Modeling Lesson

Scientific Modeling

A model is a simple system which reveals important properties of a more complex system that you wish to understand better. More than one type of model can be used to study the same complex system, each model shedding light on some different aspect of the complex system but each model has limitations on what kind of information it can give you. Some of the basic rules and limitations of modeling are presented in the following series of models.

Volcano model - basic rules and limitations of a model.
Rabbit model -how models are adapted as new data comes in.
Ideal Gas Law - using modeling to get quantitative information.

Modeling a volcano

Models are created to answer specific questions, how you design your model depends on the question(s) you want it to answer. The more specific the question, the better the chances of achieving a satisfactory answer.

A favorite question is: "Why do volcanos act the way they do?" Or, more specifically: "What happens inside a volcano that makes it shoot out lava, smoke and ash?" Many students in elementary school have seen a volcano demonstrated with baking soda and vinegar. Often demonstrators will go to great effort to make their small models look like a volcanic cone, and into the center depression will place some baking soda and vinegar. The resulting chemical reaction produces a lot of bubbles and a frothy liquid expands out of the depression and flows down the side of the cone. With a good imagination the baking soda-vinegar model looks like a real volcano, and based on this model a student could make certain predictions about real volcanoes:

Volcanoes use some sort of chemical reaction, involving something like baking soda and vinegar, to make a lot of gas bubbles.
The resulting lava is a cool and foamy liquid, smelling something like Italian salad dressing.

An important part of modeling is experimental verification ; where you compare the predictions of your model against actual observations on the complex system that you are studying. From a distance our baking soda-vinegar model looks realistic, although we see a great deal of smoke and ash is produced by real volcanoes and not by our model. Closer inspection of a real volcano reveals that the liquid flowing over the sides is very hot and not all that frothy. And the smell of a real volcanos is nothing like Italian salad dressing.

Chemical analysis of both the lava and the volcanic gases reveal very little of the chemicals that we used in our baking soda-vinegar model. Let us say that another demonstrator comes along with a different model of volcanoes, called the soup pot model. The demonstrator heats a pot of clear broth, with some noodles and veggies added, in clear bowl. As heat is added to the bottom of the bowl the veggies and the broth move from the bottom of the bowl to the top and then down again. Soon bubbles appear and as more heat is added the boiling becomes quite vigorous, spraying soup around the surrounding area and frothing over the sides of the soup pot.

A boiling pot of soup looks very little like a volcano, but the predictions it makes about volcanic behavior that come much closer to actual volcanic observations.
The boiling soup pot model answers certain questions but doesn't answer others like: where does the heat come from? Why do different volcanoes erupt differently? How much of an impact do volcanoes have on stratospheric ozone? The search for the answers to those questions requires new and different models.

In summary you can see how a model is a simple system which reveals important properties of a more complex system that you wish to understand better. Experiment verification and observation of the complex system is used to test the worth of a model. To learn more on how models can be changed as the questions or observations change, consider the story of the rabbit model provided by Dr.Bob Panoff, of the NCSA.

The Rabbit model

Let us say that you live in a society that suddenly decides that the local rabbit population would be a good food source. Your society knows little about rabbits, so it assigns several hunters to go out and make observations. You are assigned to interpet their data and build a model which can be used to predict what will happen to the rabbit population as time goes by.

What will happen to the rabbit population as time passes? Even before your hunters return with their observations its easy to see that the number of bonny bunny babies (hereafter referred to as ''kits'') that will be born depends on the number of rabbits you start with. Using the letter 'b' to represent 'birth-rate' and "d" to represent the change in a value, you can write a very simple equation:

b = dR/dt ...where R represents the number of new rabbits and

"t" is time.

A simple graph of this equation would allow you to calculate ''R'', the number of new rabbits when ''b'' and'' t'' are known. Note that because this model presumes a constant birthrate as time pases, the slope of the graph is unchanging.

Constant birthdate

As you consider your first model you realize that one of your starting assumptions: a constant birthrate, is wrong. The kits grow to become parents, as the number of parents grows so does the birthrate. In fact it looks like the population growth will be exponential...

New birth rate

A quick graph of this equation yeilds truely frightening results: it will be just a matter of time before civilization, as you know it, will be over run with rabbits!! You consider which goverment agencies to contact and how to phrase the press-releases: ''Hare-raising senario predicts society to fall in a flood of fur!!'' Calming down a bit, you realize that you should do nothing until you get experimental verification of your model by the hunters out in the field.

Unfortunately in the real world of competing researchers and competing newsreporters, 'newsleaks' of exciting but un-verified models make front-page news. When flaws are found in the models the press loudly discards the model as a farce or worse, an intentional fabrication. This is not the right way to use models!

Wisely, you have waited for data from the field. Hunters have recorded changes in the rabbit population over the course of several years and their findings are shown below:

Rabbits over a three year period

You are facinated by this unexpected leveling off of the population. It strikes you that you have not accounted for an increasing death rate due to the depletion of food. You communicate with your hunters and they verify the gradual depletion of the surrounding grass as the population increases.

Note how a model can guide researchers in what to look for.

At the end of the third year your hunters start to report a sharp decline in the rabbit population which gradually levels off from the third through the ninth year:

new rabbits

They also notice changes in several other populations, the golden eagle and owl populations; populations observed to be predetors on the rabbits:

New Rabbits and Preditors

You alter your model to account for two populations, predator and prey, that are interdependent on each other. This requires coupled equations, shown below in simplified form. The answer to our initial question:''What will happen to the rabbit population as time passes?'' appears to be growing a lot more complex.

Equations for determinign the new Rabbit and Predator populations

A final important point There is a tendency by both public and media to moralize models: models that are verfied are 'good' until they fail in a certain situation, in which case they become 'bad'. A model is a tool, and even when a tool fails it tells you something about a situation that you didn't know before. In the event of a unexplained observation, a model can be modified or completely replaced but its failure reveils new knowlege about the situation that is being modeled and that is good. In an effort to keep models simple, assumptions can be made and factors neglected, hopefully without serious reduction in the prediction accuracy. This priciple is exemplified in the stiring saga of the Ideal Gas Law.

Ideal Gas Law

Leading up to the time of the American Civil war, chemists were trying to come up with a model that would answer the specific question: How many molecules of gas are in a given volume? Obviously the number of molecules depended on the volume of the gas. Researchers had already produced several equation which described how changes in pressure and temperature affect the volume of a gas:

Boyles Law: P * V = a constant , when temperature is held constant

Charles Law: V/T = a constant , when temperature is expressed in Kelvin units and

pressure is constant.

A ''Law'' is a term applied to a model that, within its limitations, has been repeatedly

verified for a number of different situations.

A previous model of gases, ''The Kinetic theory of gases'', had made a number of accurate predictions about a gases physical and chemical behavior, by modeling a gas as a mass of moving, colliding particles. The kinetic theory of gases appears elementary to us now, but before 1810 the very concept of atoms and molecules was still questioned by many. In addition to pressure and temperature, there were two other factors to consider in the determination of the volume of a gas: the volume of the molecules themselves and the amount of intereaction that molecules had when they collided with each other.

The amount of interaction is important because the more time molecules spent interacting in each collision, the fewer collisions they have per unit time. Pressure is really the measure of molecular collisions per unit time, so increasing the interaction means decreasing the internal pressure exerted by the gas; and as a gas exerts less pressure against its suroundings, its volume decreases.

The Vander Waals equation considers all these factors as it models gas behavior:

Gas law constant which varies with the units used for Pressue and Volume

...where ''n'' is the moles (a unit of 6*10e23 molecules or atoms), ''a'' is the attractive-force coefficient and ''b'' is the size of the gas molecule or atom. Well, ''a'' and ''b'' are difficult to determine now, and in 1810 their determination was out of the question.

Modeling gas behavior was vastly simplified by the work of Avagadro and Cannizzaro. In order to explain chemical reactions between gases, in 1808 Avaogadro theorized that equal volumes of all gases, at the same temperature and pressure, contain an equal number of molecules.

In other words the different sizes and reactivities of different kinds of gases can be ignored!

But because Avagadros model lacked experimental verification; it was ignored for over fifty years until a fellow Italian, Stanisslao Cannizzaro, revived and applied it to finding molecular weights. Cannizzaro made more of an impact not only because he wrote more clearly, but because by this time there was experimental vertfication for his model. Once it was shown that the size and chemistry of a gas can be ignored in the determination of molar-volume Vander Waals equation could be quickly summarized down to our present Ideal Gas Law.

P V = n R T

Within its limitations the Ideal Gas Law can accurately predict the behavior of real gases. For example consider the changes in the volume of methane(CH4) as the pressure on it changes:

pressure(measured in atmospheres) 1 atm 1.5 atm 2.0 atm 4.0 atm 8.0 atm

V obs (the observed volume of CH4) 13.3liter 10.0 liter 5.0 liter 2.4 liter

V ideal(the predicted volume of CH4) 13.3liter 10.0 liter 5.0 liter 2.5 liter

But at far more extreme pressures differences between actual and predicted values start to appear:

graph sowing pressure vs Ratio Vobs

This is because at more extreme pressures factors which the Ideal Gas Law chooses to ignore become important enough to undermine the models accuracy but for the usual lab conditions the Ideal Gas Law closely predicts the physical behavior of a gas.

In summary an ideal gas is a model, a fictional creation which has no molecular volume and no molecular interactions. Like many simple models, it makes accurate predictions where certain factors can be ignored; more complex models are needed when these factors can no longer be ignored.

Author: Brien Sparling

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Curator: Jill Dunbar	Last Update: May 30, 2001
NASA Official: Walt Brooks