Boltzmann Bucks—Helping Students Conceptualize Entropy

Boltzmann Bucks provided to students

Each year, my honors chemistry class eventually gets to the point where thermodynamic quantities and the relationships between them are introduced. For many students, the very nature of the ideas within thermochemistry often creates a sense of overwhelming abstraction that is difficult to overcome. Of all these ideas, the concept of entropy has given both my students and me the greatest trouble. This year, I was determined to change that.

I wanted to better understand the concept of entropy myself and search for more effective methods teaching it to novice chemistry learners. Fortunately, thanks to Theresa Marx and Erica Posthuma-Adams, I was introduced to an engaging activity that really improved how my students (and me) think about entropy—the Boltzmann Bucks game. If you are looking to go beyond using traditional, arguably misleading, definitions of entropy involving “disorder” and “messy bedroom” analogies, this activity can serve as a wonderful opportunity for students to more accurately conceptualize entropy.

As our thermochemistry unit approached, I started to take a closer look at how and why the concept of entropy was derived in the first place. Until that point, certain aspects of how I taught entropy could be summarized by the following definitions and ideas:

Entropy is…

  • A measure of disorder
  • A measure of the dispersal of matter and energy
  • A measure of chaos

Changes in entropy

  • Increase in moles from reactants to products (increase in entropy)
  • Release of heat—exothermic (increase in entropy)
  • Lower-energy to higher-energy state of matter s → l → g (increase in entropy)

Direction and Entropy (Spontaneity)

  • Reactions that result in an overall increase in entropy tend to be favorable
  • The increase in entropy accounts for the irreversibility of natural processes

Whenever a student would ask a thoughtful question such as, “what do you mean by disorder?”, I would resort to common examples and analogies such as,

  • Your bedroom always goes from clean to messy (low to high entropy)
  • Entropy is like an incomplete jigsaw puzzle (incomplete puzzle has higher entropy than completed puzzle)
  • Waterfalls always flow downhill
  • At 20 0oC, ice melts
  • When a tire is punctured, air always leaves from the tire to the surroundings

Regardless of the analogy or example, they were simple to use due to their familiarity with students’ past experiences and I could get students to consistently predict when entropy was increasing or decreasing throughout a process. However, I never truly felt as though my students understood the overall concept of entropy and its explanatory role for why natural processes tend to go in one direction and not the other. As a teacher, it felt more like I was just giving them a word to know, telling them when it is increasing or decreasing, and then just informing them that the universe tends to favor certain directions for processes. It was unlike any other feeling I have had teaching a specific concept in chemistry; completely disconnected from understanding but somehow tricking myself and my students into thinking understanding was taking place.

To relieve my own personal stress with this feeling of inadequate understanding, I started to look more closely at the concept of entropy through various resources. It wasn’t long before I realized an essential characteristic of entropy that had been completely vacant from my own understanding and, subsequently, my teaching—its simple relationship to probability.

Suddenly, terms that I had heard before but never took the time to fully comprehend, such as microstates and distributions, started to bring a sense of clarity to a topic that had always been fuzzy to me. But developing an understanding of these terms, their relation to probability, and how it all fit together to describe the concept of entropy was not an easy task for me. So how was I supposed to get my students to arrive at a similar revelation?

Instead of scouring the Internet for some kind of activity, I reached out to my PLN of educators and, as usual, they did not disappoint. It turned out that there was a JChemEd article, Give Them the Money: The Boltzmann Game, a Classroom or Laboratory Activity Modeling Entropy Changes and the Distribution of Energy in Chemical Systems,1 describing an activity that appeared to be exactly what I was looking for. While I strongly encourage you to give the article a read for more details, I will try to summarize the activity and some of the key points made in the article.

What is the game supposed to model?

What the Boltzmann Game models is how energy is distributed in real chemical systems. In particular, the authors suggest that this game simulates the harmonic oscillator model of vibrational excitation, which describes how energy is quantized and constantly being exchanged between the molecules of a system.

How does this game compare to other activities meant to model this concept?

"what seems to be lacking are simple quantitative activities that students can participate in that effectively convey the essential probabilistic nature of entropy"~Hanson and Michalek on why they developed the activity

The inherent value of the Boltzmann Game stems from the lack of simple, inexpensive, and engaging classroom activities that appropriately model the concept of entropy. More specifically, the authors suggest, “what seems to be lacking are simple quantitative activities that students can participate in that effectively convey the essential probabilistic nature of entropy".

OK, so what does it look like?

After a brief conversation (optional) with students about the random nature of how energy is exchanged and distributed at the particle level, students are told they will be modeling this concept by playing a game that is grounded in probability and randomness—rock, paper, scissors. Here is a synopsis of how the game is set up, played (see figure 1), and recorded:

Game Setup

  • All students will form teams of two. It doesn’t matter whom you are paired up with since you won’t be together for long.
  • Every student receives one Boltzmann Buck (B$) (found in supporting information), which represent 1 packet (quantum) of energy.
  • Students will form 2 circles; 1 circle within the other and each pair of students will need to decide who will be in the “inner ring” and who will be in the “outer ring.”
  • The “inner ring” students from all teams form a circle facing outward while the “outer ring” students will form a circle facing their partner.
  • The result should be two circles, with the two students in each team looking at one another.

Figure 1: Rules of Rock-Paper-Scissors


After all rounds were completed, students came back to the classroom and were told to copy the data into their notebooks. The following table reflects the data we gathered that day (figure 2).

Figure 2: Data from my class

Without any class discussion, I asked my students to answer the following questions:


What does energy have to do with probability?

Even though we had previously discussed the idea that energy is exchanged randomly, I could see that statement not fully being grasped prior to the game. However, after playing the game, many students were trying to make sense of the relationship between energy and probability in meaningful ways. Though I wish I had recorded these answers, the following response from one student taken directly from the JChemEd article was similar to what I heard from my own students: “Probability comes into play with energy because there is a chance that you will gain or lose your energy, and there is a chance of both for every molecule that’s floating around other ones just like in the game. You could gain, lose, or simply stay the same.”

Why isn’t the most probable distribution of money one where all players have the same quantity of money?

What I loved about this question was how many students intuitively came up with a similar explanation. The majority of them realized that the number of ways for the money to be distributed equally between all players was incredibly small compared to other potential outcomes. It simply was too improbable. Understanding the role of probability with respect to the distribution of money was essential if they were going to connect how this game modeled the probabilistic feature of entropy.

Figure 3: One Boltzmann Buck is given to each student at the beginning of the activity.2 (Reused with permission) 

After a brief class discussion about their answers, I followed up with another question to see if they could apply new information to their model.

Would the energy distribution be any different if more energy was added (I give you more B$)?

While this question was not as widely understood initially by my students, several of them realized that if I inserted more money (energy) into the game, there would be a greater distribution (a higher average) and the number of players with zero would decrease.

It was at this point that I introduced the thermodynamic terms microstate and distribution. In short, I talked about how the number of distinct ways energy can be distributed throughout a system is known as a microstate. In our game, this would be like describing how much B$ each person had at any given point in time. Since calculating the possible number of ways each person could have B$ can quickly get out of hand, our game focused on collecting information on the distributions of B$ (energy). As the data suggested, some distributions of B$ (energy) were more probable than others. This increase in probability of energy distribution directly corresponds to an increase in entropy. Based on this information, I finally asked them to describe how entropy is intimately related to probability in their own words.

Though we did not get in to the details of how we could quantify the number of microstates using Boltzmann’s entropy formula (S = k·lnW), I thought it would be useful for students to calculate the number of ways (microstates) a particular distribution of money could be made. By doing this, they could visibly see an actual quantity for a particular distribution and compare it to other distributions. Through comparisons, they could literally see that some distributions were simply far more probable than others. To determine these quantities, I showed them how they would need to involve the use of factorials.

For example, the number of ways to achieve the distribution from my “most probable distribution” column in Figure 3 would be calculated as follows:

This distribution can occur in roughly ninety billion ways; far more than any other distribution.

Calculating the number of ways a particular distribution could occur was also useful because it forced students to realize that different distributions were not a matter of possibility, but probability. When discussing topics such as entropy and spontaneity, we often describe one direction of a certain process as impossible. For example, we point to extreme examples like dropping a glass cup and watching it shatter in to a million pieces. We declare that the reverse process, the million pieces coming back together to form the original glass cup, is impossible! Or how energy always flows from warmer objects to colder objects and the reverse is impossible. However, once we understand the probabilistic nature of entropy, we make the shift from declaring particular directions as impossible and instead, view them through the lens of being so incredibly improbable that we can be reasonably confident it will never happen. Viewing the directionality of processes this way may seem trivial since the end result is essentially the same, but simply declaring one direction as impossible without adequate explanation as to why we believe this completely removes any need for understanding why certain processes tend to go in one direction and not the other.

So How Did This Game Impact Understanding of Entropy?

Getting my students engaged with this activity and the subsequent conversations that followed, opened up doors for the types of questions I could ask my students with respect to explaining why spontaneous reactions tend to increase in entropy. In the past, I would ask a simple question like, “which state of matter has a higher entropy—solid or liquid?” Nearly all students would answer correctly but when pushed to explain their answer, they would simply rely on concrete explanations such as, “the particles in a liquid are more dispersed.” In our recent exam, I asked the exact same question and got answers that absolutely blew me away (see figure 4) compared to previous years. Students were including ideas such as probability and comparing the possible number of microstates and distributions of matter in a liquid compared to a solid. This was something I had not seen before and it was obvious their overall explanatory abilities regarding entropy had improved.

Figure 4: Student quotes after completing the activity

Additionally, this new understanding helped them more easily conceptualize spontaneity and make predictions without simply resorting to algorithmic “tricks” I had relied upon in the past. More students were viewing problems related to entropy and spontaneity in a fundamentally different way than before.

The Boltzmann Game not only helped my students construct a more meaningful and accurate definition of entropy, it provided a memorable experience for students connect with their explanations regarding entropy in a more effective way. As stated by the authors, entropy is seen as a measure of “the number of ways a state can have the same overall distribution of energy”, and any differences of entropy are measures of the “relative probability of two possible distributions”. Statements like this would have gone right over the heads of my students had we not played this game. This realization was a game changer for me and I will most certainly do it again. Additionally, I plan to incorporate different uses of the game that were suggested by the authors such as modeling the exchange of energy between a system and its surroundings.

Log in to have access to Supporting Information: a pdf of Boltzmann Buck images from the original JCE article and a handout I use with students as we work through the activity.

1. Hanson, R. Michalek, B. Give Them the Money: The Boltzmann Game, a Classroom or Laboratory Activity Modeling Entropy Changes and the Distribution of Energy in Chemical SystemsJournal of Chemical Education. Vol. 83 (4), April 2006, p. 581.

2. The Boltzmann Buck is reused with permission from the Supporting Information of the activity outlined in the Hanson,  Michalek cited above. Copyright 2006 American Chemical Society.

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Comments 9

Mitchell McQueen | Fri, 12/04/2020 - 12:07

I enjoyed the article. I am sure I am missing the obvious, but how did you get the most probable distribution data?

Ben Meacham's picture
Ben Meacham | Fri, 12/04/2020 - 13:19


For some reason, I don't think I made that part very clear in the article so totally legit question! I've included an image that should provide some clarity (sorry for the formatting). Basically, I just put in an example distribution and calculated the total # of ways that particular distribution could happen. Whatever distribution gave me the greatest possible # of ways (most microstates), that's what I used for the most probable distribution. 

I made the function that I used in my spreadsheet visible on the image below. Hope that helps!

Stephen Mulick's picture
Stephen Mulick | Fri, 01/28/2022 - 13:14

Is there any larger meaning to the fact that the most probable distribution in that image is a Fibonacci sequence?

Thomas Mitchell | Fri, 01/21/2022 - 12:04

Hello!  Thank you for this activity, I thought it was really great, but I have a question.  At the start of each "round" do you "reset" and redistribute everyone to have 1 dollar?  It seems, or, seemed when my students did it, that a few people ended up with more money, throwing off the distrubution.  I suppose that helps explain how some particles might have more energy in a situation, but still, it was a question I had.

Perhaps doing a coin flip instead of Rock-Paper-Scissors might make it more random? (tho not as fun, certainly)

Also: is there a formula to calculate the "most likey" distribution values for the different energies?  Guessing and checking is challenging since you have to balance the amount of money you have with the number of people who might have that amount (for example, you can't have 5 people with 3$ if you only have 14 students)

Thanks again for the activity!

~Mr. Mitchell (Funny! Same name!)

Ben Meacham's picture
Ben Meacham | Sat, 01/22/2022 - 09:29

Hi Thomas, thanks for the questions. 

1) At the start of each "round" do you "reset" and redistribute everyone to have 1 dollar?

No. The purpose of each "round" is simply to check in at that moment in time what the current distribution of $ is. In theory, you can make each round last for however long you wish; there's no "correct" amount of time that needs to go by for each round. I usually tell students to stop playing after about 2-3 mins, do a quick count of how many people have a certain amount of $ and then instruct them to keep playing before I check in with them after another 2-3 mins have gone by. Nothing is "reset" after each round and there is no redistribution of $ after each round. 

You had mentioned that when you did it, you noticed that a few people ended up with more money. If I understand what you're saying correctly, then this is precisely what should happen based on what's most probable. 

I suppose you could do a coin flip for the game instead of rock-paper-scissors. I'm not entirely sure whether this would result in it being "more random" though since both games are effectively decided by chance. That being said, since rock-paper-scissors involves more possible pathways, instead of being binary like a coin flip, I would imagine the use of rock-paper-scissors would be more likely to result in the intended outcome of the analogy to entropy. I may be wrong about this but that's just my intuition. 

2) Is there a formula to calculate the "most likely" distribution values for the different energies?

From the original JChemEd paper, I calculate the most probable distribution using the following equation.

W = number of ways a particular distribution can happen

There is naturally a bit of "guess and check" to it but since I know how many students are in my class playing the game, I can just base my equation off of that. If some students end up being absent, then I adjust this number. Personally, I just have this autocalculate within a spreadsheet to make it all a bit easier (see image above in the previous comment). 

I hope that provides some clarity to your questions. If not, let me know. Thanks for the feedback and inquiring about the activity!

Thomas Mitchell | Sat, 01/22/2022 - 13:38

Hi Ben,

Thanks for the reply!  I see what you mean about the rounds and keeping the distribution "flowing" as it were, and how that models the different possible states, (plus it's just easier than redistributing each time!) so thanks for the clarification.

For the most likely distribution, maybe I'm thinking about it incorrectly.  I guess my question was more like how do you get the values of, in your example, 8 students with 0$, 5 with 1$, etc.  This is particularly confusing for me since the total "dollar" value of this distribution is more than the number of students, and presumably more than the number of dollars they start with (8*0$ + 5*1$ + 3*2$ + 2*3$ + 1*4$ + 1*5$ = 26$, and there are 20 students in the example).  I see that the formula above calculates the *total # of ways* this particular distribution can be arranged, but I don't see how to get the 8 people, 5 people, etc *of* the distribution.  Perhaps that is where guessing and checking is the only solution?  Seems like there should be a formula... but in trying to look it up I found this, and wasn't sure how to proceed.

Maybe there isn't an easy way to do it, and it's not so hard to guess-and-check, but I just thought I'd ask to see.

Thanks again!

~Thomas Mitchell

Ben Meacham's picture
Ben Meacham | Tue, 01/25/2022 - 15:52

I think I better understand what you're saying now, thanks for clarifying. To make this a bit easier, I just made a relatively quick video response instead of typing out everything. I figured this would allow me to more clearly respond to what you're saying. Hope it helps! Click HERE for the video.

Stephen Mulick's picture
Stephen Mulick | Mon, 04/18/2022 - 01:28

Hi Ben,

Thanks for sharing and putting this all together. I've been tinkering with my own spreadsheet, modeled after yours, and I think I agree with Thomas - there's a conservation of dollars you have to consider. The most probable distribution column you're listing is technically impossible; there can't be those numbers of students with those dollars, because there aren't that many dollars in circulation. I think that's where Thomas's multiplying comes in - he's trying to satisfy that the 20 students are holding a total of 20 B$. I can't quite wrap my head around the paper that he linked, but by checking both number of students and dollar amounts and trying to maximize the number of ways, I do think I've managed to get a 'best' answer through guessing and checking. It works out to 10, 4, 3, 2, 1 students with 0, 1, 2, 3, and 4 B$, respectively. Hope this helps!

Ben Meacham's picture
Ben Meacham | Mon, 04/18/2022 - 15:16

Hey Stephen,

You're correct, there needs to be a conservation of dollars when considering the most probable distribution. I did this activity again a couple of months ago and ran it by some of my physics colleagues and that very same issue was addressed. Like you described, I would determine the most probable distribution by doing a quick "guess and check" while still ensuring the numbers I had in my distribution reflected the conservation of dollars. This distribution will depend on how many students you have participating. For example, in one of my classes, there were only 18 students present. This ended up producing a distribution of 8, 5, 3, 1, 1 students with 0, 1, 2, 3, and 4 B$, respectively. Thanks for the feedback!