Undoubtedly, few mathematical equations have had a larger impact on the human race than those described in Michael Guillen's book "Five Equations that Changed the World: The Power and Poetry of Mathematics" (Hyperion, NY, 1995, ISBN: 0-7868-8187-9). The book provides an entertaining review of the lives of five scientists and their philosophical journey toward the discovery of their most famous equations. These five scientists are:
- Isaac Newton and the Universal Law of Gravity (1600's)
F = g x M x m ÷ d^2
- Daniel Bernoulli and the Law of Hydrodynamic Pressure (1738)
P + r x v^2/2 = Constant
- Michael Faraday and the Law of Electromagnetic Induction (1831)
G x E = -dB/dt
- Rudolf Clausius and the Second Law of Thermodynamics (1868)
DELTA S(universe) > 0; and
- Albert Einstein and the Theory of Special Relativity (1905)
E = m c^2.
While it may be difficult for most to comprehend the mathematical derivation of these equations, nearly everyone can understand their impact on today's society. Surely, had Newton been present in the mission control room when Neil Armstrong first set foot on the moon, he would have done cartwheels all the way home (after, of course, being kicked out of the room for running around yelling "I'm the King on the World" and berating every technician in sight). Likewise, Bernoulli would have experienced great joy when the Wright Brothers applied his (and Nikolai Zhukovsky's) work to the flight of the Kitty Hawk. I'm sure that Michael Faraday (as well as Gene Roddenberry) would smile humbly as we flip up the covers on our cellular phones to converse with our friends in New Delhi, India, or Sydney, Australia. But alas, each fell victim to Clausius' equation that describes the universe as a predator of "life and lifelike behavior, always inclined toward death and destruction."
As I read this book, I wondered what equations effect us the most in our journey toward the development of the perfect material. Each of us in the thermal processing industry have, in some way or another, been involved with a process intent on modifying the natural state of a material. After all, that's what thermal processing and heat treatment are all about - making a more useful material out of a useless one.
My thoughts led me the to conclusion that there are basically five or six equations that drive each of us in this industry to do what we do. In short, our mission is to modify properties by modifying structure. But, finding a single equation to represent all aspects of this microstructure/property relationship is a daunting task.
The selection of the equations below is, of course, debatable; but from a strict materials engineering perspective, each of these equations provides a keystone for the fundamental direction that we take when we process a material to its final form or structure. My curiosity did not prevent me from seeking input form others. For example, Dave Van Aken, our Engineering Concepts editor, made valid points with regard to the Boltzmann equation, the Paris equation, and the Orowan equation. While many would question my exclusion of heat transfer equations, I felt there was not a single equation that simultaneously represented every type of heat transfer (convection, conduction, and radiation). In addition, heat transfer does truly not describe a state, but rather a "means to an end" by which the results of one of the five equations below can be changed. Other thermodymanic or electrical equations have been excluded because of their close relationship to the more fundamental equations of Clausius and Faraday.
My five equations are:
- Robert Hooke and the Law of Force-Displacement Proportionality (1600's)
F = k x or s = E e
- Adolph Fick and the First Law of Diffusion (1855)
J = -D[dc/dx]
- J. Willard Gibbs and the Free Energy Criterion for Phase Equilibrium (1875)
DELTA (G) = DELTA (H) - T x DELTA (S)
- Svante Arrhenius and the Time-Temperature Rate Relationship (1889)
1/t = A exp(-Q/RT); and
- W.L. Bragg and the Law of Crystallographic Diffraction (1912)
n(lambda) = 2d sin(theta).
Each of these equations provide the fundamental basis for what heat treating is all about, whether it is changing the ultimate strength of the material, the stability of a material, the rate of a reaction, or the modification of structure.
Have fun thinking about your five equations and enjoy the complicated poetry of mathematics. I would be interested to hear your selections. IH
"The funny thing is that everything is science fiction at one time or another."
- Gene Roddenberry
D. Keith Patrick