Yet they have to buy a text book which costs about US$175 (In Canada). In our two classes there are 1100 students. Multiply by the average of 5 courses students usually take and you can see that books are a major expense. And this is just one class. Do the math, and students shell out a lot.

Then there are the tons of trees that are sacrificed. Our text for freshman chemistry weighs 3 kg (6.6 lbs)! Money-conscious students usually sell their texts and the second hand market blossoms. After about three years, the book is down to $50 or so and the book companies start to lose their market. If it were a regular book, then when a print edition runs out more are printed if needed, but not in the text book market. Although new editions appear with claims of new improvements and corrections, the dominant reason for a new edition is to get the market back. If the book was selling like hot cakes, then new editions are rarely contemplated.

You cannot copy protect hard copy.

The text that the team teaching our course chose is General Chemistry by Petrucci et al and is in it 10th edition! The only changes I noted from the 9th were the addition of a few more problems.  The general rule to justify a new edition is that it should have about 20% new content. This is rarely the case.

Companies desperate to hold onto their market share, inundate teachers with the plethora of “resources”, such as a web site (usually not good and not used much by students anyway); other multimedia; solution manuals; and study guides. These are considered a necessary burden by text book companies, but they jack up the price.

It should not be this way and today, with computer technology, the heavy environmentally unfriendly hard copy text books should be rejected by teachers in favour of ebooks, if available.

The developing world needs these books too, but cannot afford them, so they ignore copyrights and make them out of cheap paper and sell them for a few dollars.

I believe it is long overdue to do away with hard copy text books altogether, along with their high cost, and adopt ebooks.

Think of the advantages: no paper, nothing to ship, can be updated so users always have the latest edition, integrated into the internet, easy to copy protect, and can be sold for a fraction of the price of hard copy. No resale market.

Moreover the usual hard copy text contains way too much material for a two semester course–with about a thousand pages. Ebooks can readily be broken up into modules which are focused on the material needed for a course. As one of the authors of Physical Chemistry by Laidler Meiser and me, we got the copyright from Houghton Mifflin and have made it into a copy protected pdf with built in multimedia popups, and an extensive free solution manual on-line. Although the full ebook is available, there are six modules covering thermo, electrochem, kinetics, quantum, stat. mech., and solids and liquids. The starting price is only $14.99, (a 5 month license) a far cry from $150 plus. A semester course is usually four months, so a 5 month license fits the needs of many students.

Copy protection is essential and we have developed an MCHPDF viewer that takes any number of books in pdf format and secures them.  That is the PDF viewer acts like a library of copy protected ebooks, with license expiry and updates built in. Give us your pdf ebook and we will give you royalties much greater than anything offered by a text book company.

However one of the most exciting aspects of ebooks is to reach the developing world. Many just cannot afford paper. Even if individuals cannot afford to buy their own copy, governments, organisations and universities can instead obtain a fixed number of licenses at a reasonable cost, making the ebooks available to millions of people.

Today it is natural for students to read and study from a computer. When I compare ebooks with hard copy, I can find nothing to favour the latter over the former. What’s your take on this?

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http://www.youtube.com/watch?v=wFe2zu2116I

After rolling 2, 3, 4, 10 and Avogadro’s dice, as seen in the entries below, it becomes clear that the most random states (most number of ways of rolling a number) always dominate while those with fewer arrangements occur less frequently:

1 Entropy: Randomness by rolling two dice

2 Entropy: Randomness by rolling three dice

3 Entropy: Randomness by rolling four dice

4 Entropy: Randomness by rolling ten dice

5 Entropy: Randomness by rolling Avogadro’s dice

In this final entry of randomness and entropy, the concept of an ensemble is discussed.

We are using a die to represent a particle that has six states that come up randomly.  Hence we have treated systems with 2, 3, 4, 10 and Avogadro’s constant (let’s use 10²³) of particles (dice) and have shown that as the number increases, the total number of accessible states, is given by 6ⁿ. Clearly the number of states in Avogadro’s case is  6¹⁰²³ : an enormous number!! If you start to roll this many dice, every roll gives an outcome in exactly the same way as for 2 or 3 dice: just add up all the rolls (it will take a bit of time!! but it is still a specific number).  However when your roll, and if you could add them all up, the answer would be always very close to 3.5×10²³ (see end). This is clearly because the total number of accessible states, W, can be replaced by the total number of random accessible states, W_random that all have outcomes clustered around 3.5×10²³ .

Each time you roll that many dice, it is unlikely they will come up in any arrangement other than the ones that give the value of 3.5×10²³ .

We call the collection of all those arrangements that give the most probable outcome an Ensemble. In French ensemble means  a collection.

The ensemble concept is useful in statistical mechanics.  So now let us think of a gas moving around in a container at 300 K.  We could take repeated snap shots of the gas, and every time we would see the particles frozen in different position.  We could also measure the speed of each particle (in principle).  Each snap shot is like a roll of Avogadro’s dice. Below there are three of the many different arrangements and these will arise because each is consistent with a value of 300 K.

If the were not consistent with a temperature of 300 K such a state would be very improbable and can be neglected.

Consider he following is an ordered state:

This state is possible but it is not consistent with 300 K.  In fact if the 5 balls are distinguishable, then there is only one way to arrange these.  Since Boltzmann’s equation of the entropy is

S=klnW_random

then since in that ordered state W = 1, then the entropy of that state is zero (ln(1)=0) which means perfect order.

In contrast, let us calculate the entropy of rolling 6¹⁰²³ dice (and treat them as particles). In this case the entropy is given by S = kln(6¹⁰²³) = 1.8xkx10^{23 .} Let us use the value of Boltzmann’s constant, k=1.3806503×10^-23 J K^-1 which gives the entropy of S = 2.48 J K^-1 . This is the value of the randomness of the gas.

Suppose we have a system of two bulbs joined together with a closed stop cock. On one side the gas has 10²³ particles, each with 6 states giving 6¹⁰²³ accessible states. Let us suppose that the empty bulb is the same size and is under the same conditions, so when the stop cock is opened, the number of accessible state increases to 6¹⁰²³ x  6¹⁰²³ . For this larger system there are many, many more accessible states and the gas will move into the evacuated bulb and occupy those states randomly. The entropy must increase.  I will not do that calculation, because we need to take into account that the particles are indistinguishable, but it was from these considerations that Boltzmann determined his equation (Probabilities multiply but the entropy adds, so the only function that has this property is the logarithm.)

When Boltzmann was explaining this, Poincaré pointed out that for a finite system (fixed number of dice) it will eventually return to its original state (Poincaré recurrence theorem or ergodic theory).  Apparently to this, Boltzmann replied:

“You should live so long!”

Most probable roll:

• 2 dice:  (12-2)/2+2=7
• 3 dice: (18-3)/2+3=10.5 or 10 and 11 because we deal with integers only
• 10 dice: (60-10)/2+10=35
• 10²³ dice: (6×10²³- 10²³)/2+10²³ =  3.5×10²³

The interactive software used in this video is part of the General Chemistry Tutorial and General Physics Tutorial, from MCH Multimedia. These cover most of the topics found in AP (Advanced Programs in High School), and college level Chemistry and Physics courses.

Related Posts:

• Entropy (Part 5): Randomness by rolling Avogadro’s dice
• Entropy (Part 4): Randomness by rolling ten dice
• Entropy (part 3): Randomness by rolling four dice
• Entropy (Part 1): Randomness by rolling two dice
• Thermodynamics and Physical Chemistry Song by Flanders and Swann

I thought I would expand a bit more on my recent post on stress at exam time. The transition from high school to university is both exciting and challenging.
Likely you got through high school quite easily; memorized a lot; and/or crammed at the last minute, yet still did well.  University is not like that.  You have to work consistently throughout a course, keep up, and not leave your study to the last minute.  You have to prepare for a university course like a runner prepares for a race. Since I teach physical chemistry, I will try to bring together some things from those courses that I hope will help students hone their study habits. You have to teach yourself to learn.

Ask yourself “Why science?”.  Here are some possible reasons:

• Science is rewarding and a career in that area seems fulfilling.
• The course is a pre-requisite for later courses; otherwise you would not take it.
• You are in pre-med because your mother wants you to be a doctor, but you would rather be in arts, etc.

It is your life and your education.  Choose the program you want, not what others tell you to take. Don’t ignore advice but think what you want before you make choices.  You are over 18 and an adult.

When students do not do well on mid-terms, they often think they are not smart enough for university. If you did well in high school and passed the university entrance requirements, you have the brains and are capable.  So relax and trust yourself. All you need to do is know how to keep that material in your head.

Becoming efficient

If you have a system that works for you, don’t change it. On the other hand, maybe some of the suggestions here might help.

Freshman classes are usually large and the exams are often multiple-choice and you might not be used to them.  I will say a few words about multiple choice exams further on.

A “course” is a “path” through the material which your prof. decides is important and attempts to teach you.  Exams come from that material, so listen to him/her. You need to organize that material, see how it fits in the big picture and manage your time properly: that is, you must be efficient.

You do not have to study all the time.  University is a lot more than passing exams and good grades.  You meet new people and develop socially, you join clubs that interest you, and you go out and enjoy yourself. After four years, you have the ability to think critically and the intellectual resources for a good and rewarding career.

Here are some suggestions.

Efficiency.

• Preview the material before going to class.  You likely have the course notes, so quickly look them over.  Previewing puts the principle concepts in your mind, so you can better understand the material of the lecture.  If you do not have much time, read the conclusions first, then glance at paragraphs in the chapters, remembering that the key points are in the first line or two of a paragraph.  Get a bird’s eye view of what will be covered.

• Studies show that those who preview before class study 30% less and score 7% higher!

• Go to class. You cannot teach yourself new and challenging material alone.  Even Einstein had teachers, so attend class.  If you miss a class, the next might be difficult to understand.  A lot is covered in one lecture, and it is easy to fall behind. As examinations draw near, listen to what your professor emphasizes because s/he knows what is on the exam.

• Review the material within 24 to 48 hours of the lecture.  If you do this you will retain the information much better.  Studies show that if you review soon after a lecture, you retain 80% a week later, whereas without review, after a week you only retain 5%.

• Do assignments:  similar questions are often repeated on exams. If you get stuck on a problem, and need to refer to the solution manual, make sure that you can do the question later without the solution manual. Do not do a lot of questions badly, do a few relevant (from those assigned) questions well.

• Read the text book. However do not read the text first when the material is new.  Read it after the lectures and you have done assignments. With that experience, reading the text will pull the material together.

• Midterms:  after getting your results back, understand where you went wrong so as not to repeat the error.

• Do old exams if available

If you keep up with your courses, then studying for finals is much easier because all you need to do is review and revise material you have already learned.

Dividing up your time.  

Know the material the Prof. emphasizes.

Allot time to courses.  Some take up more time, but do not neglect the easier courses.

When studying, a lot of time is spent trying to get the material into your head, but not a lot of time is devoted to getting it out.  Exams ask you to get it out.

First, do not be a passive learner.  Children watching Sesame Street learn passively and this is not enough at the university level. Writing out notes and ideas is dynamic and helps to consolidate them. Articulating the material out loud helps a lot, so discuss.  Ask your prof or teaching assistant questions, they really are happy to answer. Many students come to me and say. “I know this is a stupid question but…”  Students do not ask stupid questions.  If you do not understand something, make sure you get it because it is likely that gem of knowledge will be on the exam.

Discuss ideas with your fellow students. Working in a group is an example of peer learning. It is effective because you talk and discuss the more difficult concepts without the intimidation of your prof. You can give your views and hear the ideas of others.

At exam time, reduce your knowledge into one or two pages.  Each sentence corresponds to a summary of a topic which in your mind, those you have decided are good candidates for questions.

Look after yourself

Studies show that 1 hour of study when you are fresh in the morning is equal to 1.5 hours of study in the evening.  You should do easy things at night.  Memorization is best done before going to sleep.  Do the challenging things in the day time.

Each of us is different, but every 50 minutes or so take a break.  Concentration span can also be extended by changing topics. Ensure you have privacy when you need it. Turn off your cell phone, close the door and leave FaceBook alone while studying.

Again, studying is like preparing for a race.  You do not only run.  You look after your body.  Get enough sleep.  You can:

• Exercise—and you study better.
• Eat right, but before studying or exams do not eat heavy meals.
• Before an exam do not pull all-nighters.
• Before an exam, review what you have learned.
• Before an exam, set your alarm clock, so do not sleep through it.

Some tips on exams.

• Exams require you to think clearly, so you must know the material and be in good shape.
• Don’t arrive late, as you will get nervous.
• Don’t arrive too early as you will talk to friends who will say “have you studied this?” and that could unnerve you if you haven’t.
• Settle down, relax and trust yourself. You have prepared, and your prof really wants you to do well.  There will be no deliberate “tricky questions”.

Multiple choice exams

A few pointers:

• Read the directions carefully.
• Read the question carefully, and if you do not see through it after a few moments thought, go on and find questions you can do.  Then come back to those others. Doing other questions often give hints for the ones you found difficult at first.
• Do not waste time on those you cannot do right away.
• Approach a question logically.
• After completing the exam go over the questions again and check.  However it is frustrating when a correct answer is changed to a wrong one.  Before changing an answer, figure out what was wrong in the first choice. If you cannot find the error, check over carefully your new answer.  One of them must be wrong, find out which.
• Look at the possible answers that you can choose from and eliminate the ones that are obviously incorrect.

How some profs set exams:

• Not everything can be tested, but the key points a prof stresses are good indications of exam questions.
• Each question tests a particular concept. Make your list and study questions from those your prof assigned.
• The objective is not to stump you, but to test the material so the better you know it, the better your grade.
• There are often a few conceptual questions which require at least two logical steps. These separate those that memorize from those who understand.
• One or two more challenging questions give the better students a chance to prove themselves.
• When the choices are numerical, the wrong answers are obtained by the prof making common errors (units:  maybe the question uses grams but the formula needs kilograms).
• Use of, “none of the above”, usually means the prof cannot think of a 5^th choice!  So you need a good reason to choose “none of the above”. Remember, however, it is often a valid answer.

Good luck.

Related Posts:

• Is Learning Chemistry difficult? Are you stressed about it?
• Entropy (Part 6): Randomness and ensembles
• Entropy (Part 5): Randomness by rolling Avogadro’s dice
• Entropy (Part 4): Randomness by rolling ten dice
• “We’ve lost our Nobel Prize winner!!”

http://www.youtube.com/watch?v=fzcSX0yYr-Y

After rolling 2, 3, 4 and 10 dice, as seen in the entries below, it becomes clear that the most random state (most number of ways of rolling a number) always dominates while those with fewer arrangements occur less frequently:

Entropy (Part 1): Randomness by rolling two dice

Entropy (Part 2): Randomness by rolling three dice

Entropy (Part 3): Randomness by rolling four dice

Entropy (Part 4): Randomness by rolling ten dice

What about Avogadro’s constant say 10²³ dice? The trend is now clear.  There as so many random states that they completely swamp all others. We find a single spike:

In other words, with this many dice, you can roll them as much as you want, and the chance that there is an outcome other than the one that corresponds to the position of the spike is so unlikely you can safely ignore them. Just for ease of writing, consider a ten sided dice. Then for 10²³ such 10 sided dice the chance they all come up 1 is, 10^{-1023 . This number is so tiny that it is all but zero. We can ignore these states.}

The blow-up of the base of the figure shows that a few non-random states are possible and these are fluctuations.

The jump to entropy being a measure of the randomness is easy and the famous expression on Boltzmann’s head stone can be understood, where W is, again, the total number of accessible random states, and k if Boltzmann’s constant. I like to make an analogy between Planck’s constant that determines the smallest quantum of energy and Boltzmann’s constant that determines the smallest change in entropy.

Note that as soon as states couple, add more dice, then the number of random states increases, and the system moves into a new most probable outcome. The same happens with entropy. Open the stop cock and the evacuated bulb is filled as the particles move in to occupy those newly accessible states. W increases enormously and so does the entropy, according the Boltzmann’s equation.

Including non-random states makes no difference, (unless we are at a situation where fluctuations are large, like at phase transitions), but that is not the point here.

Collections of all those random states are called ensembles and they are discussed next.

The interactive software used in this video is part of the General Chemistry Tutorial and General Physics Tutorial, from MCH Multimedia. These cover most of the topics found in AP (Advanced Programs in High School), and college level Chemistry and Physics courses.

Related Posts:

• Entropy (Part 4): Randomness by rolling ten dice
• Entropy (part 3): Randomness by rolling four dice
• Entropy (Part 2): Randomness by rolling three dice
• Entropy (Part 6): Randomness and ensembles
• Entropy (Part 1): Randomness by rolling two dice

You need “stress” in your life.  No stress would mean you would stay in bed all day. Well I’m a chemistry prof and like physical chemistry, and not a psychologist, but over the years you get to know students worries.

There are two types of stress. There is bad stress (“I’m scared”, “I’m dumb” “It is too hard?”) and there is good stress (“Great day, gotta get up!”, “I really want to understand stuff” “I am looking forward to tonight, so got to look good.”)

Use the good stress.

Be motivated. If you hear a pop song and you like it, I bet you know it by heart after hearing it 2 or 3 times. You have to want to learn a subject. So do not let chemistry intimidate you, it is not that hard, but you must be organized.

At the beginning, PAY ATTENTION to the definitions. I shouted, sorry, but they are the key. Then you apply the definitions to problems, which is almost completely what is tested at the beginning.

Chemistry is fun too, but you have to make it fun. It answers lots of questions in daily life and makes you more aware of things from household chemicals, global warming and green house gases, to stuff on labels and side effects from pharma products.

Learn with your friends. Peer learning is great. Also you can talk to your cat and tell it how NaOH and HCl neutralize each other–that is articulation helps retaining material. Do the assigned problems only (not the others) and do them properly. If you have trouble, look at the answer in the solution manual, but then do it again without looking. You can preview the text material first, but please read it again after you have studied the material because then the text makes sense.

Keep up. Do not cram. Imagine you get a 4 month job and you say to your boss, “Hey pay me for the four months, but I will do all the work in the last 2 weeks!!” Keep up, understand the stuff as you go and then review before the exam.

When you get stuck, do not give up. You can bet it is that bit of knowledge that will be on the exam, so work on things you have trouble with, not on the things you already know.

Do not believe your teachers. Question everything they say. Then mull over the ideas. When things do not make sense to you, you have not understood them. All of science makes sense. It has to. It is Nature and Nature is, really, amazing: Nature created you, and you are pretty amazing!!

A student has responsibility to learn, that is your job and you can do it well.

Good luck,

Chemistry fun: I have some posts about chemistry, like why do chickens in the tropics have thin shells, and the thermodynamics of weight loss, and the Kinetics of sobering up.

original image from sun dazed on flickr

Related Posts:

• Exams: Teach yourself to learn.
• Millikan’s Oil Drop Experiment
• Discovery of the electron
• Entropy (Part 6): Randomness and ensembles
• Entropy (Part 5): Randomness by rolling Avogadro’s dice

http://www.youtube.com/watch?v=-99Fi-BKXbE

In order to illustrate the concept of randomness as it pertains to entropy, in a series of entries different numbers of dice have been rolled.

Entropy 1: Randomness by rolling two dice

Entropy 2: Randomness by rolling three dice

Entropy 3: Randomness by rolling four dice

A die with six random states is used to illustrate a particle, so as the number of dice increases, so the number of states increases. For n dice  there are 10ⁿ different ways they can be rolled.   The roll that comes up most frequently is the one that has the most number of arrangements.  As the number of dice increases, that random states becomes more and more likely as seen in the above entries for 2, 3 and 4 dice.  Now we jump to 10.

For 10 dice there are over 60 million arrangements (6¹⁰) and Figure 4 shows the outcomes for 30,000 rolls. This can be compared to Figures 1 to 3.

For ten dice, the chance of a number lower than 20 or greater than 60 is negligible.  The chance of rolling 10 one’s is one over 60 million. The most random states are dominating.  This is only for 10 dice. Next case will be Avogadro’s dice which have 6¹⁰²³ states, which is a lot more than 60 million.

Figure 1

Figure 2

Figure 3

Figure 4

The interactive software used in this video is part of the General Chemistry Tutorial and General Physics Tutorial, from MCH Multimedia. These cover most of the topics found in AP (Advanced Programs in High School), and college level Chemistry and Physics courses.

Related Posts:

• Entropy (Part 5): Randomness by rolling Avogadro’s dice
• Entropy (part 3): Randomness by rolling four dice
• Entropy (Part 2): Randomness by rolling three dice
• Entropy (Part 6): Randomness and ensembles
• Entropy (Part 1): Randomness by rolling two dice

I helped Bob with some of the preparations and organization.  Interestingly in 1993, I organized the 11^th Canadian Symposium at McGill University with again about 200 participants.

But back in 1971, theoretical chemistry was considered to be quantum chemistry with the goal of using computational methods to solve the Schrodinger Equation for chemical bonding.  Although this important part the theoretical chemistry has made tremendous gains today, Bob wanted to bring into the conference more formalism: statistical mechanics, group theory and spectroscopy.  For example Brian Wybourne, then at U of Chaterbury in NZ gave a brilliant talk on group theory. Chuck Curtiss was there.  He was my academic grandfather and he worked in kinetic theory. Russian born Ilya Prigogine, who became a Belgium citizen, was then at the U. of Texas, Austin and he got the Nobel prize in chemistry a few years later, (1977).

At that time Prigogine was well known.  He was a short rotund man with European airs, and he arrived with a flourish at the first coffee break, wearing his jacket draped over his shoulders, shaking hands and holding court. In the following sessions at question period, he frequently rose to give a long discussion of how in his talk, all the questions raised by the present speaker would be resolved.  In fact I was lost in his talk with so many integrals and symbols.

Charles Coulson (Oxford University) gave a brilliant after banquet speech.  I wish I had a copy but the message was not to let computational aspects of Theoretical Chemistry get in the way of conceptualization of science.

Robert S. Muliken

However the person there who did have a Nobel Prize was Robert S. Muliken from the University of Chicago (physics). Muliken played a primary role in the development of Molecular Orbital Theory.  He was 74 then.

At that time, it was customary in a week’s symposium to have an outing. Although it was mid-August, Bob, an avid skier, had planned a trip to Whistler mountain where the participants would take the gondola up to the Round House and spend a couple of hours in the clean mountain air and enjoy the surrounding scene of mountains and glaciers.  However the day before the outing, by chance I called Whistler and found that gondola was closed for maintenance, and at that time it was the only way up.

We scrambled and decided to divert the outing to Hope, B.C. where there is a walking trail of several kilometers, along which the participants could marvel at the huge B.C. fir trees.  So off we went.  Everyone was advised to return to the buses at an appointed time, but when Bob did the head, he discovered Muliken was not present. He came out of the bus and said to me,

“We’ve lost our Nobel Prize winner!!”

A quick search of the area did not turn him up.  Bob went on a jog in the opposite direction around the trial and I went looking around in other places. Bob returned, panting, with no luck and I had not found him either. Maybe he had fallen into a gulley along the way.  What were we to do?  We were getting worried.

Eventually we found him.  He had decided the walk was too much for him on that hot day, and so he found a quiet spot behind the convenience store and had fallen asleep.

Related Posts:

• Entropy (Part 6): Randomness and ensembles
• Exams: Teach yourself to learn.
• Entropy (Part 5): Randomness by rolling Avogadro’s dice
• Is Learning Chemistry difficult? Are you stressed about it?
• Entropy (Part 4): Randomness by rolling ten dice

http://www.youtube.com/watch?v=llS6YcdMcXs

The difficulty students have in understanding that entropy is a measure of randomness can be illustrated by rolling dice. Two and three dice are treated in the first two entries,

Entropy (Part 1): Randomness by rolling two dice

Entropy (Part 2): Randomness by rolling three dice

In this entry four dice are rolled.

The basic idea is that a physical system has many different arrangements (states) of particles which are consistent with some macroscopic quantity, like the temperature. Boltzmann found that out of all possible ways those particles can be arranged, only those that are consistent with the actual temperature need be considered.  The chance of any other arrangements is negligible in comparison.  Rolling dice illustrates this nicely.

One die is considered to be a particle with 6 states, so n dice have have 6ⁿ possible arrangements: thus two dice have 36, three have 216 and four dice have 1,296 arrangements. Four dice have outcomes of 4 to 24 and the most probable roll is the one with the most randomness (different ways of forming the same outcome), just like Boltzmann realized for a physical system.

Figure 1

Figure 2

Figure 3

Even so, rolling four dice does not allow us to ignore the ordered states. Certainly you can roll four dice and the values of 4 and 24 do occur from time to time, but the most probable outcome is 14. Clearly there are many more ways of rolling four dice to give a 14 than any other outcome. The outcome 14 is the most random state for four dice.

For four dice the distribution is even sharper than for three. The plot in Figure 1 is for 10,000 rolls of 4 dice and figures 2 and 3 give the probability distributions for two and three dice respectively. Although this is not quite statistical they are pretty good. Note again the same conclusions but now there are 64=1,296 arrangements and the chance of a 4 or 24 outcome is very small, less than 0.08%. Rolling a 14 uses up most of the arrangements and is the most probable. Compare the chance of rolling a two in Figure 3 and a three in Figure 2. For more dice, these ordered states become less probable relative to the most random state.

In the next entry I jump to 10 dice with over 60 million arrangements. Still 6¹⁰ is a long way from Avogadro’s number of dice with 6¹⁰²³ particles.

None-the-less we can begin to see the most probable outcome corresponds to the most number of arrangements: this is the state with the most randomness.

The interactive software used in this video is part of the General Chemistry Tutorial and General Physics Tutorial, from MCH Multimedia. These cover most of the topics found in AP (Advanced Programs in High School), and college level Chemistry and Physics courses.

Related Posts:

• Entropy (Part 5): Randomness by rolling Avogadro’s dice
• Entropy (Part 4): Randomness by rolling ten dice
• Entropy (Part 6): Randomness and ensembles
• Entropy (Part 2): Randomness by rolling three dice
• Entropy (Part 1): Randomness by rolling two dice

http://www.youtube.com/watch?v=3rYhjZ5Ct3o

In blog entry Entropy (Part 1): Randomness by rolling two dice, it is suggested the difficulty students have in understanding that entropy is a measure of randomness can be approached by rolling dice. In the first entry two dice were rolled but in that case there are only 36 arrangements and 10 outcomes (rolls from 2 to 12). This does not show that the most random state dominates (i.e. the one with most number of arrangements consistent with a roll of 7) . To show that more dice need be rolled.  In this entry three dice are shown to have more randomness in the outcomes (3 to 18).

For three dice we see that the probability of the least ordered states, 3 and 18, have a much lower chance of occurring than the most probable outcomes which are 10 and 11.

For three dice there are 6³=216 possible arrangements, so the chance of a 3 or an 18 outcome is 1/216.  But there are 27 arrangements that give a 10 or 11, with a probability of 27 times greater than rolling a 3 or 18.

Figure 1

Figure 2

Figure 1 shows a plot of the outcomes relative to the the probability.  In a series of 3,000 rolls, the number of times a given outcome occurs is recorded, and this is divided by the total number of rolls.  For enough rolls the distribution becomes Gaussian and each column is the probability of getting an outcome for a given roll.

Figure 2 (click to enlarge) shows the distribution of possible arrangements for the different outcomes. Clearly in comparison to two dice (whence six arrangements gives a 7 and only one gives a 2 or 12), the distribution is more peaked in the center than the distribution of two dice (compare with Figure 3).

The number of accessible states for two dice is 36 and for three it is 216, but this is a long way from rolling Avogadro’s constant of dice. In that case the distribution is so peaked, only the random states need be retained. To move towards that limit, in the next entry four dice are rolled followed by ten.

Figure 3

The interactive software used in this video is part of the General Chemistry Tutorial and General Physics Tutorial, from MCH Multimedia. These cover most of the topics found in AP (Advanced Programs in High School), and college level Chemistry and Physics courses.

Related Posts:

• Entropy (Part 5): Randomness by rolling Avogadro’s dice
• Entropy (Part 4): Randomness by rolling ten dice
• Entropy (part 3): Randomness by rolling four dice
• Entropy (Part 6): Randomness and ensembles
• Entropy (Part 1): Randomness by rolling two dice

Students have trouble with the concept of entropy.  We tell them that entropy is a (quantitative) measure of the randomness of a system but what does that really mean, and how do we explain it clearly?  We can discover entropy as a state function like Carnot did, and then study it by measuring different systems.  Of course we find entropy always increases for spontaneous processes leading to the second law of thermodynamics.

To me, entropy is a substance as tangible as energy.  One can use equilibrium statistical mechanics and either minimizes the energy or maximize the entropy to arrive at the same conclusions. Entropy is the essence of the second law, so it is essential that the concept of randomness be clear.

There are many ways we try to explain the notion of the most random state.  For example two bulbs connected by a stopcock are common; one evacuated and the other full; open the stopcock, talk about the number of accessible states increasing, and the system moves to occupy them–the random state. Then we explain that there is a small but finite probability that the gas stays on one side even with the stopcock open, but the chance is so small it never happens. I think such approaches tend to confuse students. They need to visualize randomness.

So what I do is roll dice. I start with two, then move to three, four, ten and then Avogadro’s constant of dice, and roll them randomly.  In all cases, when the number of rolls becomes statistical, the distribution is Gaussian or Normal. As the number of dice increases the random states start to dominate until the chance of less-random, ordered, states becomes negligible.

I think this is easier to visualize.  So in this blog I will roll 2 dice and in the next four I will roll 3, 4, 10 and Avogadro’s constant of dice.  Finally I will say a few words about ensembles.

Consider a die represents a particle with 6 states and any one of those states comes up randomly. So one die has 6 equally probable outcomes.  However two dice have 36 ways of rolling but only 10 outcomes (2 to 12). Therefore there are more ways to roll some outcomes that others.  Whereas there is only one arrangement (snake eyes) that give a 2, there are 6 arrangements that give a 7.  [(6,1), (1,6),(5,2),(2,5),(4,3),(3,4),] We will denote these arrangements by W, like Boltzmann did. So for two dice W = 36.

Also the concept of arrangements being consistent with an outcome becomes clearer. There are six for a roll of 7, five for a roll of 6 or 5, and so on. The analogy is then made to an ideal gas: for a temperature of, say, 300 K, the number or arrangements of the molecules from the total number possible are consistent with that temperature is analogous to asking how many arrangements there are for a roll that gives a 7.

So when the 2 dice are rolled, the state with the most randomness, 7, comes up 6 times more often than a 2 or a 12. The more random a state is, the more probable it becomes because it has more arrangements that are consistent with a 7 than any other roll.

The result is that as the number of dice increases, then at some point the probability of the random states starts to dominate all others and finally one can replace all the states by only the random states.  That is, in the limit of a large number of dice, W is essentially equal to W_random.  But this is not true for two dice with only 36 arrangements. To get to that point a lot of dice (or a lot of molecules) are needed .

Here are the results:

Figure 1

Figure 2

Rolling two die 2000 times (Figure 1 (click graph to enlarge)):  This is not quite statistical as you can see from the bars that are not exactly on the horizontal lines (this is luck in gambling), but it makes the point.  Notice the number of hits is greatest for the number 7 (chance is 6/36) and smallest for a 2 or 12 (chance of 1/36).

In figure 2, you can see the “frequency table”: the total number of rolls is recorded and the number of hits of a certain number is noted. The probability is simply the ratio of the two.

In the next entry I will roll three dice.

In the meantime, check out this Song by Flanders and Swan on Thermodynamics

The interactive software used in this video is part of the General Chemistry Tutorial and  General Physics Tutorial,  from MCH Multimedia. These cover most of the topics found in AP (Advanced Programs in High School), and college level Chemistry and Physics courses.

Related Posts:

• Entropy (Part 6): Randomness and ensembles
• Entropy (Part 5): Randomness by rolling Avogadro’s dice
• Entropy (Part 4): Randomness by rolling ten dice
• Entropy (part 3): Randomness by rolling four dice
• Thermodynamics and Physical Chemistry Song by Flanders and Swann