A Lesson on Entropy
Pretending that I have not prepared for the lecture, I would take my cellphone and dial an emergency number. It is my direct line to Saint Peter. So the lecture begins....
"Hello, is this Saint Peter?", I ask. "This is Angel de Dios, no, I am not one of your guys, I am just a teacher here at Georgetown." With the recent conclave, I had to explain to Saint Peter that I was not about to ask him for the secret behind black versus white smoke. I knew he was not about to share that secret with me. I was calling to seek help for an introductory lecture on entropy. I was hoping that I would get a chance to talk with the late Professor Ludwig Boltzmann.
Saint Peter says, "Yes, Boltzmann is up here." He then offers Boltzmann's direct phone number in case we get disconnected, and I grab a piece of chalk to write the number:
Hallo. Guten Morgen. Ist da Professor Boltzmann? Hello. Good morning. Is this ProfessorBoltzmann?
Wunderbar. Hier ist Angel De Dios. Ich lehre gerade einen Kurs in Allgemeiner Chemie an der Georgetown Universitaet. Great. This is Angel de Dios. I am currently teaching a course in General Chemistry at Georgetown University.
Ich rufe Sie, Professor Boltzmann, an, weil ich Schwierigkeiten habe, eine Vorlesung ueber Entropie zu halten. Koennen Sie mir dabei helfen? Das waere sehr nett. The reason why I am calling, Professor Boltzmann is that I am having difficulty in beginning a lecture on entropy. I was wondering if you would be able to help. Thank you, in advance, Professor Boltzmann.
Meine Studenten sind dieses Jahr sehr aufmerksam und fleissig. Ja, Professor Boltzmann, sie haben hart gearbeitet und wir haben Kapitel wie Kinetik, Thermodynamik, Quntentheorie und Saeure-Base Chemie behandelt. My students this semester have been very diligent. Yes, Professor Boltzmann, they have been studying hard as we have covered topics in kinetics, thermochemistry, quantum mechanics, and acid-base chemistry.
Professor Boltzmann, ich hatte gehofft, dass Sie uns Entropie als eines der wichtigsten
chemischen Themen naeherbringen koennten. Professor Boltzmann, I was hoping that you
would be able to guide us as entropy is one of the most important concepts in chemistry.
Meine Studenten sind sehr empfaenglich, insbesondere wenn ich sie ins Konzept mit
einbinden kann. Ich bin sicher, Professor Boltzmann, dass Sie der beste Ansprechpartner dafuer sind. My students are very perceptive especially if I am able to engage them in theconcept. I am sure that you would be the best person to help, Prof. Boltzmann.
Oh, ich verstehe. Sie sind schon ueber meinen Kurs informiert. Ich hatte nicht gewusst, dass Petrus schon FIOS hat. Das ist ja toll. Dadurch waren Sie also in der Lage, meine Webseite zu finden und das Kapitel ueber Entropie zu korrigieren. Herzlichen Dank fuer ihre Hilfe, Professor Boltzmann. Oh, I see. You are aware of my course. I did not know that St. Peter up there subscribes to FIOS. That is really good. So you were able to hack into my website and change my lecture notes on entropy. So they should be ready now. Thank you very much, Prof. Boltzmann.
Ich werde nun zu der Webseite gehen und meine Vorlesung beginnen. Vielen DankAnd the lecture is as follows:
nochmal, Professor Boltzmann, Für konnten sie nicht Sie lieben. Aber noch, war Ihre
Liebe zutreffend. Und als keine Hoffnung nach innen gelassen wurde. Auf dem starry,
starry Nacht. Sie dauerten Ihr Leben, wie Geliebte häufig. Aber ich könnte Ihnen,,
Professor Boltzmann: Diese Welt wurde nie bedeutet Für ein so schön wie Sie.
I will then go to the website and start my lecture. Thank you again, Professor Boltzmann, For
they could not love you, but still your love was true, and when there was no hope left inside
on that starry starry night, you took life as lovers often do, but I could have told you, Professor
Boltzmann, this world was never meant for one as beautiful as you.
Molecules are Stupid
Molecules tend to follow simple rules of probability. To begin our discussion of entropy, let us go through a brief description of probability. Here is the first example: (I have two boxes and two balls)
Here are the possible scenarios: "Let me count the ways."
Case 1: Both balls are in Box A
Case 2: Both balls are in Box B
Case 3: There is one ball in each box
Thus, we have three cases:
Case number of ways 1. Both balls in box A 1 2. Both balls in box B 1 3. Each box has a ball 2
With three balls and two boxes, we have the following:
Case number of ways All three balls in Box A 1 Two balls in A, one in B 3 Two balls in B, one in A 3 All three balls in Box B 1
With four balls and two boxes, we have the following:
Case number of ways All four balls in Box A 1 3 balls in A, 1 ball in B 4 2 balls in A, 2 balls in B 6 1 ball in A, 3 balls in B 4 All four balls in Box B 1
With twenty balls and two boxes, we have the following:
Balls in Box A Balls in Box B number of ways 20 0 1 19 1 20 18 2 190 17 3 1140 16 4 4845 15 5 15504 14 6 38760 13 7 77520 12 8 125970 11 9 167960 10 10 184756
With 68 balls, the number of ways in which each box has 34 balls is 2.8 x 1019. Compare this huge number with the one and only way of having 68 balls in Box A. What happens with a mole (6.02 x 1023) of balls? Clearly, if it is only a matter of probability, there is no chance for having all 6.02 x 1023 balls in one box.
By the way, there is an equation that allows us to calculate the number of ways for any arrangement of any number of balls in any number of boxes:
Molecules as Balls
Here, P is the number of ways for N balls in x boxes, in which the occupancy is: nA is the number of balls in box A, nB is the number of balls in box B.
Energy Levels as Boxes
Suppose we have 7 atoms (or molecules) and we introduce quantum mechanics into the picture, that is, the energies of these atoms are quantized. For simplicity, we will assume that the allowed energy levels are 0, e, 2e, 3e, 4e, 5e, 6e, 7e.... We can think of these energy levels as boxes. The total energy of the system is, however, restricted. It should always equal 7e. Thus,
n0 + n1 + n2+ n3 + n4 + n5 + n6 + n7 = 7The first equation takes care of the number of atoms (or molecules) while the second equation comes from the total energy requirement. Here then are the possible arrangements:
en1 + 2en2 + 3en3 + 4en4 + 5en5 + 6en6 + 7en7 = 7e
Case n0 n1 n2 n3 n4 n5 n6 n7 Number of microstates
1 6 1 7 2 5 1 1 42 3 5 1 1 42 4 5 1 1 42 5 4 2 1 105 6 4 1 1 1 210 7 4 1 2 105 8 4 2 1 105 9 3 3 1 140 10 3 2 1 1 420 11 3 1 3 140 12 2 4 1 105 13 2 3 2 210 14 1 5 1 42 15 7 1Boltzmann suggested that if one can observe such an assembly over a long period of time, each microstate will occur with equal probability and one will find that the number of occurrences for any particular set of distribution is proportional to the number of microstates that correspond to that set. For example, in the above system, there are a total of 1716 ways or microstates. The probability of case 1 occurring, for example, is 7 in 1716. The probability of case 2 happening is 42 in 1716. The probability that case 10 is the case is 420 in 1716 (the highest of all).
So, entropy is introduced. Case 10 in the above table is the most probable distribution. I try to avoid the word "disorder" in describing entropy. There is nothing "disordered" in the way molecules distribute energy among themselves. The box with the lowest energy will always be occupied. But there are no gaps, no molecule is allowed to monopolize and take all of the energy....
Of course, I wonder if students do appreciate entropy after this lecture or do the students simply remember Avogadro's number....