From Fireworks to Quantum Mechanics

While observing the fireworks over the 4th of July weekend, I couldn’t help but think of quantum mechanics. “What’s the connection?” you ask. The very reason for color is best understood through the eyes of quantum mechanics. Particularly if you wish to produce predetermined colors as part of a fireworks show. Commonly in the chemistry lab one will perform the “flame test” to determine which elements may be in the sample you are testing. Depending on the color of the flame you can deduce which compounds may be in your sample. It is a very quick way of making practical use of this phenomenon. But how does it work? Photons of a particular energy are released corresponding to each elements particular line spectrum. Don’t worry! Keep reading, I jumped ahead. Allow me to explain.
When white light (such as sunlight) is passed through a prism, we see all the colors of the rainbow (ROY-G-BIV). If we were to use just the light emitted from a particular elemental source we would see only particular spots of the rainbow. This was the observation. This indicates that only certain energies are allowed for the electrons in particular atoms. In other words, the energy of the electrons are quantized. Changes in energy between discrete energy levels will produce only certain wavelengths of emitted light. For example, a given change in energy from a high to a lower level would give a wavelength of light that can be calculated from the Planck equation. ΔE=hν where the Greek letter nu is representative of the frequency and h is planck’s constant. Nu is directly interchangeable with the wavelength ν=c/λ. Ok, enough with the math talk. What does all this really mean in English? Picture the electrons in orbitals around the nucleus of the atom. When these electrons become excited (like during the explosion of the gunpowder in a firework) they get to absorb energy (that’s the photon). The electron absorbs the energy and gets promoted into a new higher energy level. Obviously it will not maintain being excited for long (it’s not energetically favorable) and it will fall back down to the ground state (where it started). When the electron goes back to the ground state it emits a photon (energy) corresponding to the amount of energy between the higher energy level and the ground state. This amount of energy corresponds to a particular frequency which corresponds to a particular wavelength which our eyes can interpret as a particular color. Each element has a particular set of orbitals with a “signature” line spectrum. Hence, different elements can emit different colors (wavelengths). Quantum mechanics helps to explain this phenomenon. The rise of quantum theory has helped to explain many such observations that just don’t seem to make sense using Newtonian mechanics. Since you have now been introduced to quantum mechanics (perhaps not for the first time), you should know what the new James Bond movie title means. “Quantum of solace” A quantum being a quantized (discrete) amount of something. So we have a quantum of solace which would be the smallest amount of solace (consolation) acceptable. A bit of trivia for you.

From the Atomic to the Macroscopic

Picture 18 mL of water. Ok, if your like most people this will be nearly impossible since we tend to relate volume completely different in our everyday lives here in the United States. What I mean to say then is picture 0.6 fl. oz. of water. So in this small sample how many molecules of H₂O do we have? How do we know? . . . Fine then! What can we related to 0.6 fl. oz.? A can of soda is about 12 fl. oz. Still not picturing it? Ah, the need for the metric system as a U.S. standard . . . I digress. I have been withholding it from you. What is something we can picture 18mL of water in? Think of it this way, most finger nail polish bottles are ~0.6 fl. oz. So let’s start this all over again.

Picture a fingernail polish bottle full of water. How can we ever determine the amount of molecules contained therein? Consider a pencil lead. How many atoms are in the visible tip? These seem to be very fundamental questions. Indeed, I assume you know how to find the answer. Maybe, knew how to find the answer (perhaps this would be a better phrasing). Never fear, I will refresh you.  But first, why do we even care? Consider impressing your date tonight with this wealth of information. Maybe impressing your family and friends to show your intellectual superiority would be in order? Is that not enough to get you motivated? Umm, what if you wanted an easy way to compare different substances? This concept would be your fascination in such a case. Consider a chemical reaction in which you are relating differing amounts of substances. You couldn’t ever work with the reaction in a coherent manner without this information. Now are we ready to examine this a little closer? Perhaps not, but we will look at it anyway . . . I assume that if you are even reading this blog you must be slightly interesting in scientific things. Let’s get back to the concept at hand.

How many atoms in that visible pencil lead? How many molecules in that bottle of water? Place your guesses now . . .

It became necessary to develop a way to examine this concept in a standardized manner. Johann Josef Loschmidt developed a method in ca. 1865.[i] His method was used to estimate the number of atoms in a given volume of gas. This laid the ground work for additional experiments to be conducted. The fact remains that atoms are so tiny that the number needed to make a visible sample is enormous. In fact, even the smallest speck of dust visible to the naked eye contains at least 10^17 atoms.[ii] Our brains cannot really make sense of this gargantuan number with proper comparison (that will come later). Because such a large number of atoms are contained in such a small sample, a unit like grams is all too cumbersome. This is the primary reason that a standard was reached. The main unit we use is call an atomic mass unit (amu). An amu is defined as exactly 1/12 the mass of an atom of Carbon-12 and is equal to 1.660539×10^-24 grams. This definition allows us to say:

Mass of one Carbon-12 atom = 12 amu (exactly).

Now you may be starting to see that we can relate a particular number of amu to pencil lead. Since a pencil lead is generally graphite (an allotrope of carbon, aka graphite=carbon). With the given relationship between grams and amu we can find out how many atoms we have by just weighing a sample (and using our trusty periodic table, what you don’t have a periodic table hanging right next to your desk?!?).  So, the periodic table says that Carbon has an atomic mass of 12.01 amu (if your worried about why this number is not exactly 12 feel free to leave a comment). Let’s just weigh the pencil lead tip. We find an analytical balance and we find the lead to weigh 15 mg (that is 1.5 x 10^- 2 g). How many atoms is that?

1.5×10^-2g (1amu/1.660539×10^-24g)(1 C atom/12.01 amu) = 7.5 x 10^20 C atoms

Now we know how many atoms in that pencil tip! It is easy to get an idea about the huge number of atoms in a sample of something. Chemists use the unit known as the mole (mol). One mole of any element is the amount whose mass in grams is equal to its atomic mass. Whenever you have the same number of moles of different elements, you also have the same number of atoms. Now unimaginable ways of relating sample together emerge.

How many atoms in a mole? Experiments have shown that one mole of any element contains 6.022141×10^23 atoms. Just like 1 dozen oranges is 12 oranges, a mole of oranges would be 6.022141×10^23 oranges. How about that water? If we have 18 mL of water or 0.6 fl. oz. if you prefer, we have 18 g of water (using the density).

18g(18g/mol)=1mol of water = 6.022141×10^23 molecules

That simple fingernail polish bottle of water contains that many molecules of water. How can I imagine a number that big? If we were to spread 6.02×10^23 marbles over the entire surface of the Earth we would produce a layer about 3mi thick. If Avogadro’s number of pennies were placed side by side in a straight line, they would encircle the Earth 300 Trillion (3×10^14) times.[iii] Consider this comparison[iv]:

20,000 is the average college debt in the US
300,000,000 is the population of the US
6,000,000,000 is the population of the entire earth
14,960,000,000,000 centimers from Earth to the sun
142,006,167,000,000,000 is the supposed age of the earth in seconds
1,260,000,000,000,000,000,000 is the liters of water on the earth
602,200,000,000,000,000,000,000 Avogadro’s number

That is the number of molecules of water in just 18mL. The best guess of the number of grains of sand on the Earth is somewhere between 10^20 to 10^24. You may never look at a pencil lead or a glass of water the same again.

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[i] Loschmidt, J. (1865), “Zur Grösse der Luftmoleküle”, Sitzungsberichte der kaiserlichen Akademie der Wissenschaften Wien 52 (2): 395–413

[ii] McMurray, John E.; Fay, Robert C. (2010), “General Chemistry Atoms First”, Upper Saddle River, NJ: Pearson Prentice Hall, p. 47

[iii] Brown, Theodore; LeMay, H.; Bursten, Bruce (2006), “Chemistry the Central Science”, Upper Saddle River, NJ: Pearson Prentice Hall, p. 91

[iv] McMurray Ibid, p. 49

Fundamental Principles of Science

When thinking of science the first scientific principle that comes to many peoples minds are the laws of thermodynamics. Just as thoughts on the existence of God are foundational to theology (see Principia Theologica), the foundational thoughts of science generally conjure up vivid imagery of scientific laws and white lab coats.

Why is it that this is the place that the public comes to first when entertaining thoughts of science? Perhaps they feel a sense of definitiveness when discussing scientific matters. Maybe science provides a place where the average person can turn to derive truth about the world around us.  Somehow, if a single piece of evidence that was scientifically found (that is found via the laws of scientific inquiry) is cited, the discussion has a new sense of authority derived from Ph.D.s in a lab somewhere. Apparently, the common man has the idea that those in the process of unaided (unaided in the sense that this is the forefront and the material cannot be gained through the written word) discovery in the lab are able to achieve such a high standard of pure fact as to not be questioned. Even when questioned these scientists work on projects that are so immense and complex that if this is what they say, who are we to really cast doubt on their conclusion. These people have spent their life (or at least a large part) studying the very principle on which the conclusion is based and must be closer to correct than my thought process can attain. This is an easy way for the average person to categorize many aspects of science and scholarship in general. Whether this is a warranted approach to take, I leave to you to question.

How does this relate to thermodynamics? Thermodynamics is often a set of scientific ideas that is brought to light when science is discussed by the common man to provide a sense of factual authority. The foundational character of these laws is obvious and indeed they have been well tested and proven correct, to the point of being granted the status of a scientific law, the laws of thermodynamics. So next time you are tempted to invoke the laws of thermodynamics to support your argument as to why the wooly mammoths of the northern hemisphere preferred to roam gregariously or your thoughts about how the future of sub-oceanic dwellings will rely on renewable resources you will have an idea as to if the laws of thermodynamics truly apply to your discussion.

What does the word thermodynamics even mean? Well you might not be acquainted with a formal definition of thermodynamics but it comes down to the mechanical action produced by heat. How does this have such wide applicability? The various fundamental laws of thermodynamics are so foundational to many scientific principles you will see how it is possible for it to creep up in discussions. While there are certainly limits on what classical thermodynamics can explain it provides a robust description of the properties of matter in bulk such as pressure, temperature, volume, electromotive force, magnetic susceptibility, heat capacity etc.[1] Most are familiar with the three laws of thermodynamics and are happy to just negate the fourth. Ok there really isn’t a fourth law just what is known as the zeroth law which makes four laws of thermodynamics.

Thermodynamics was essentially created by Carnot (you may be familiar with the Carnot engine). He laid out his formulations in his paper entitled “Reflections on the Motive Power of Heat.”[2] Much has been put forth in the field of thermodynamics since that time. Anyone who has studied statistical mechanics can tell you that there are many facets to this subject and it can be quite difficult.

The four laws of thermodynamics are seemingly broad and general. Maybe they are even self-evident. Here is what we have:

0^th law of thermodynamics: If A is in thermal equilibrium with B, and B is in thermal equilibrium with C, then A is in thermal equilibrium with C.[3] Stated another way that is if A=B and B=C then A=C. In mathematics, this is known as the property of transitivity.[4] You are probably wondering why anyone would start the numbering at zero. Well, you’re not alone. It seems that this law was more elementary than the other established laws of thermodynamics at the time and so the proper place for it was not the fourth law but the zeroth law. What does it mean to be in thermal equilibrium? It is a situation in which two objects would not exchange energy by heat or electromagnetic radiation if they were placed in thermal contact. As you may have guessed, temperature is the property that determines whether an object is in thermal equilibrium with other objects.

1^st law of thermodynamics: The internal energy of an isolated system is constant. This is also seen in the formula U=q+w. where U is the change in internal energy, w is the work done on a system, and q is the energy transferred as heat to a system.[5] Here we could get into many discussions about definitions, which I am going to skip (feel free to comment if you would like further explanation).

2^nd law of thermodynamics: The entropy of an isolated system increases in the course of a spontaneous change. That is the change in the total entropy (S) is > 0.[6] This is the law most often cited. You will hear it invoked in everything from evolution to ice crystals. It was explained to me during the course of my education that the entropy of a system relates to the amount of information that is available to the system. You can think of this in terms of possible states that something could inhabit. If there are a large number of possible states then the entropy is large. Undoubtedly, this does injustice to some aspect or area to which the second law is applied. It provides a convenient construct with which to work.

3^rd law of thermodynamics: The absolute value of the entropy of a pure solid or a pure liquid approaches zero at 0K. One could state this mathematically as lim(S)=0 at T goes to 0. This is the Planck formulation of the third law.[7] The third law is usually overlooked in most texts. By far, attention is placed on the first and second laws.

Other concepts have been put forth as candidates as fourth, fifth, and sixth laws but they remain unaccepted in general and are more controversial. I am not aware of any scientist who questions the 0 through 3^rd laws.

We have covered the laws of thermodynamics and hopefully broadened our understanding of these principles of science. Next time you are tempted to utilize a scientific law to support your argument please try to use it in context and please be prepared to demonstrate how, from the very law itself, you can rightly use it.

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[1] Klotz, Irving M.; Chemical Thermodynamics; W. A. Benjamin, Inc.: New York, NY, 1964

[2] Carnot, S.; Reflexions sur la puissance motrice du feu, Bachelier, Paris, 1824.

[3] Serway, Raymond A.; Jewett, John W. Jr.; Physics for Scientists and Engineers; 6^th ed. Thomson Brooks/Cole: Belmont, CA. 2004 p. 582

[4] Papantonopoulou, Aigli; Algebra pure and applied; Prentice Hall: Upper Saddle River, NJ. 2002 p. 8

[5] Atkins, Peter; de Paula, Julio; Physical Chemistry; 7^th ed. W. H. Freeman and Co.: New York, NY. 2002 p. 35

[6] Ibid, p. 92

[7] Planck, M. Thermodynamik, 3^rd ed., Veit & Co., Leipzig, 1911 p. 279

A Scientific Investigation into Religious Experience

Over on Principia Theologica (PT) we have begun a series of posts on the existence of God. While PT is primarily concerned with the principles of theology here on Principia Scientia we are interested in the scientific aspects of the existence of God. We will attempt therefore to expound this subject from the point of science.

Where do you begin to approach this subject using a scientific point of view? How do we scientifically measure God, or His existence? Well, this is a really strange question. This question really falls out of the range and scope of most typical scientific approaches. Indeed (depending on your definition of science) this question cannot be addressed head on via science. What means do we have to understand the existence of God or even religious experience through the scientific method? One cannot utilize the scientific method to investigate non-scientific hypotheses. This is, essentially, where we have to leave the issue (some dramatic conclusion huh?).

Recently I came across an article[i] in the March 24, 2009 issue of Proceedings of the National Academy of Sciences. This article details information about the cognitive and neural foundations of religious belief. It shows which parts of the brain are active when engaged in religious thought. Does this mean that religious experience is all in your head?!? In some sense it is and in another sense it is not. Keep the fact that this is a scientific investigation in the forefront of your mind. This means we are using the scientific method to investigate the neural and cognitive effects that religious experience has on our brain. It is just that and nothing more or less. So, any ideas about using this information to support a hair brained theory about how religion and God only exists in our brains as some sort of figment of imagination and not in reality are utterly unsupported here. What is seen here relates particular questions about religious experience to brain activity. For instance, when a statement about the anger of God was related to subjects the regions of the brain that detect emotion through facial expression and language were active. Statements concerning the love of God stimulated the areas connected with positive emotions and suppression of sadness. There is the power of positive thinking for you! When additional statements relating to doctrine were conveyed to subjects the regions of their brain that help to decode abstractness and metaphor were activated. Experience with God seems to trigger the area involved with memories and imagery of self in action. Figure 3 (given below) from the paper¹ shows the regions of the brain activated during this part of the experiment. Notice the level of cognitive engagement when pondering religious postulates. While this study focused only on Western Christian beliefs it is thought that different religious beliefs will have a similar neural response.

Some people regard those with religious beliefs as narrow minded. I submit to you that the opposite is true; nearly the entire brain is active and participates in our religious involvement (see the images below for a visual aid). Pascal Boyer stated in an essay included in the journal Nature, “that all it takes to imagine supernatural agents are normal human minds processing information in the most natural way.”[ii] He goes on to say later in the article, “Some form of religious thinking seems to be the path of least resistance for our cognitive systems. By contrast, disbelief is generally the result of deliberate, effortful work against our natural cognitive dispositions – hardly the easiest ideology to propagate.”² He is implying that we need to squelch our brain function (or deny it) in order to actively be “religiously uninvolved”. That gives new meaning to being an atheist. On the other hand, those who are religious should have renewed vigor in pursuing the development of their mind and seeking truth. If you have some spare time and access to PNAS you should take the time to read this article, its not quite brain surgery or rocket science for that matter.

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[i] Kapogiannis, D.; Barbey, A.; Su, M.; Zamboni, G.; Krueger, F.; Grafman, J., “Cognitive and neural foundations of religious belief”, Proc. Natl. Acad. Sci. USA, 106 (2009) 4876-4881. See http://www.pnas.org/content/106/12/4876.abstract

[ii] Boyer, P., “Religion: Bound to believe?”, Nature 455 (2008) 1038-1039