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Empirical Math Model: Ideal Gas Law

January 29, 2013 - 9:54am


To appreciate the distinction between curve fitting and what it means for a tool to be truly predictive it might help to consider how the ideal gas law was developed.

In 1663, Robert Boyle performed a series of experiments at room temperature and observed that pressure (P) and volume (V) of a gas obeys a simple mathematical relationship; as pressure increases, volume decreases by the same proportion implying the product, PV, is constant. More than 100 years later, in 1787 and again in 1802, Jacques Charles and Joseph Louis Gay-Lussac demonstrated that the temperature (T) and volume (V) of a gas also obeys a simple mathematical relationship; as temperature increases, volume increases by the same proportion implying that the ratio, V/T is constant. Another 10 years after that in 1811, Amedeo Avagadro demonstrated that volume (V) and the number of molecules (n) of a gas obeys a simple mathematical relationship; as more molecules are added, the volume increases by the same proportion implying that the ratio, V/n is constant. As the different pieces of this puzzle came together over a period of 200 years, we arrived at the ideal gas law, PV=nRT, where P is pressure, V is volume, T is temperature, n is # of molecules and R is the universal gas constant.

The history of the ideal gas law is a great example of the development of an empirical math model. The behavior of gases was observed at specific pressures and temperatures revealing a simple mathematical relationship between the relevant variables in the experimental data. There was really no deeper understanding about various physical processes governing the behavior of a gas. Nonetheless, the empirical math model was sufficient to nicely fit experimental data for temperatures and pressures commonly encountered in ordinarily life.

Indeed the simple math model could then be used to successfully predict what we should observe at pressures and temperatures for which we had no data. However, at more extreme pressures and temperatures, the ideal gas law fails to predict the behavior of real gases by significant margins. This experience demonstrates both the beauty and the pitfalls of an empirical math model. On the one hand, it is simple and easy to use and serves to usefully predict behavior in many commonly encountered situations. On the other hand, it is an arbitrary application of a mathematical expression to fit experimental observations quite possibly devoid of any basis in physical reality and can therefore easily fail to predict behavior outside of very narrow ranges of applicability.

For more extreme temperatures and pressures, the ideal gas law fails miserably to explain what is observed in real-world experiments. The true behavior of a real gas over a wide range of temperatures and pressures is governed by a number of physical processes including thermodynamics and electromagnetics ultimately having to do with the advanced area of physics known today as equations of state of matter. A physics model considers all of these physical phenomena to characterize the behavior of the gas according to what actually happens in the real world.  The more accurately it reflects the real world, the more predictive the code can be considered.