over which the force acts:

Pressure =

The SI unit of pressure is the Pascal. 1Pascal = 1 N / m2 = 1 kg m-1s-2 . However, there will be times when you'll be given pressure in non-SI units. The table summarizes the most common pressure units and their conversion factors.

UnitRelationship
Pascal (Pa)

1= 1

Torr

atm

1 mm Hg at 0 o C

atm

atmosphere (atm)

101.3×105 Pa

Bar

1×105 Pa

1 Pound per Square Inch (psi)

0.0680atm

If you've ever checked your tires' pressure, you've probably encountered pounds per square inch, or psi. The other units are less familiar: they arose because gravity exerts a downward force on the atmosphere, which consequently exerts a pressure on the Earth's surface and whatever else happens to be there. On a calm day at sea level, the force gravity exerts is1.01325×105 N per 1 m2 . Since pressure = force / area, this gives a pressure of 1.01325×105 Pa. One standard atmosphere (atm) is defined as exactly 1.01325×105 Pa.

So how did we figure out that standard atmospheric pressure is1.01325×105 Pa in the first place? Atmospheric pressure was first measured with a barometer. A barometer consists of a large dish and a long glass tube that is sealed at one end. The tube and dish are filled with mercury (HG) or some other liquid, and the tube is inverted into the dish. If all this is done without any air entering the tube, a column of liquid will remain above the dish.

Figure %: The Mercury Barometer

When the tube full of mercury is inverted in the dish, the mercury level will drop. It will continue to drop until the pressure generated by the column's weight equals the atmospheric pressure. Since we know the column's height h , the density of mercury ρ , and the acceleration due to gravity g (9.81m s-2 ), we can calculate the atmospheric pressure P .

baroeq

P = ghρ

At 0 o Celsius, it turns out that the atmosphere can support a column of mercury 760 mm tall (the unit mm Hg is thus equal to 1/760). 1 atm = 760 mm Hg only at 0 o Celsius since the ρ of Hg changes with temperature. It's a pain to recalculate ρ at different temperatures, but the units of Torr come to the rescue. 1 Torr = 1/760 atm at all temperatures. The Bar is related to the Torr, but is not quite as useful. 1 bar = 1×105Pascals.

Students are often given a non-Hg barometer or conditions where g does not equal 9.8 m/s2 . Don't let these types of questions phase you; realize which variables are changing, convert everything down to SI units, and plug them all in to @@Equation@@. There are examples in the problem section.



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