Paper ID #29075

A Visual and Intuitive Approach to Teaching and Learning the Concept ofThermodynamic Entropy

Dr. Daniel Raviv, Florida Atlantic University

Dr. Raviv is a Professor of Computer & Electrical Engineering and Computer Science at Florida AtlanticUniversity. In December 2009 he was named Assistant Provost for Innovation and Entrepreneurship.

With more than 30 years of combined experience in the high-tech industry, government and academiaDr. Raviv developed fundamentally different approaches to ”out-of-the-box” thinking and a breakthroughmethodology known as ”Eight Keys to Innovation.” He has been sharing his contributions with profession-als in businesses, academia and institutes nationally and internationally. Most recently he was a visitingprofessor at the University of Maryland (at Mtech, Maryland Technology Enterprise Institute) and at JohnsHopkins University (at the Center for Leadership Education) where he researched and delivered processesfor creative & innovative problem solving.

For his unique contributions he received the prestigious Distinguished Teacher of the Year Award, theFaculty Talon Award, the University Researcher of the Year AEA Abacus Award, and the President’sLeadership Award. Dr. Raviv has published in the areas of vision-based driverless cars, innovative think-ing, and teaching innovatively. He is a co-holder of a Guinness World Record a co-author of five books.

Dr. Daniel Raviv received his Ph.D. degree from Case Western Reserve University in 1987 and M.Sc. andB.Sc. degrees from the Technion, Israel Institute of Technology in 1982 and 1980, respectively.

Mr. Daniel Ryan Barb, Florida Atlantic University

Daniel Barb is an undergraduate student studying Mechanical Engineering at Florida Atlantic Univer-sity. He spent six years in the United States Navy working in a nuclear power plant aboard a fast attacksubmarine.

c©American Society for Engineering Education, 2020

A Visual and Intuitive Approach to Teaching and Learning the Concept of

Thermodynamic Entropy

Daniel Raviv and Daniel Barb

College of Engineering and Computer Science

Florida Atlantic University

Emails: ravivd@fau.edu, dbarb@fau.edu

Abstract

We all experience entropy and Thermodynamics in day-to-day life, whether or not we

recognize it. To some extent, most people have a somewhat intuitive knowledge regarding the

interaction and transfer of energy. Despite this, many students struggle with Thermodynamics in

classroom settings. The equations and mathematics can be overwhelming and frustrating, in part

because it is often hard to visualize thermodynamic interactions. This is especially true for the

concept of entropy, which many consider to be one of the toughest thermodynamic principles to

comprehend.

Today’s students have a plethora of distractions available to them. If students feel bored

or frustrated with the material, oftentimes they will browse the Internet on their laptops or pull

out their phones to entertain themselves. They learn differently, more intuitively, experiencing

short attention spans. Therefore, the material and presentation methods should be clear, intuitive

and engaging.

The primary focus of this paper is to help students to understand, and instructors to

explain, the fundamental concept of entropy at the macro level by utilizing intuitive and

example-based approaches. This paper seeks to cut through the volumes worth of material

written on the subject in order to simplify the topic of entropy to something that is clear and easy

to understand. To accomplish this, the paper contributes by (1) introducing examples of

spontaneous processes that most people should already understand, (2) providing a brief review

of the general operations of heat engines and the Carnot cycle, (3) framing the Carnot cycle in

relation to entropy, (4) discussing non-ideal heat engines, (5) showing analogies to help the

reader understand the significance of the ratio Q/T as a definition for entropy, (6) adding some

brief notes on entropy that are beyond the general scope of this paper, and (7) presenting

brainteasers designed to engage students in the classroom. We relate these examples to the idea

of the Carnot cycle and to some quantitative formulas. The point of this approach is to provide

students with examples that translate textbook explanations to real life and help in

comprehension of the material. We believe that when using these intuitive examples students

tend to better understand the concept of entropy.

This paper should be considered as reporting on work in progress. The presented

information is meant to be supplemental in nature and not to replace existing textbooks or other

teaching and learning methodologies. The work in this paper has been presented to 35 students in

a Thermodynamics classroom setting. Following the presentation, it has been assessed and

received very positive feedback. This visual, intuitive and engaging approach to teaching and

learning has also been tested in the past for topics in Statics (explaining center of gravity),

Calculus (explaining integration and explaining derivation by chain, product, and quotient rules),

Differential Equations, Control Systems, Digital Signal Processing, Newton’s Laws of Motion,

and Computer Algorithms. In all of these cases, students highly praised the approach and found

it to be very effective for learning.

I. Introduction

Peter Atkins writes in his book, The Laws of Thermodynamics: A Very Short

Introduction, “No other scientific law has contributed more to the liberation of the human spirit

than the second law of thermodynamics” [1]. In another work, he writes, “The Second Law has a

reputation for being recondite, notoriously difficult, and a litmus test of scientific literacy. …Not

knowing the Second Law of Thermodynamics is equivalent to never having read a work by

Shakespeare” [2]. Simply put, understanding entropy is crucial to understanding engineering.

Entropy provides a universal cause – it tells us why things do or don’t happen.

Many books have been written on the topic of entropy. Unfortunately, it is still viewed as

a confusing topic for many Thermodynamics students and professors. This paper is about an

example-based approach to teaching entropy that explains the topic in a simple, visual, and

intuitive way. We want to use this example-based approach to improve student understanding of

entropy and show that it doesn’t have to be a notoriously difficult topic. Helping readers

understand this topic more clearly will provide a foundation for future related topics they will

encounter, and will help build better engineers.

A. Our Contribution

This paper deals only with the Second Law of Thermodynamics, and in particular the

concept of entropy as an approach to understanding it. The focus of this paper is on helping

students visually and intuitively to understand the ratio Q/T and its change through a working

process as a way to define and measure entropy. We feel that the new analogies provided for

explaining this ratio offer a new and effective method for teaching the subject of entropy.

The content of this work has been shared with 35 students in a Thermodynamics course

in classroom setting and assessed following the lecture. The results, as summarized in the

Appendix, indicate that this teaching method is effective in helping students understand the

concept of entropy.

In our efforts to effectively communicate the concept of entropy we have made some

small sacrifices in scientific accuracy, and sidestepped more complicated explanations.

B. Assumptions about Prior Knowledge of the Students

In order to effectively focus on the idea of entropy, this paper makes assumptions

regarding the reader’s prior knowledge. They include a basic understanding of the Zeroth, First,

and Second Laws of Thermodynamics, the concept of equilibrium, the meaning of an isolated

system, and the distinction between state and path functions. Additionally, this work assumes

that the reader has a basic background understanding of the Carnot Cycle and the concept of

reversibility. This is expanded upon later to explain it through the lens of entropy.

C. Background on Entropy

There are several ways to look at the Second Law of Thermodynamics. One is through

the lens of entropy. “Entropy is simply a quantitative measure of what the Second Law of

Thermodynamics describes: the dispersal of energy in a process in our material world” [3].

Energy naturally tends to disperse as heat, and it is important to understand that the total entropy

of an isolated system never decreases over time.

Sadi Carnot, born in 1796, developed the first successful theory of the maximum

efficiency of heat engines. While Carnot’s work received little attention during his lifetime, Lord

Kelvin and Rudolf Clausius later built on his theory to create the basis of the study of

Thermodynamics.

Heat engines are used to convert heat energy to mechanical energy, and this mechanical

energy can be used to do work. In a heat engine with maximum theoretical efficiency, known as

the Carnot Engine, all the energy that is not converted to work is expelled to a lower temperature

heat sink; for this theoretical heat engine, no energy is lost (e.g. due to mechanical friction or

other losses). Because no energy is “wasted,” the process is reversible – that is, entropy of this

process does not increase.

In the mid-1800s, a German physicist named Rudolf Clausius gave a name and a

definition to the concept of entropy. He said that entropy describes the amount of energy

dissipated in a working process. This description, while correct and helpful, is incomplete, and

was created before we had an in-depth understanding of molecular processes (for reference, the

neutron wasn’t discovered until 1932).

In the late 1800s, Ludwig Boltzmann built further upon the idea of entropy by developing

the field of Statistical Mechanics. His view of the Second Law of Thermodynamics was that

dynamically ordered states are infinitely improbable [4]. In other words, ordered systems are

Statistics problems. Because of Boltzmann’s work, for many years entropy was explained to be a

measure of the disorder of a system. That description, often accompanied with the “messy room”

analogy, is wrong and has been removed from most of today’s textbooks. Nonetheless, his work

in Statistical Mechanics helps explain the properties of atoms and the behavior of systems as a

whole, and is considered to be his greatest achievement.

The First Law of Thermodynamics states that energy cannot be created or destroyed, only

changed in form. The Second Law of Thermodynamics – the law of entropy – puts constraints on

The First Law. That is, The Second Law gives meaning to time and the direction of energy flow

in the context of thermodynamics. One way to think of entropy is as a measurement of the

availability of useful energy in a system. While energy cannot be created or destroyed, as the

system approaches equilibrium the energy of that system becomes less available for use.

D. Paper Organization

This paper is organized in the following way: (1) introducing examples of spontaneous

processes that most people should already understand, (2) providing a brief review of the general

operations of heat engines and the Carnot cycle, (3) framing the Carnot cycle in relation to

entropy, (4) discussing non-ideal heat engines, (5) showing analogies to help the reader

understand the significance of the ratio Q/T as a definition for entropy, (6) adding some brief

notes on entropy that are beyond the general scope of this paper, and (7) presenting brain teasers

designed to engage students in the classroom.

II. Intuitive Examples of Spontaneous Processes

The purpose of this section is to share examples of real-life spontaneous process that

students can already intuitively understand. They can provide students with a starting point for

learning about entropy based on what they already understand. In all the intuitive examples

shown below, the systems move towards a state of equilibrium, and in the process of reaching

equilibrium the entropy of the system increases. Once equilibrium is reached, no more changes

occur in the system at a macro level. This is because equilibrium is the state of maximum

entropy of a system.

A video on the topic from Caltech put it very clearly: “Because equilibrium is a state in

which all temperatures are equal, it’s a place where no machines operate. No geysers spray, no

volcanoes erupt, no water falls. No big engines want to do any work, and no little engines can.

Everywhere, nature seems impelled towards the state of equilibrium” [5].

A. Dilution – Tea Bag in Water

When placing a bag of tea in a cup of hot water, the tea tends to seep into the water. As

this process occurs, the water appears “darker” closer to the bag of tea, and lighter further away

from it. This is because the highly concentrated tea dissolves slowly into the water. After a long

time, the concentration of tea in the water will be the same throughout. At this point, there is no

concentration difference, equilibrium is reached, and there is no longer a driving force for the

transfer of tea.

Figure 1: Bag of tea in water

B. Height Difference – Two Tanks of Water

Consider the tanks shown in Figure 2. Tank A is filled with more fluid than tank B,

resulting in a difference in height of the fluid. The two tanks are separated by a shut valve. When

the valve is opened, the fluid will flow from tank A to tank B. As this happens, the height in tank

A will lower and the height in tank B will rise.

Figure 2: Two tanks separated by a shut valve

After some time the final fluid height in both tanks will be equal at a new height,

somewhere between the initial heights of tanks A and B, as shown in Figure 3.

Figure 3: Two tanks with equalized fluid height

When the valve is first opened, the height difference, which results in a pressure

difference at the connecting pipe, acts as a driving force that pushes fluid from the higher to the

lower pressure chamber. Once the height (and therefore the pressure) between these two tanks is

equalized, this driving force no longer exists and the fluid height will not change anymore. On a

macro scale, we can consider that at this point there is no movement of fluid between the two

chambers, i.e., the system reaches an equilibrium.

C. Temperature Difference Between Hot and Cold Plates

Imagine two iron plates in contact – one is at a high temperature T1 and the other is at a

low temperature T2, as shown in Figure 4.

Figure 4: Two iron plates in physical contact

When the plates are attached to each other, the faster moving atoms in the hotter plate

collide with the atoms in the colder plate. Their energy is imparted and heat is transferred from

the hot to the cold plate. Assuming that the plates are identical in size, composition, and that no

energy is transferred to or from the surroundings, eventually they will reach an equilibrium

temperature of T3, which is approximately halfway between T1 and T2, as shown in figure 5.

Figure 5: The two iron plates have reached an equilibrium temperature

D. Hot Coffee Cup on a Counter

Many people feel like they need coffee in the morning to “get going.” Often times the

coffee is too hot to drink immediately, so it makes sense to set it on the counter and wait for it to

cool down a bit. The hot coffee initially transfers heat to the counter and the air surrounding the

cup. If they forget about their coffee for long enough, the coffee cools down to “room

temperature,” the same temperature as the counter and the surrounding air.

Figure 6: Coffee cools to the temperature of its surroundings

Note that the energy transferred from the coffee cup has been dispersed to the

surroundings, but the surrounding is so large relative to the coffee cup that we don’t notice the

infinitesimally small increase in the ambient temperature.

E. Diffusion

Imagine two sides of a tank separated by a divider. Side “A” is filled with a specific gas.

Side “B” is a theoretically perfect vacuum. This tank is shown in Figure 7.

Figure 7: Side A is filled with gas at a high initial concentration while Side B is a vacuum

Once a portion of the divider is removed, the molecules on side A will tend to move

towards the area of lower concentration (i.e., side B). Over a period of time the concentration

difference between the two sides gets smaller and smaller, and eventually approaches zero, i.e.,

approaches equilibrium, as shown in Figures 8 and 9.

Figure 8: Sides A and B both have an approximately equal final concentration

Figure 9: Reaching equilibrium

III. Brief Review of Heat Engines

Figure 10: Steam engines power old-fashioned trains

As previously mentioned, once a system reaches equilibrium, no work can be done by the

system. For any heat engine to operate, a difference in temperature must be present. Heat

engines, like the one that powers the steam engine in Figure 10, work by turning the flow of heat

into mechanical energy. Some of the input energy is converted to work, and the rest is expelled

to a low temperature reservoir, as shown in Figure 11. This can be written in equation form as

follows:

𝑊 = 𝑄𝑖 − 𝑄𝑜

In this equation, W is the work, while Qi is the energy of the high temperature reservoir

(“energy in”), and Qo is the energy of the low temperature reservoir (“energy out”).

For a heat engine to operate, there must be two reservoirs (a “source” and a “sink) at two

different temperatures, so heat can flow from high to low temperature. When the difference in

temperature between the heat source and heat sink is larger, more flow of heat occurs.

Figure 11: A heat engine converts some energy to work, and transfers the rest to the low

temperature heat sink

The Carnot cycle is the ideal cycle for the heat engine, and provides the maximum

possible theoretical efficiency. Like any other heat engine, it converts energy from a high

temperature source into work, and expels the rest to a lower temperature sink. However, in the

Carnot Cycle no energy is “wasted” due to mechanical friction, etc. A perfectly efficient engine,

shown in Figure 12, is called a “Carnot engine” and undergoes the Carnot cycle. The Carnot

cycle is reversible, which means the system can theoretically be put back to its original state with

no change in the system or its surroundings. More precisely, “A change is said to occur

reversibly when it can be carried out in a series of infinitesimal steps, each one of which can be

undone by making a similarly minute change to the conditions that bring the change about” [6].

It is important to note that in Carnot’s engine the entropy of the system does not change.

Figure 12: Diagram showing the Carnot Engine

Unfortunately, in the real world, heat engines are not perfectly efficient. In the process of

a heat engine’s operations, energy is lost to friction, ambient losses, and more. This “wasted”

energy means that less energy is converted to work. Additionally, in a real engine, shown in

Figure 13, the entropy of the system increases throughout the system’s operation.

Figure 13: Diagram showing a real heat engine

Note that as compared to Figure 12, Figure 13 has a smaller arrow representing the work

output, and a larger arrow showing Q0. This is because in a real engine, less work is done and

some energy is “wasted.” Note also that the arrow representing So for the real engine is larger

than the arrow representing Si in the same figure, whereas in Figure 12 (Carnot’s engine) the size

of the entropy arrows is the same. This is because in real engine entropy increases throughout the

working process whereas in an ideal Carnot engine the entropy is constant.

IV. Entropy in Relation to the Carnot Cycle

The efficiency of any heat engine can be calculated using the following equation:

𝜂 =𝑊

𝑄𝑖=

𝑄𝑖−𝑄𝑜

𝑄𝑖= 1 −

𝑄𝑜

𝑄𝑖 (1)

Where 𝜂 is the efficiency of the heat engine, and as mentioned earlier W is the work done

by the engine, Qi is the heat expended from the source, and Qo is the heat sent to the sink.

Carnot made a huge discovery when studying ideal heat engines. This discovery states

that when the process is reversible, the efficiency equation can also be written as equation 2

below:

𝜂 = 1 − 𝑇𝑙𝑜𝑤

𝑇ℎ𝑖𝑔ℎ (2)

Where Tlow is the absolute temperature of the heat sink and Thigh is the absolute

temperature of the heat source. By comparing equation (2), which is true only for ideal heat

engines, to equation (1), which is true for all heat engines, we find that for the ideal heat engine:

𝑄𝑜

𝑄𝑖=

𝑇𝑙𝑜𝑤

𝑇ℎ𝑖𝑔ℎ (3)

Rearranged, this can also be written as:

𝑄𝑜

𝑇𝑙𝑜𝑤=

𝑄𝑖

𝑇ℎ𝑖𝑔ℎ (4)

This result rewritten in the following box is an amazing discovery regarding a reversible

process by Carnot! Let’s explain.

In the Carnot Engine, the ratio of heat taken in to the heat expelled is the same as the ratio

of the two absolute temperatures needed to drive the heat engine. This discovery means that in

Carnot’s Engine, the ratio of Q/T is constant, i.e., invariant! Caltech’s entropy video cleverly

phrased this concept as follows: “Something that goes in is the same when it comes out. If it’s

not the heat, and if it’s not the temperature, what is it? It’s the heat divided by the temperature at

which it flows” [5]. According to Rudolf Clausius, this ratio is entropy. In Carnot’s Engine,

𝑸𝒐

𝑻𝒍𝒐𝒘=

𝑸𝒊

𝑻𝒉𝒊𝒈𝒉

entropy flows in at a high temperature, does work, and this same amount of entropy leaves at a

low temperature. Thus, an ideal engine conserves not only energy, but also entropy.

V. Non-Ideal Heat Engines

As previously discussed, in non-ideal heat engines the entropy increases throughout the

working process. This change in entropy between two states can be determined using:

𝛥𝑆 =𝑄𝑜

𝑇𝑙𝑜𝑤−

𝑄𝑖

𝑇ℎ𝑖𝑔ℎ (5)

where ΔS is the change in entropy between the states, Q is the heat energy of the system at its

initial and final states, and T is the absolute temperature of the system at its initial and final states

(in Kelvin). In a non-ideal engine, the change in entropy is always positive.

Because the entropy equation only considers the initial and final states of the system (that

is, the initial and final value of Q/T), entropy is a state function rather than a path function. As

Professor Sushkov put it, “Since entropy is determined by the state of the body it is perfectly

natural that changes in it should be determined fully by the initial and final stages and do not

depend on the way in which changes are made” [6].

VI. Considering the Ratio – Analogies to Relate to Entropy

A. Sneeze in Library vs. Crowded Street

In The Laws of Thermodynamics: A Very Short Introduction, Peter Atkins writes, “The

analogy I have used elsewhere to help make plausible Clausius’ definition of the change in

entropy is that of sneezing in a busy street or in a quiet library. A quiet library is the metaphor

for a system at low temperature, with little disorderly thermal motion. A sneeze corresponds to

the transfer of energy as heat. In a quiet library a sudden sneeze is highly disruptive: there is a

big increase in disorder, a large increase in entropy. On the other hand, a busy street is a

metaphor for a system at high temperature, with a lot of thermal motion. Now the same sneeze

will introduce relatively little additional disorder: there is only a small increase in entropy” [1].

Figure 14: Sneeze in a library vs. sneeze in a crowded street

This analogy, illustrated in Figure 14, provides a succinct visual analogy emphasizing the

significance of entropy as a ratio. Based on this example, here are some new analogies along the

same line.

B. Light in Hotel Room – Day vs. Night

A lightbulb emits approximately the same amount of light at any point throughout its

lifespan. Despite this, the effectiveness of a light bulb (i.e., its usefulness in a qualitative sense)

changes based on the amount of ambient light. Consider a hotel room with a light during both

day and night times, as shown in Figure 15.

Figure 15: A light bulb shining in a hotel room, during both day (left) and night (right) time

While the amount of light given off by the light bulb at day and night time is the same,

the effect on our visual system, i.e., ability to “see” is totally different: it is almost not affecting

the overall light during day time, but it is very meaningful at night time.

C. Change in Hourly Wage

Consider two employees: One makes $10 per hour and the other $100 per hour. An

hourly raise of $1 to each is the same in absolute terms, but is very different in terms of the effect

on each of the employee’s lifestyle. The person earning $100 per hour will barely notice the 1%

raise. On the other hand, the person who makes $10 per hour will be meaningfully affected by a

10% raise.

While both raises are the same in dollar amounts, they are very different in terms of

percentage. The percentage of the raise is a ratio of the raise to the original wage. In this case it is

clear that the ratio counts more than the absolute amount of the raise. This ratio is generally more

important when considering wage increases, which is why companies usually consider raises as a

percentage of what the employee is currently getting paid.

D. Disease in Airplane

If you surveyed random people regarding what they don’t like about flying, many may

list the germs as a big factor. In a plane many people are crammed into a relatively small “tube,”

which could result in diseases spreading much more easily.

In an airplane with just a couple of passengers, there is far less concern regarding the

spread of germs. Likewise, the risk of spreading disease is minimal when people are spread over

a very large space. If we consider the ratio of people to the size of the space they’re in, it is

therefore neither the number of people nor the volume of the contained space that is the

deterministic factor for the spread of germs. Instead, the spread of germs can be better measured

from the ratio of the number of people (e.g., in the plane) to the volume (e.g., internal volume of

the plane).

VII. A Few Notes on Entropy - Beyond the Scope of This Paper

The focus of this paper has been fairly narrow – it has centered on the macro view of

entropy and how to understand it in a fundamental way. The following are a few topics that are

not the focus of this paper, but are discussed briefly due to their relevance to the topic of entropy.

A. Micro and Macro States – Ways to Look at Entropy

Up to this point, our discussion of entropy has largely focused on the “macro” level of a

system – that is, viewing the system as a whole. Entropy can also be viewed on a “micro” level,

which focuses on the microscopic, molecular level.

1. The Arrow of Time

In the macro state of entropy, it is possible to tell the direction of time. As an example,

consider someone dropping a basketball, like in Figure 16 below.

Figure 16: Person dropping a basketball [7]

Almost anyone can tell which photo is the “before” and which one is the “after.” An

object will not spontaneously “jump” into the air and into the man’s hands. From the set of

images in figure 16, finding the arrow of time is simple. As the entropy of the ball increases, the

ball bounces less and less as in Figure 17.

Figure 17: The path of a dropped basketball

On the other hand, consider the individual molecules of a system. At all times the

molecules are moving around and bouncing off of each other. Two consecutive side-by-side

snapshots will not reveal enough information to discern the order of the photos. This is because

the arrow of time is not apparent at the micro level. Figure 18 shows molecules under a

microscope at two different times (left and right). It is impossible to tell which image came

before the other.

Figure 18: Molecules under a microscope at two different times

2. Boltzmann’s Entropy Equation

Boltzmann developed an equation for calculating the entropy of an ideal gas. This

equation is a probability equation and is significant in the field of statistical mechanics. The

equation is:

𝑆 = 𝑘𝑏𝑙𝑛 (𝑊)

where S is the entropy of the system, W is the number of different ways in which the molecules

can be arranged, and kb is the Boltzmann constant. Simply put, Boltzmann’s entropy equation

develops a relationship between the entropy of a system and the number of ways that

atoms/molecules of that system can be arranged. A good example to visually explain this micro

level concept is given in a PBS Space Time video [8]. A few snapshots are shown below.

“Go” is a board game that has been around for more than 2500 years. It is most popular

in East Asia, and the standard format has a board that is a 19x19 grid containing 361 points, but

it is possible to play on smaller grid sizes as well.

Imagine 180 black pieces distributed in a random arrangement on the board. Each

arrangement can be considered as a micro state of the system. There are approximately 2x10107

possible arrangements of pieces. Many of these arrangements are fairly “uniform,” and to the

untrained eye look very similar, as shown in Figure 19.

Figure 19: Three random board distributions (from [8])

On the other hand, there is only one arrangement, shown in Figure 20, that has all 180

pieces on the same side of the board. This micro state is 2x10107 times less likely to occur than

one of the other, smooth-looking micro states.

Figure 20: All pieces on one side of the board (from [8])

While some piece arrangements look more or less ordered to the untrained eye, they have

the same entropy state as other, random-looking arrangements. This is because any given

“ordered” looking arrangement is just one microstate, and is just as likely as any other given

microstate. There are simply very many “disordered” looking microstates, which is why they

appear more.

VIII. Brain Teasers

The following are a couple of brain teasers designed to engage students in the classroom

setting and get them to apply their knowledge of entropy to the real world. The “Isolated room

with fridge” brain teaser forces students to apply their understanding of heat processes and how

no heat engine is perfectly efficient. The “Soda in the fridge” brain teaser forces students to

consider what an isolated system means.

A. Isolated room with fridge

You are in an isolated room that does not allow any energy transfer into or out of the

room. The room has a fridge that can be plugged-in or unplugged. Your goal is to minimize the

temperature of the room.

Should you:

A. Plug the fridge in, but leave the fridge door closed

B. Plug the fridge in and open the door

C. Leave the fridge unplugged?

Answer: Leaving the fridge unplugged will not increase, the temperature of the room.

At first glance, a logical option may be to plug the fridge in and open the door. With a

little help from the Second Law of Thermodynamics we know that this is not true. Because no

heat engine is ideal, cooling the air requires energy expenditure. Since the fridge is not ideal, any

cooling effect it performs will be outweighed by the amount of heat added to the room through

its working process.

B. Soda in the fridge

Modern refrigerators allow fairly precise selection of internal temperature. For example,

you could choose to have the fridge maintain an internal temperature of 39° F. Later, you can

place a room temperature soda into the fridge and, after waiting a while, the soda’s temperature

will lower to 39° F. The reason that the temperature of the soda decrease, while the temperature

of the fridge remains unchanged is that refrigerators expend energy to cool its content and in the

process transfers heat from the fridge to the surroundings of the fridge.

IX. Assessment

Refer to Appendix A.

The methods and examples shown in this paper were presented and assessed in a

Thermodynamics class taught by the first author of this paper at Florida Atlantic University. A

questionnaire was distributed to students at the end of the lecture in order to gauge how receptive

students were to the intuitive and engaging learning techniques presented in this paper. The

responses are based on a class presentation that included examples from this paper. 35 students

filled out the questionnaire. A summary of the questionnaire and raw results can be found below.

Figures A.1 and A.2 show the self-identified demographic breakdown of participating

students. There is a fair amount of diversity, which is representative of the diversity at FAU and

its engineering programs. Most students were 23 years old or younger, but there were a handful

of older students, including four older than 33.

Questions regarding student opinions about the importance of the lecture or learning in

general had five options: very important, important, moderately important, slightly important,

and not important. Questions regarding opinions about the effectiveness of various presentations

or learning methods had the options: strongly agree, agree, neutral, disagree, or strongly

disagree.

When asked about the “importance of being introduced to Thermodynamics through

visual examples,” 50% of the students responded with “very important” and an additional 38%

responded with “important”. Only one student marked “slightly important,” and no student

marked “not important.” To a question regarding the “importance of being introduced to

Thermodynamics concepts through hands-on activities, 34% responded “very important,” and an

additional 28% responded “important.” Only three students responded “slightly important” and

three felt that hands-on activities were not important.

Students were mixed, though slightly favorable, on their opinions of the importance of

traditional and PowerPoint presentations. When asked about the “importance of learning using

traditional presentations,” only 3 students marked either very important or not important. 40%

marked important, 26% marked important, and 26% felt that traditional presentations were only

slightly important. Answers to the question regarding the importance of learning

Thermodynamics through PowerPoint presentations were similar.

Students subjectively found that visual presentations helped them understand the concept

of entropy. This is apparent by the responses to the question: “How effective was the

presentation on Thermodynamics today?” and “What could have helped you to understand the

concept of entropy in Thermodynamics better?” most students wrote that they found the

presentation effective, specifically the visual portions of the presentation.

These results, when taken as a whole, show that many students feel that they learn best

through visual and engaging approaches to education. They generally are neutral to the idea of

traditional and PowerPoint presentations, and prefer to be taught rather than teach themselves.

X. Conclusion

The explanations in this paper are meant to introduce the concept of entropy in visual and

intuitive ways. The paper is not comprehensive and does not attempt to replace current teachings

and textbooks. The explanations intentionally do not include numerical examples and proofs in

order to avoid intimidating students, but rather focus on the basic understanding of the concept

itself. We hope that it will serve as an extra resource for helping students understand a concept

that is widely regarded as difficult.

XI. Acknowledgements

The authors thank Dr. Myeongsub Kim from Florida Atlantic University for allowing us to

present this work to his Thermodynamics class. Also, the Center for Writing at Florida Atlantic

University provided great feedback as well.

XII. References

[1] P. Atkins, “The Laws of Thermodynamics: A Very Short Introduction,” Oxford

University Press, 2010.

[2] P. Atkins, “Four Laws That Drive the Universe,” Oxford University Press, 2007.

[3] F. Lambert, “Entropy is Simple – If We Avoid the Briar Patches!” 2013.

[4] R. Stein and J. Powers, “The Energy Problem,” World Scientific, 2011.

[5] Cal Tech, “Episode 47: Entropy – The Mechanical Universe,” YouTube, 2016.

[6] V. Sushkov, “Technical Thermodynamics,” Gordon and Breach, 1965.

[7] freakjointz, “Bouncing Ball Basketball Reference 24fps,” 2013.

[8] PBS Space Time, “The Misunderstood Nature of Entropy,” YouTube, M. O’Dowd, 2018.

Appendix – Assessment Results

The following set of graphs is a compilation of assessment results.

Figure A.1: Self-identified ethnic distribution

Figure A.2: Self-identified age distribution

Figure A.3: Student feedback on importance of understanding Thermodynamics concepts

Figure A.4: Student feedback on importance of visualizing Thermodynamics

Figure A.5: Student feedback on importance of being introduced to Thermodynamics concepts

through visual examples

Figure A.6: Student feedback on importance of hands-on activities in Thermodynamics

Figure A.7: Student feedback on importance of class engagement in Thermodynamics

Figure A.8: Student feedback on importance of traditional presentations for learning

Figure A.9: Student feedback on importance of PowerPoint presentations for learning

Thermodynamics

Figure A.10: Student feedback on importance of learning Thermodynamics through textbook

reading

Figure A.11: Student self-assessment of mathematics competency

Figure A.12: Student opinions on self-teaching