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Lecture: Randomness

The secret is avoid the “r” word.

When I taught equilibrium I never brought up anything about randomness or the role it played in the process of determining K values. I derived the mass action expression from the rate equations as follows.

Rate(forward) = Kf[A]a[B]b     Rate(reverse) = Kr[C]c[D]d

At equilibrium Rate(forward) =  Rate(reverse) Therefore:

Kf[A]a[B]= Kr[C]c[D]d   and Kf/Kr =  [C]c[D]d/ [A]a[B]b

You don’t find this very often in text books and for good reason. In fact some students will point out that something is wrong with all this when they see quite large K values for equilibriums where the forward reaction in endothermic. Based on energy alone this does not make much sense and there is also the problem in the fact that some reactions seem to go to completion. If the Kinetic energy distribution curve is in fact asymptotic there should always be at least some molecules with enough energy to make the reverse reaction occur. There has to be something else that is pushing these equilibriums away from what would make sense based on energy considerations alone. If this all comes out as you go along then you are in great shape and the need to introduce the drive towards maximum randomness as a new idea in the world is at hand.

A model of the interplay of Maximum Randomness and Minimum Energy

When my oldest daughter (Shelly) was in elementary school there was much fun to be had with her about the world. Shelly had a creative mind and was curious about most everything, but also a little stubborn. She came home from school one day talking about how many sides different geometric figures had. Squares have four side, triangles have three and if you have five sides it’s called a pentagon, etc. I asked her how many sides a circle had. She thought this over for a moment and said with certainty that circles don’t have any sides. I took a piece of paper and made it into a big square (four sides dad) and I cut each corner off and asked her how many sides do we have now. She counted eight and again I cut off each corner and gave it back to her. She started counting on her way to sixteen then stopped as she started to see the outcome of this little adventure. She gave me an angry look, crumpled up the paper and walked away muttering “dad, circles don’t have sides”.

Randomness and it’s power in our lives is much harder to grasp than how many sides a circles might have or not have. In talking to Shelly (still in elementary school) about this phenomenon I proposed that she put one hundred pennies in an old cigar box all heads up and tape it shut. Randomness will not like this order and work to move to a more disordered state. After setting it up Shelly wanted to look in just a few minutes later, but I suggested more time would be needed for randomness to work it’s will on the pennies. For the next few days she would ask “can we open it now” and I put her off saying more time was needed. Eventually she forgot about the cigar box on a shelf in her room. When I thought of it and she was not around I would go in and give the box a good shake or two (temperature you know) and over time randomness did in fact have it’s way with the pennies. Several weeks later when Shelly thought about the box again we opened it and found 40+ tails.  Shelly was truly taken aback and the drive of randomness became real for her. I hope I talked to her about my shaking the box, but as I write this I can’t remember if I did or not. This is all well and chemistry students can think of many situations where both the drive towards minimum energy and maximum randomness work together to produce what they see. In fact Shelly’s room during her high school years was a near perfect example of the natural state of minimum energy and maximum randomness. Every thing on the floor and no order what so ever. She and I had many not so good talks about the conditions in her room, but I finely gave in with the compromise that the door to her room would always be closed. Even our own mortality can be discussed with the thought we are not at all natural when viewed in the cold hard light of achieving minimum energy and maximum randomness. Dust to dust and high school students are a bit uncomfortable when they begin to see that the battle of our bodies working against these two natural drives show signs of loosing by our mid to late twenties.

This to is all well and good, but what we are looking for is a case where the drives work against each other for control of some system. I was a junior at St. Cloud State in 1963 when I became enamored with a girl I would see on campus every now and then. I wanted to meet her and started plotting how an introduction could happen. None of my friends knew her (or thought she was so hot) and even though I worked in the women’s dinning hall (she never came there) no opportunity arose to make her acquaintance. Jon told me his normal mode was to somehow learn the name, find out the home town, get a map, and memorize all the towns and highways around. Then just walk up and say “aren’t you from so and so on highway such and such north of this and that”. When you owned up to your game (sooner rather than later) it could be done with a compliment and most often be forgiven. I’m getting no where and then by chance I’m sitting in the Press Bar on West St. Germain St. (St. Cloud Minnesota) and in she walks with two girl friends. I don’t know how it will work out, but I just walk over and ask if they would help me out with a little experiment. I quickly add that it’s a game of chance and if I should lose I would buy the round of drinks they had just ordered. You will see that it would be very very hard for you to loose in that it’s like a coin toss except you win heads or tails. After some laughter they agree and I produce a paper match book and take out a match. If you look carefully at a paper match they have a tan side and a side with some blue so it can be flipped gray or blue like a coin. Now, if the blue side is up after the toss I will buy all three of your drinks and if the grey side shows I will buy just one drink for the girl that has stolen my heart. A paper match can, unlike a coin, stand on it’s edge. It took some care, but I stood the match on edge to demonstrate the possibility. If that should happen I would have my drink with you and we would get to know each other a little. Share home towns, majors, year in school, likes, dislikes etc. I know I’m budging in on your party, but what are the chances of the match landing and standing on edge anyway? All I needed to do now was to bend the match and toss it on the table. One way to think about the bent match is that unlike before where energy didn’t care which side came up now energy is in the game.

In my first years of teaching I realized that this could be a good demonstration (model) for the interplay of maximum randomness and minimum energy working to control a system. This would be a situation that is not too complicated and that students can see. One hundred enlarged paper match sticks (one side marked and one side plane) when thrown in the air an allowed to flutter down should give an even split of plane side up versus colored side up. This is true only because the potential energy of either landing orientation is the same. For the results shown we got 47 and 53 which is convincing enough for our needs in this model. However, if we bend our enlarged paper match sticks in half we generate five possible landing orientations.  There is one “teepee”, two “leg up” orientations as well as two “on side” positions. Randomness would call for a result of 20 in each orientation while minimum energy would prefer all to be down on the side with the center of mass only 1/8 inch off the floor. Teepee has the most potential energy and the two leg ups would fall in between with the center of mass approximately 1/4 of the way up the leg or 0.375 inches off the floor. All one has to do is stand in the center of class and toss the one hundred bent bits of paper and as they flutter down the battle between minimum energy and maximum randomness settle on a compromise result. Neither gets it’s way, but both have influence on the models result. The drive towards minimum energy pushes the system towards the “on the side” orientation yet maximum randomness prevents energies total control and even gets six percent to land teepee style. Now, wouldn’t it be great if we could get the floor to vibrate with enough energy to produce all three orientations. We would then get to watch a physical dynamic equilibrium in action! Alas, I could never figure out how to get the floor to thusly vibrate. Just to keep the idea of an equilibrium constant in the thought process we could calculate one if we designate some orientations as reactants and others as products. We can make it an endothermic or exothermic reaction and the K value would be calculated from numbers rather than concentration. I thought I might throw the bent bits of paper at the floor to demonstrate the result at a higher temperature. We should shift the system towards the higher energy orientations of teepee and leg up. Again, alas the result actually shifted towards the “on side” orientation due to low energy bounces that upsets the leg up and teepee positions moving some of them to “on side”. I guess I have to let go of the rope right about here!

After this randomness vs minimum energy lesson some students brought in a game called “Pass the Pig” where two pigs (small plastic critters) are use instead of dice to make points and win the game. The pigs have several landing positions with great names like side out, joweler, standing, etc. The inventors of the game did pretty well in setting the points earned by each landing orientation where those with high potential energy getting more points and throwing a pair of “side outs” cost you your turn. The students appreciated the fact that they now had insight into the scoring vs the drives towards minimum energy and maximum randomness. In this game you are always hoping randomness dominates when it’s your turn. Just a note, the young women of my dreams and I dated only once! No chemistry there!!