Gas Laws Save Lives:

The Chemistry Behind Airbags

Stoichiometry and the Gas Constant Experiment

Authors: Rachel Casiday and Regina Frey

Department of Chemistry, Washington University

St. Louis, MO 63130

Key Concepts:

Safety of Airbags (Note: This section contains an animation.)

Chemical Reactions to Generate the Gas to Fill an Airbag Decomposition of Sodium Azide (NaN 3 ) Reactions to Remove Harmful Products Reaction Stoichiometry

Ideal-Gas Laws (Macroscopic Picture) PV=nRT Estimating the Pressure to Fill an Airbag Acceleration Force Pressure Deflation

Kinetic Theory of Gases (Microscopic Picture) Pressure as the Result of Molecular Collisions with Container Walls Average and Root-Mean-Square Speed of Molecules Maxwell-Boltzmann Distribution

Protection in a Collision Newton's Laws Airbags Decrease the Force on the Body Airbags Spread the Force Over a Larger Area

Undetonated-Airbag Disposal: Safety Considerations

Introduction: Airbags Improve Automobile Safety

The Safety Advantage of Airbags

The development of airbags began with the idea for a system that would restrain automobile drivers and passengers in an accident, whether or not they were wearing their seat belts. The road from that idea to the airbags we have today has been long, and it has involved many turnabouts in the vision for what airbags would be expected to do. Today, airbags are mandatory in new cars and are designed to act as a supplemental safety device in addition to a seat belt. Airbags have been commonly available since the late 1980's; however, they were first invented (and a version was patented) in 1953. The automobile industry started in the late 1950's to research airbags and soon discovered that there were many more difficulties in the development of an airbag than anyone had expected. Crash tests showed that for an airbag to be useful as a protective device, the bag must deploy and inflate within 40 milliseconds. The system must also be able to detect the difference between a severe crash and a minor fender-bender. These technological difficulties helped lead to the 30-year span between the first patent and the common availability of airbags.

In recent years, increased reports in the media concerning deaths or serious injuries due to airbag deployment have led to a national discussion about the usefulness and "safety" of airbags. Questions are being raised as to whether airbags should be mandatory, and whether their safety can be improved. How much does the number of deaths or serious injuries decrease when an airbag and seat belt are used, as compared to when a seat belt is used alone? How many people are airbags killing or seriously injuring? Do the benefits of airbags outnumber the disadvantages? How can airbags be improved?

As seen in Figures 1 and 2, airbags have saved lives and have lowered the number of severe injuries. These statistics are continuing to improve, as airbags become more widely used. Nevertheless, as the recent reports have shown, there is still a need for development of better airbags that do not cause injuries. Also, better public understanding of how airbags work will help people to make informed and potentially life-saving decisions about using airbags.

Figure 1 This bar graph shows that there is a significantly higher reduction in moderate to serious head injuries for people using airbags and seat belts together than for people using only seat belts.

Figure 2 Deaths among drivers using both airbags and seat belts are 26% lower than among drivers using seat belts alone.

Overview of How Airbags Work

Timing is crucial in the airbag's ability to save lives in a head-on collision. An airbag must be able to deploy in a matter of milliseconds from the initial collision impact. It must also be prevented from deploying when there is no collision. Hence, the first component of the airbag system is a sensor that can detect head-on collisions and immediately trigger the airbag's deployment. One of the simplest designs employed for the crash sensor is a steel ball that slides inside a smooth bore. The ball is held in place by a permanent magnet or by a stiff spring, which inhibit the ball's motion when the car drives over bumps or potholes. However, when the car decelerates very quickly, as in a head-on crash, the ball suddenly moves forward and turns on an electrical circuit, initiating the process of inflating the airbag.

Once the electrical circuit has been turned on by the sensor, a pellet of sodium azide (NaN 3 ) is ignited. A rapid reaction occurs, generating nitrogen gas (N 2 ). This gas fills a nylon or polyamide bag at a velocity of 150 to 250 miles per hour. This process, from the initial impact of the crash to full inflation of the airbags, takes only about 40 milliseconds (Movie 1). Ideally, the body of the driver (or passenger) should not hit the airbag while it is still inflating. In order for the airbag to cushion the head and torso with air for maximum protection, the airbag must begin to deflate (i.e., decrease its internal pressure) by the time the body hits it. Otherwise, the high internal pressure of the airbag would create a surface as hard as stone-- not the protective cushion you would want to crash into!

Movie 1 Please click on the pink button below to view a QuickTime movie showing the inflation of dual airbags when a head-on collision occurs. Click the blue button below to download QuickTime 4.0 to view the movie.

What about the Gas Used to Fill the Airbag?

Chemical Reactions Used to Generate the Gas

Inside the airbag is a gas generator containing a mixture of NaN 3 , KNO 3 , and SiO 2 . When the car undergoes a head-on collision, a series of three chemical reactions inside the gas generator produce gas (N 2 ) to fill the airbag and convert NaN 3 , which is highly toxic (The maximum concentration of NaN 3 allowed in the workplace is 0.2 mg/m3 air.), to harmless glass (Table 1). Sodium azide (NaN 3 ) can decompose at 300 o C to produce sodium metal (Na) and nitrogen gas (N 2 ). The signal from the deceleration sensor ignites the gas-generator mixture by an electrical impulse, creating the high-temperature condition necessary for NaN 3 to decompose. The nitrogen gas that is generated then fills the airbag. The purpose of the KNO 3 and SiO 2 is to remove the sodium metal (which is highly reactive and potentially explosive, as you recall from the Periodic Properties Experiment) by converting it to a harmless material. First, the sodium reacts with potassium nitrate (KNO 3 ) to produce potassium oxide (K 2 O), sodium oxide (Na 2 O), and additional N 2 gas. The N 2 generated in this second reaction also fills the airbag, and the metal oxides react with silicon dioxide (SiO 2 ) in a final reaction to produce silicate glass, which is harmless and stable. (First-period metal oxides, such as Na 2 O and K 2 O, are highly reactive, so it would be unsafe to allow them to be the end product of the airbag detonation.)

Gas-Generator Reaction Reactants Products Initial Reaction Triggered by Sensor. NaN 3 Na

N 2 (g) Second Reaction. Na

KNO 3 K 2 O

Na 2 O

N 2 (g) Final Reaction. K 2 O

Na 2 O

SiO 2 alkaline silicate

(glass) Table 1 This table summarizes the species involved in the chemical reactions in the gas generator of an airbag. Note: Stoichiometric quantities are not shown.

Questions on Chemical Reactions Used to Generate the Gas

Write a balanced chemical equation for the first reaction in the airbag gas generator (the decomposition of NaN 3 ).

Write a balanced equation for the net gas-generating reactions (the combination of the first and second reactions).

The Macroscopic Picture of Gas Behavior: Ideal-Gas Laws

Calculation of the Amount of Gas Needed

Nitrogen is an inert gas whose behavior can be approximated as an ideal gas at the temperature and pressure of the inflating airbag. Thus, the ideal-gas law provides a good approximation of the relationship between the pressure and volume of the airbag, and the amount of N 2 it contains. (The ideal-gas law is PV = nRT, where P is the pressure in atmospheres, V is the volume in liters, n is the number of moles, R is the gas constant in L · atm/mol · K ( R = 0.08205 L · atm/mol · K), and T is the temperature in Kelvin.) A certain pressure is required to fill the airbag within milliseconds. Once this pressure has been determined, the ideal-gas law can be used to calculate the amount of N 2 that must be generated to fill the airbag to this pressure. The amount of NaN 3 in the gas generator is then carefully chosen to generate this exact amount of N 2 gas.

Estimating the Pressure Required to Fill the Airbag

An estimate for the pressure required to fill the airbag in milliseconds can be obtained by simple mechanical analysis. Assume the front face of the airbag begins at rest (i.e., initial velocity v i = 0.00 m/s) , is traveling at 2.00x102 miles per hour by the end of the inflation (i.e., final velocity v f = 89.4 m/s) , and has traveled 30.0 cm (the approximate thickness of a fully-inflated airbag).

The airbag's acceleration (a) can be computed from the velocities and distance moved (d) by the following formula encountered in any basic physics text:

v f 2 - v i 2 = 2ad. (1)

Substituting in the values above,

(89.4 m/s)2 - (0.00 m/s)2 = (2)(a)(0.300 m)

a = 1.33x104 m/s2.

(2)

The force exerted on an object is equal to the mass of the object times its acceleration (F = ma) ; therefore, we can find the force with which the gas molecules push a 2.50-kg airbag forward to inflate it so rapidly. 2.5 kg is a fairly heavy bag, but if you consider how much force the bag has to withstand (see Figure 5), it becomes apparent that a lightweight-fabric bag would not be strong enough. Note: In the calculation below, we are assuming that the airbag is supported in the back (i.e., all the expansion is forward), and that the mass of the airbag is all contained in the front face of the airbag.

F = ma

F = (2.50 kg)(1.33x104 m/s2)

F = 3.33x104 kg · m/s2 = 3.33x104 N. (3)



(4)

Pressure is defined as the force exerted by a gas per unit area (A) on the walls of the container (P = F/A), so the pressure (in Pascals) in the airbag immediately after inflation can easily be determined using the force calculated above and the area of the front face of the airbag (the part of the airbag that is pushed forward by this force). Note: The pressure calculated is gauge pressure.

The amount of gas needed to fill the airbag at this pressure is then computed by the ideal-gas law (see Questions below). Note: the pressure used in the ideal gas equation is absolute pressure. Gauge pressure + atmospheric pressure = absolute pressure.

Deflation of the Airbag

When N 2 generation stops, gas molecules escape the bag through vents. T he pressure inside the bag decreases and the bag deflates slightly to create a soft cushion. By 2 seconds after the initial impact, the pressure inside the bag has reached atmospheric pressure.

Questions on the Macroscopic Picture of Gas Behavior: Ideal-Gas Laws

A certain model of car is equipped with a 60.0-liter, 2.00-kg airbag that inflates at 2.00102 miles per hour in the "h" direction (see figure below) and is 25.0 cm thick when fully inflated, as shown below.



Compute the area of the front face of the airbag when it is fully inflated. Show your calculation, including proper units. HINT: Think of the airbag's shape as a cylinder whose height is 25.0 cm, the thickness of the inflated airbag, as shown in the diagram below. Remember, for a cylinder, V=bh, where b is the area of the base and h is the height of the cylinder. Assume that most of the mass is contained in the front face of the airbag. Compute the gauge pressure (in atmospheres) inside the airbag when it inflates at 2.00102 miles per hour. Show your calculation, including proper units. You may need to check the table of physical constants and conversions in your chemistry book to find some of the numbers you need. Calculate the mass (in grams) of sodium azide required to generate enough nitrogen gas to fill the airbag at the pressure you calculated in part (b). Assume that the temperature of the gas is 25.0oC. Show your calculation, including proper units. (Note: the answer you determined in part (b) is gauge pressure. The ideal gas equation uses absolute pressure. Gauge pressure + atmospheric pressure = absolute pressure. Assume the atmospheric pressure is 1 atm.)

An automotive engineer proposes using a new fabric for airbags, which is cheaper but 30% heavier than the conventionally used nylon fabric. How would the implementation of this higher-weight airbag affect the amount of sodium azide required to fill the airbag in the same amount of time? Briefly, justify your answer.

The Microscopic Picture of Gas Behavior: Kinetic Theory of Gases

Thus far, we have only considered the macroscopic properties (i.e., pressure and temperature) of the gas in an airbag from the point of view of the ideal-gas law, which is derived from experimental observations (i.e., empirically). Now we turn to a theoretical model to explain these macroscopic properties in terms of the microscopic behavior of gas molecules. The kinetic theory of gases assumes that gases are ideal (i.e., no interactions between molecules, and the size of the molecules is negligible compared to the free space between the molecules), but treats each molecule as a physical body that moves continually through space in random directions.

In a microscopic view, the pressure exerted on the walls of the container is the result of molecules colliding with the walls, and hence exerting force on the walls (Figure 3). When many molecules hit the wall, a large force is distributed over the surface of the wall. This aggregate force, divided by the surface area, gives the pressure.

Figure 3 This is a schematic diagram showing gas molecules (purple) in a container. The molecules are constantly moving in random directions. When a molecule hits the container wall (green), it exerts a tiny force on the wall. The sum of these tiny forces, divided by the interior surface area of the container, is the pressure.

An important relationship derived from the kinetic theory of gases shows that the average kinetic energy of the gas molecules depends only on the temperature. Since average kinetic energy is related to the average speed of the molecules (E K = mu2 / 2, where m=mass and u is the average speed) , the temperature of a gas sample must be related to the average speed at which the molecules are moving. Thus, we can view temperature as a measure of the random motion of the particles, defined by the molecular speeds.

We see from the kinetic theory of gases that temperature is related to the average speed of the molecules. This implies that there must be a range (distribution) of speeds for the system. In fact, there is a typical distribution of molecular speeds for molecules of a given molecular weight at a given temperature, known as the Maxwell-Boltzmann distribution (Figure 4). This distribution was first predicted using the kinetic theory of gases, and was then verified experimentally using a time-of-flight spectrometer. As shown by the Maxwell-Boltzmann distributions in Figure 4, there are very few molecules traveling at very low or at very high speeds. The maximum of the Maxwell-Boltzmann distribution is an intermediate speed at which the largest number of molecules are traveling. As the temperature increases, the number of molecules that are traveling at high speeds increases, and the speeds become more evenly distributed (i.e., the curve broadens). A useful indication of a typical speed in the Maxwell-Boltzmann distribution is the root-mean-square speed (u rms ), which depends on the temperature and the molecular weight of the gas according to the formula



(5)

where R is the gas constant in J/mol�K (R = 8.3145 J/mol · K), T is the temperature in K, and M is the molecular weight in kg/mol.

Figure 4 The Maxwell-Boltzmann distribution can be shown graphically as the plot of the number of molecules traveling at a given speed versus the speed. As the temperature increases, this curve broadens and extends to higher speeds.

As seen in Figure 4, there is a unique distribution curve for each temperature. Temperature is defined by a system of gaseous molecules only when their speed distribution is a Maxwell-Boltzmann distribution. Any other type of speed distribution rapidly becomes a Maxwell-Boltzmann distribution by collisions of molecules, which transfer energy. Once this distribution is achieved, the system is said to be at thermal equilibrium, and hence has a temperature.

Questions on the Microscopic Picture of Gas Behavior: Kinetic Theory of Gases

Helium is a noble gas that is not used to fill airbags because there is no convenient and economical way to generate a large amount of He gas quickly. From your examination of the Maxwell-Boltzmann distributions in Figure 4 in the tutorial and the formula for u rms , predict whether the root-mean-square speed of the gas molecules in an airbag at 25.0oC would increase, stay the same, or decrease if He were used instead of N 2 . Briefly, explain your reasoning.

On one graph, sketch the Maxwell-Boltmann distributions for He and N 2 .

How Does the Presence of an Airbag Actually Protect You?

Newton's familiar first law of motion says that objects moving at a constant velocity continue at the same velocity unless an external force acts upon them. This law, known as the law of inertia, is demonstrated in a car collision. When a car stops suddenly, as in a head-on collision, a body inside the car continues moving forward at the same velocity as the car was moving prior to the collision, because its inertial tendency is to continue moving at constant velocity. However, the body does not continue moving at the same velocity for long, but rather comes to a stop when it hits some object in the car, such as the steering wheel or dashboard. Thus, there is a force exerted on the body to change its velocity. Injuries from car accidents result when this force is very large. Airbags protect you by applying a restraining force to the body that is smaller than the force the body would experience if it hit the dashboard or steering wheel suddenly, and by spreading this force over a larger area. For simplicity, in the discussion below, we will consider only the case of a driver hitting the steering wheel. The same arguments could, of course, be applied to a passenger hitting the dashboard, as well.

Recall from Equation 3,

F = ma, where .



(6)

F is the force on the body, and if F>0, the body is accelerating; if F<0, the body is decelerating. In this case, v f = 0 m/s (when the body's motion is stopped), v i is the velocity of the body at the time of collision, Δt is the time interval for the body to go from v i to v f , and m is the mass of the body. Hence, in this case,

(i.e., the body is decelerating) and the force exerted by the steering wheel (i.e., an immovable object) on the body to bring it to rest is

. (7)

This force from the steering wheel causes the injuries to the body in an accident.

Recall, Δt (in Equation 7) is the time interval for the body to come to rest from its velocity at the instant of collision. Increasing ΔDt (i.e., increasing the time over which the decelerating force is applied) lowers the force exerted on the body. If there is no restraining device (i.e., no airbag or seat belt), then Δt is very small and the body hits the steering wheel instantaneously. Hence the force is large and injuries are severe. If there is a restraining device (e.g., an airbag), ΔDt increases (i.e., the airbag reduces the rate of deceleration). Therefore, the force on the body is smaller and fewer injuries result.

Newton's third law ("For every action, there is an equal and opposite reaction.") tells us that the body must exert a force on the steering wheel that is equal, but opposite, to the force exerted by the steering wheel on the body. Why, then, does the steering wheel not appear to move when the body exerts this force on it? The steering wheel is attached to the car, and so the mass of this object is much larger than the mass of the body that hits it. Hence, although the force is equal, the larger mass accelerates much less according to Equation 3, and the motion is imperceptible.

Similarly, when an airbag restrains the body, the body exerts an equal and opposite force on the airbag. Unlike the immovable steering wheel, the airbag is deflated slowly. This deflation can occur because of the presence of vents in the bag. The force exerted by the body pushes the gas through the vents and thus deflates the bag. Because the gas can only leave at a certain rate (recall the kinetic theory of gases), the bag deflates slowly, and therefore Δt increases.

Additionally, airbags help reduce injuries by spreading the force over a larger area. If the body crashes directly into the steering wheel, all the force from the steering wheel will be applied to a localized area on the body that is the size of the steering wheel (Figure 5a), and serious injuries can occur. However, when the body hits an airbag, which is larger than a steering wheel, all the force from the airbag on the body will be distributed (spread) over a larger area of the body (Figure 5b). Therefore, the force on any particular point on the body is smaller. Hence, less serious injuries will occur.

Figure 5a When a body hits the steering wheel directly, the force of this impact is distributed over a small area of the body, resulting in injuries to this area. The area that hits the steering wheel is shown in red. Figure 5b When a body is restrained by an airbag, the force of the impact is distributed over a much larger area of the body, resulting in less severe injuries. The area that hits the airbag is shown in orange.



The objective of the airbag is to lower the number of injuries by reducing the force exerted by the steering wheel (and the dashboard) on any point on the body. This is accomplished in two ways: (1) by increasing the time interval over which the force is applied, and (2) by spreading the force over a larger area of the body (Figure 5).

Questions on How the Presence of an Airbag Actually Protects You

A 65.0-kg man is traveling at 15.0 m/s when he experiences a head-on collision.

Calculate the force (in Newtons; 1 N = 1 kgm/s2) exerted on him when he is restrained by an airbag that increases Δt to 1.5 s. What does the sign of the force calculated in part (a) imply about the change in velocity of the body?

Warnings are posted against using rear-facing car seats in the front seats of vehicles with passenger-side airbags. Consider a 5-foot tall adult who weighs 140 pounds. If this person has turned to talk to passengers in the back seat at the moment of impact, would you expect the airbag to be protective? Explain your reasoning.

Why do airbags have a mass of 2.5 kg (a fairly large mass)? Would a lighter bag (an ordinary garbage bag is the extreme example) function as well? Briefly explain your answer.

Additional Considerations: Undetonated-Airbag Disposal

Thus far we have discussed how airbags function to protect us when there is a head-on collision. But the vast majority of airbags in cars, fortunately, are never deployed within the lifetime of the automobile. What happens to these airbags? Typically, cars are flattened and recycled at the end of their lifetime, and the airbags are never removed from the cars. This can be hazardous, because these airbags still contain sodium azide, whose presence during the automobile-recycling process endangers workers, and can damage recycling equipment and the environment.

How does this happen? Sodium azide can react in several ways when it undergoes the conditions of the recycling process itself. The first step of this process is to flatten the automobile hulk. Once the car is flattened, it is impossible to see whether or not it contains an airbag. If the container holding the NaN 3 is damaged during flattening, then NaN 3 , which is potentially mutagenic and carcinogenic, can be released into the environment. (Recall, the maximum concentration of NaN 3 allowed in the workplace is 0.2 mg/m3 air.) The next step in recycling cars is to shred them into fist-sized pieces so that the different types of metal can be separated and recovered. Sodium azide released during this process may contaminate the steel, iron, and nonferrous metals recovered at this stage. Of greater concern, however, is the large amounts of heat and friction generated by the shredder. Recall that NaN 3 reacts explosively at high temperatures; hence, there is a risk of ignition when airbags pass through the automobile shredder. This danger is amplified if sodium azide comes in contact with heavy metals in the car, such as lead and copper, because these may react to form a volatile explosive. The pieces of the car may also pass through a wet shredder. Here, another danger arises because if the NaN 3 dissolves in water, it can form hydrazoic acid (HN 3 ):

NaN 3 + H 2 O ---> HN 3 + NaOH.

HN 3 is highly toxic, volatile (i.e., it becomes airborne easily), and explosive.

What can be done to prevent these reactions of sodium azide in undetonated airbags? Somehow, the airbags must be prevented from going through the automobile-recycling process. Warning devices that would alert recyclers to the presence of an undetonated airbag in flattened car hulks have been tested, but these are generally expensive to implement, and they would need to be in every automobile airbag. Also, it is extremely difficult or impossible to remove an airbag from a car that has already been flattened, and so the question of what to do with these flattened cars containing airbags remains unanswered. This will become an increasingly large problem, as airbags have recently become mandatory equipment in new automobiles. Hence, the proportion of cars with airbags in recycling plants will increase. A better solution is to remove the airbag canister before the car is sent for flattening or recycling. This is cheaper, simpler, and more efficient, and allows the car to be recycled safely. This strategy is already used for other hazardous components of cars, such as lead-battery cases. However, there is an added incentive for removing batteries that is not yet applicable for removing airbags from cars before recycling. The lead from batteries can be re-sold, but currently there is no market value for airbag canisters. Thus, strictly-enforced laws or a market-based incentive system may be required to ensure that airbags continue to protect our safety, even after the lifetime of the automobiles that contain them.

Questions on Undetonated-Airbag Disposal

Suggest a way to flatten cars for recycling so that airbags in vehicles would all deploy.

If you wanted to use a different set of chemical reactions to generate gas for airbags, what characteristics should the reagents have?

Summary

Airbags have been shown to significantly reduce the number and severity of injuries, as well as the number of deaths, in head-on automobile collisions. Airbags protect us in collisions by providing a cushion to decrease the force on the body from hitting the steering wheel, and by distributing the force over a larger area. The cushion is generated by rapidly inflating the airbag with N 2 gas (from the explosive decomposition of NaN 3 triggered by a collision sensor), and then allowing the airbag to deflate.

Fundamental chemical and physical concepts underly the design of airbags, as well as our understanding of how airbags work. The pressure in the airbag, and hence the amount of NaN 3 needed in order for the airbag to be filled quickly enough to protect us in a collision, can be determined using the ideal-gas laws, and the kinetic theory of gases allows us to understand, at the molecular level, how the gas is responsible for the pressure inside the airbag. Newton's laws enable us to compute the force (and hence the pressure) required to move the front of the airbag forward during inflation, as well as how the airbag protects us by decreasing the force on the body.

Additional Links:

The Insurance Institute for Highway Safety maintains an informative site about airbag safety, including QuickTime movies that demonstrate the effectiveness of airbags.

The National Highway Traffic Safety Administration gives updates on airbag safety, especially regarding new laws allowing certain people to purchase on-off switches for their airbags. The site also includes QuickTime crash-test videos.

This site from Technical Services Forensic Engineering shows a photograph of the collision sensor that triggers airbag deployment.

The University of Wisconsin Why Files site also has an informative page about airbags.

References:

Bell, W.L. "Chemistry of Air Bags," (1990) J. Chem. Ed. 67 (1), p. 61.

Crane, H.R. "The Air Bag: An Exercise in Newton's Laws," (1985) The Physics Teacher, 23, p. 576-578.

Cutler, H. and E. Spector. "Air bags and automobile recycling," (1993) Chemtech,23, p. 54-55.

Insurance Institute for Highway Safety. "Airbag Statistics." 7 October 1998.< http://www.hwysafety.org/airbags/airbag.htm>.

Madlung, A. "The Chemistry Behind the Air Bag: High Tech in First-Year Chemistry," (1996) J. Chem. Ed., 73 (4), p. 347-348.

Newton's Apple: Teacher's Guides. "Airbags and Collisions." 22 June 1998. <http://www.ktca.org/newtons/9/airbags.html>.

Acknowledgements:

The authors thank Dewey Holten, Mark Conradi, Michelle Gilbertson, Jody Proctor and Carolyn Herman for many helpful suggestions in the writing of this tutorial.

The development of this tutorial was supported by a grant from the Howard Hughes Medical Institute, through the Undergraduate Biological Sciences Education program, Grant HHMI# 71199-502008 to Washington University.

Copyright 1998, Washington University, All Rights Reserved.

Revised October, 2000.