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Physics 535
HW: ENTROPY
Name
1. In question 1, we consider processes similar to Process A from the tutorial, with a block with heat
capacity 400 J/K and initial temperature 280 K and a second block with heat capacity 200 J/K and
initial temperature 340 K.
A. In process B, the 280 K block and 340 K block change in temperature, but in the opposite direction as
in process A. In other works, the temperature of the 280 K block decreases to 260 K and the 340 K
block increases to 380 K. The blocks are still thermally isolated from all other objects.
1. Is process B allowed by the first law of thermodynamics? Explain your reasoning.
2. What is the total change in entropy of the two-block system during process B?
B. In process C, both blocks begin at 300 K but their temperatures diverge so that the block with heat
capacity 400 J/K ends up at a temperature of 280 K and the other ends up at 340 K. (Note that this
process is the reverse of process A).
1. Is process C allowed by the first law of thermodynamics? Explain your reasoning.
2. Using your result from part A.5, find the change in entropy for each block and for the two-block
system during process C.
C. The following questions refer to processes A (from the tutorial), B, and C.
1. For those processes that you would expect to observe, was the total change in entropy of all
systems involved greater than, less than, or equal to zero?
2. For those processes that you would not expect to observe, was the total change in entropy of all
systems involved greater than, less than, or equal to zero?
Adapted with permission from Michael Loverude
ENTROPY
Physics 535
2. Consider a macroscopic system of two blocks that are in thermal contact. The two-block system is
isolated from the rest of the universe. Initially, a hotter block is at a temperature TH and a colder block is
at a temperature TC. In the final state of the system, the two blocks are at the same intermediate
temperature TI.
A. Compare the following quantities for the initial state and final state:
Entropy for the block that starts at TH
Sf > Si
Sf < Si
Si = Sf
not enough information
Entropy for the block that starts at TL
Sf > Si
Sf < Si
Si = Sf
not enough information
Combined entropy for the two block system
Sf > Si
Sf < Si
Si = Sf
not enough information
B. Consider the following statement:
“When the blocks approach equilibrium, they move to a final state that is more natural, so their level
of order increases.”
Do you agree with this statement?
Does this statement agree with the second law of thermodynamics?
Adapted with permission from Michael Loverude
Physics 535
ENTROPY
3. An ice cube with a mass of 30.0 g is dropped into a beaker of water. The initial temperature of the ice
cube is 0° C and the initial temperature of the water in the beaker is 20° C. The heat capacity of the water
in the beaker is 2000 J/K. The latent heat of fusion for water is 333 J/g. Assume that the system
consisting of the beaker and the cube does not thermally interact with its surroundings.
A. Determine the equilibrium temperature of the system.
B. Find the changes in entropy for the following processes and carefully specify the system (or
subsystem) for which you found ΔS.
a. the melting of the ice cube: (Hint: Does the temperature of the ice change during this process?)
b. the cooling of the water as the ice cube is melting:
c. the heating of the melted ice to its final temperature:
d. the cooling of the warm water to its final temperature:
C. Find the total change in entropy during the process:
D. Would you expect the reverse of this process (i.e., 30 g of ice forms spontaneously from a beaker of
water at the final temperature) to occur? Is the reverse of this process forbidden by the first or second
law of thermodynamics?
Adapted with permission from Michael Loverude
PHYS 535
– Thermo for Educators
ENGINES – HOMEWORK Name
1.Consider the cycle made up of isochores and isobars from the tutorial. Assume that the working substance
for the engine is monoatomic ideal gas and that the pressures and volumes are as shown (V2 = 3V4, P1 =
2P3). Express all results below in terms of P0 and V0.
P
A. Determine the net work done in the cycle.
1
2P0
2
4
B. Use the heat capacity of a gas at constant pressure (CP) and that at
constant volume (CV) to determine the heat entering the gas during
processes 1 and 4.
P0
3
V0
3V0
Similarly, determine the heat leaving the gas during processes 2 and 3.
C. Use your results for the previous parts to determine the efficiency of this engine.
Compare the efficiency you found to that of an engine based on the Carnot cycle in which the high
and low temperatures are the same as the highest and lowest temperatures in the cycle above,
respectively.
Adapted with permission from Michael Loverude
V
2. For each of the questions below, a diagram is shown to represent a proposed design for an engine or
refrigerator. For each such device, state whether the device in the diagram could work as shown and
explain why or why not.
A. The diagram shown, with QH = 50 J, QL = 10 J, and W = 40 J per
cycle
T H = 800 K
|QH|= 50 J
|W|= 40 J
|QL |= 10 J
T L= 400 K
B. A device similar to the above, but with QH = 50 J , QL = 35 J, W = 25 J per cycle
TH= 800 K
QH= 50 J
W= 25 J
QL= 35 J
TL= 400 K
C. The device shown at right, with QH = 40 J, QL = 20 J, and W = 60 J
TH= 800 K
QH= 40 J
W= 60 J
QL= 20 J
TL= 400 K
PHYS 535
– Thermo for Educators
ENGINES – HOMEWORK Name
1.Consider the cycle made up of isochores and isobars from the tutorial. Assume that the working substance
for the engine is monoatomic ideal gas and that the pressures and volumes are as shown (V2 = 3V4, P1 =
2P3). Express all results below in terms of P0 and V0.
P
A. Determine the net work done in the cycle.
1
2P0
2
4
B. Use the heat capacity of a gas at constant pressure (CP) and that at
constant volume (CV) to determine the heat entering the gas during
processes 1 and 4.
P0
3
V0
3V0
Similarly, determine the heat leaving the gas during processes 2 and 3.
C. Use your results for the previous parts to determine the efficiency of this engine.
Compare the efficiency you found to that of an engine based on the Carnot cycle in which the high
and low temperatures are the same as the highest and lowest temperatures in the cycle above,
respectively.
Adapted with permission from Michael Loverude
V
2. For each of the questions below, a diagram is shown to represent a proposed design for an engine or
refrigerator. For each such device, state whether the device in the diagram could work as shown and
explain why or why not.
A. The diagram shown, with QH = 50 J, QL = 10 J, and W = 40 J per
cycle
T H = 800 K
|QH|= 50 J
|W|= 40 J
|QL |= 10 J
T L= 400 K
B. A device similar to the above, but with QH = 50 J , QL = 35 J, W = 25 J per cycle
TH= 800 K
QH= 50 J
W= 25 J
QL= 35 J
TL= 400 K
C. The device shown at right, with QH = 40 J, QL = 20 J, and W = 60 J
TH= 800 K
QH= 40 J
W= 60 J
QL= 20 J
TL= 400 K
PHYSICAL REVIEW SPECIAL TOPICS—PHYSICS EDUCATION RESEARCH 11, 020118 (2015)
Identifying student resources in reasoning about entropy and the approach
to thermal equilibrium
Michael Loverude
Department of Physics and Catalyst Center, California State University Fullerton,
Fullerton, California 92834, USA
(Received 30 September 2014; published 23 September 2015)
[This paper is part of the Focused Collection on Upper Division Physics Courses.] As part of an ongoing
project to examine student learning in upper-division courses in thermal and statistical physics, we have
examined student reasoning about entropy and the second law of thermodynamics. We have examined
reasoning in terms of heat transfer, entropy maximization, and statistical treatments of multiplicity and
probability. In this paper, we describe student responses in interviews focused on the approach of
macroscopic systems to thermal equilibrium. Our data suggest that students do not use a single simple
model of entropy, but rather use a variety of conceptual resources. Individual students frequently shifted
between resources, in some cases leading to contradictory predictions. Among the resources that students
employed were some that have been previously described in the literature, including inappropriate use of
conservation. However, our results suggest that student use of resources connected to disorder are neither
simple nor monolithic. For example, many students used a previously unreported association between the
equilibrium state of a system and an increase in order, rather than disorder.
DOI: 10.1103/PhysRevSTPER.11.020118
PACS numbers: 01.40.Fk, 05.90.+m
I. INTRODUCTION
Entropy is a fundamental concept in the physical sciences
and a core idea of thermodynamics and statistical physics.
As part of a broader and ongoing project to investigate
student learning and develop curricular materials in thermal
physics, we have investigated student learning of entropy
and the approach to thermal equilibrium.
Entropy is widely known to be a difficult topic for
advanced students as well as introductory students.
Numerous instructors and authors have suggested methods
of teaching entropy, qualitatively and quantitatively, and
critiqued existing teaching of the ideas. Until fairly
recently, the research base on student understanding of
entropy has been fairly limited, and there has furthermore
been relatively little in the way of attempts to bridge
between the teaching methods and the research.
A. Previous research
Although entropy is widely considered to be challenging, there is relatively little previous research, particularly
for upper-division undergraduates. Prior studies by Duit
and Kesidou [1] and Bucy [2] helped to characterize
student conceptions of entropy and the second law of
thermodynamics; each will be touched upon further in
Sec. III C. More recently, several have examined student
Published by the American Physical Society under the terms of
the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and
the published article’s title, journal citation, and DOI.
1554-9178=15=11(2)=020118(14)
models of entropy in terms of conceptual metaphors [3].
Cochran and Heron investigated student understanding in
the context of heat engines and described several student
difficulties with entropy and the second law of thermodynamics [4]. Daane studied the role of energy degradation in
courses for preservice teachers focusing on energy models
[5]. Additional research in introductory physics courses for
life science majors has probed student resources for entropy
and spontaneity and has examined similar questions to our
study [6]. Probably the most relevant previous work for
this paper is that of Christensen et al. [7]. In that study,
Christensen et al. reported several common conceptual
difficulties with entropy, including a strong tendency of
students to conserve entropy inappropriately, and a tendency to assume that the entropy of all systems must
increase, whether or not the system is isolated. Another
recent study has used tasks adapted from [7] to probe
student understanding of entropy [8].
In addition to the systematic study of student learning,
many have taken thoughtful positions on the most appropriate ways of teaching entropy. In particular, several
critiques have questioned the notion of entropy as a
“measure of disorder” as imprecise and potentially misleading [9]. Within the domain of physical chemistry,
Lambert has released an influential series of articles of
characterizing entropy as disorder, characterizing the
notion as a “cracked crutch,” and has advocated teaching
entropy as a measure of the dispersal of energy in physical
space [10]. Further, Lambert criticizes the use of macroscopic examples like a messy dorm room or a shuffled deck
of cards. Leff has written extensively on the teaching of
020118-1
Published by the American Physical Society
MICHAEL LOVERUDE
PHYS. REV. ST PHYS. EDUC. RES 11, 020118 (2015)
entropy, including a recent and influential five-part series
covering many aspects of the concept and its application
in physics [11]. He similarly rejects the idea of entropy as
a measure of disorder as “an undesirable simplification
of a profound physical entity.” He proposes instead that
entropy is best thought of in terms either of missing
information or of “equity” or “spreading.” The latter notion,
he suggests, is essentially the same as Lambert’s preferred
idea of “dispersal.”
In recent years, many have proposed a revised instructional approach for upper-division courses on thermal and
statistical physics. This approach, often described as
“thermal physics” to make a distinction from classical
thermodynamics and statistical physics, builds up the
second law of thermodynamics as a consequence of the
statistical behavior of matter [12]. Several textbooks have
adopted this approach, including some for introductory
level courses [13] and the text for the courses in this study
[14]. We describe this approach in detail in Sec. II A.
other courses or forget the material entirely. As a result, the
tasks that we have chosen are fairly simple and do not
require extensive computation; rather, they focus on the key
ideas of thermal equilibrium and the first and second laws
of thermodynamics.
The research questions that we have considered include
the following:
• To what extent do student responses to entropy
questions suggest a consistent and coherent model,
and to what extent do responses reflect changing and
context-dependent thinking as characterized by a
resources model?
• What resources and reasoning patterns do students
access in responding to simple conceptual problems
involving entropy, the second law of thermodynamics,
and the approach to thermal equilibrium?
B. Theoretical perspectives
The current study proceeds from the assumption that
students construct understanding of scientific phenomena,
in some cases developing ideas that are in contrast with
accepted scientific viewpoints. The work is primarily
empirical and has been directed toward improving student
learning in a typical classroom setting, so we started from
the pragmatic framework of “investigating student difficulties” [15]. Despite this name, our intention was not
simply to identify specific difficulties, but rather to characterize student thinking and reasoning patterns, productive
as well as unproductive. However, as the project progressed, it became clear that student responses frequently
did not suggest stable conceptual difficulties. Reasoning
elements, such as the idea of conservation, could be
productive in some contexts and questions but lead to
incorrect predictions in others. Individual students, rather
than employing a single model repeatedly in a variety of
contexts, shifted between ideas frequently. These data
suggested the need for a resources, or knowledge-in-pieces,
perspective [16].
C. Research questions
For this paper we focus on a quite narrow portion of
the project. We restrict our focus to student predictions
and reasoning in the context of the approach to thermal
equilibrium, processes in which objects of different temperature converge to a single equilibrium temperature when
placed in thermal contact. In addition, we have chosen to
study students not during the thermal physics course in
which they first learned this material but a year or more
afterward. By performing the interviews well after instruction, we believe we have a sense of the lasting conceptual
understanding that remains after students have had the time
and opportunity to either integrate their understanding with
The goal of this portion of the project is not immediately
directed toward the development of curriculum or
approaches to teaching, though we might expect that the
findings of the work would have implications for instruction. Rather, we seek to characterize how students think
about entropy and the approach to thermal equilibrium.
II. CONTEXT AND METHODS
A. Instructional context of the work
The instructional context for this work is a variety of
upper-division courses covering thermal physics. The
primary context is a thermal physics course at California
State University Fullerton (CSUF), a large comprehensive
institution in southern California. The course, Physics 310,
follows the hybrid thermal physics approach described
above, using a popular text [14] that develops the ideas of
entropy and the second law of thermodynamics through a
statistical approach. The course meets for two 75-minute
blocks per week. Enrollments have ranged between 6 and
19, and typically a significant portion of class time is spent
on small-group tutorial exercises, some of which have been
described in other publications [17].
Most students in the course are physics majors or minors
who have completed introductory physics and several semesters of calculus. The CSUF introductory physics sequence
does not include thermodynamics, but many students
reported studying thermal physics in high school (∼5%),
in introductory physics courses at other institutions (∼20%),
or in chemistry (∼50%). A few students (10%–20%) had
previously completed a college-level math course in probability and statistics.
As described above, the course text adopts the hybrid
thermal physics approach. This approach seeks to motivate
the second law of thermodynamics through an extended
logical sequence in which students consider statistical
models of phenomena and examine the behavior of these
models as the number of particles becomes large.
020118-2
IDENTIFYING STUDENT RESOURCES IN …
PHYS. REV. ST PHYS. EDUC. RES 11, 020118 (2015)
Students first consider probability for simple two-state
systems (like coins, which can be heads or tails). In this
context they consider the key ideas of microstate, macrostate, and multiplicity, and use the fundamental assumption
that all accessible microstates of a system are equally
probable. The students then determine a means of adapting
the expression for multiplicity in the coin system to count
states for the Einstein model of a solid. This model is then
applied to a system of two interacting Einstein solids.
Students are shown that the expected macroscopic outcome
(energy shared by the solids in proportion to their respective masses) is the energy arrangement with maximum
multiplicity, and thus probability. They then consider larger
and larger systems, and apply statistical techniques to
show that it is increasingly probable that the system is near
the classical equilibrium state. As the number of particles
approaches Avogadro’s number, the probability of the
system being far from that equilibrium state becomes
vanishingly small. Thus, if solids are placed together and
allowed to exchange energy, they tend to evolve toward a
maximum probability state, in which the average energy per
particle for each solid is equal. Students are then introduced
to the Boltzmann formulation of entropy that relates entropy
(S) to the natural logarithm of multiplicity Ω:
connect the macroscopic description of the second law
of thermodynamics to the statistical version taught earlier in
the course. The students then analyze cyclic processes like
those used in engines and refrigerators and use the laws of
thermodynamics to derive limits on the performance of
those devices.
It is important to note that while the statistical definition
of entropy precedes the macroscopic idea and calculations,
the two ideas are viewed as equally important. Indeed, the
course, text, and instructor spend considerable effort in
making connections between macroscopic and statistical
pictures. Course assessments include tasks that are both
statistical and macroscopic in nature, and indeed students
are given exam questions that ask them to transition fluidly
between these concepts.
Finally, it is important to disclose that the study author
was also the course instructor for the thermal physics
course for all students in the study. For this portion of the
project, students were asked to participate in interviews
only after completing the course, when the instructor would
have no further opportunities to assign student grades.
S ¼ kB ln Ω:
After this initial statistical introduction, entropy is later
introduced as a macroscopic quantity. Students are taught
what is sometimes known as the Clausius algorithm,
relating macroscopic entropy changes dS to heat transfer:
dS ¼
dQrev
:
T
They perform qualitative and quantitative analyses of a
variety of systems using this algorithm, often involving
integration. One touchstone process that students examine
is the approach of a pair of macroscopic blocks to thermal
equilibrium, in which the heat transfers are equal and
opposite but the greater T of the hotter block leads to the
smaller absolute value of ΔS. Through this analysis,
students are led again to see that the entropy of the
Universe must increase in irreversible processes, and
TABLE I.
B. Methods
In the broader project of which this study is a part, we
have sought to document student understanding of the
target ideas using written conceptual questions and student
interviews. In this portion of the study, we describe student
responses from a set of interviews. We chose to study the
responses of students a year or more after completion of the
thermal physics course to probe the longer-term effects of
the instructional approach.
We interviewed individual volunteers, most of whom
had completed the course 1–2 years prior to the interview.
Students were compensated with a gift card. The sample of
students included eight students: 4 male and 4 female. All
were physics majors, though several had a second major or
minor in another discipline. Student ethnic identification
was two white, two Asian-American, and four Latino or
Latina. The interview participants’ course grades were
roughly representative of the course as a whole: 1 A,
3 B, 4 C. In the text below, students are identified with
pseudonyms or, in the context of student quotes, with a
one- or two-letter abbreviation as indicated in Table I.
Student pseudonyms and abbreviations.
Student pseudonym, abbreviation
Demographics
Calliope, C
Darius, D
Falcata, F
Gladius, G
Hecate, H
Jason, Ja
Jocasta, Jo
Pericles, P
Female, Latina
Male, Latino
Male, Latino
Female, Latina
Female, White
Male, White
Female, Asian
Male, Asian
020118-3
Time since completing course
1
6
1
2
2
2
2
1
yr
yr
yr
yr
yr
yr
yr
yr
MICHAEL LOVERUDE
1.
PHYS. REV. ST PHYS. EDUC. RES 11, 020118 (2015)
For each part below, two identical blocks are placed in thermal contact
and isolated from the rest of the universe. The initial temperatures of
the two blocks and proposed final temperatures are shown. For each
pair of states, state whether the transition between initial and final states
is possible, and explain why or why not.
A.
B.
C.
D.
E.
300 K 300 K
320 K 280 K
initial
final
360 K 300 K
320 K 340 K
initial
final
360 K 300 K
340 K 320 K
initial
final
340 K 280 K
300 K 300 K
initial
final
275 K 325 K
300 K 300 K
initial
final
FIG. 2 Second page of the interview prompt. Adapted from
Christensen et al. [7].
FIG. 1. The first page of the interview task. Students were asked
to respond to the questions and explain their thinking. For the first
set of four interviews, only three parts (A, D, and E) were used.
In the interviews, students were given a sheet of written
questions and asked to answer the questions while thinking
aloud. The interviewer followed a protocol that included
explicit follow-up questions asked of all students but
provided latitude to probe student responses further.
The first set of tasks involved pairs of blocks (see
Fig. 1.) Students were given the temperatures of blocks in
the initial and final states and asked to identify which of
the situations were physically possible and explain why.
For the first four interviews, only situations A, D, and E
were considered. The other situations were added for
the next four interviews at the suggestion of instructors
and a second researcher, to investigate how students
would think about a system that “overshoots” equilibrium
or one that does not fully reach equilibrium. In case C,
the intention was that the system would evolve to the
final state and stop. Our emphasis in the analysis below
is on situations A, D, and E, which were posed to all
students.
The interviewer prompted the students to give explanations but for the most part did not probe these explanations
immediately. After students had explained their responses,
they were directed to the back of the sheet, which included
the “general-context” problem from Christensen et al. [7],
shown in Fig. 2.
After the students had answered the abstract context
question, the interviewer probed the student reasoning.
After this discussion, the students were redirected to the
front page, and the interviewer asked explicitly whether the
students had thought about energy, entropy, and multiplicity or probability, and how those concepts would apply to
the problems.
Correct answers to the problems could be arrived at in a
number of ways. The simplest formulation is to recognize
that the processes must satisfy the first and second laws of
thermodynamics. As the blocks are isolated from the rest of
the Universe, the energy lost by one block must be gained
by the other; as the blocks are identical, their temperature
changes should be equal and opposite. All but one of the
five processes conserve energy and satisfy the first law.
However, one case involves a process that is the reverse of
the approach to thermal equilibrium: case A starts with
blocks at the same temperature of 300 K and ends with one
block at 320 K and the other at 280 K. The change in
entropy of the left block is positive, that of the right block,
negative. However, by considering the Clausius algorithm
we can see that the absolute value of the change for the left
block is smaller than that of the right block, and the process
results in a net decrease in entropy of the two-block system,
violating the second law of thermodynamics.
C. Data analysis
In analyzing student interview responses, a full transcript
was produced. The researchers then coded each student
sentence based on its physics content. Rather than focusing
on whether responses were correct or incorrect, each
sentence was coded as corresponding to one or more
common recurring ideas. While this project originally
was characterized by a student difficulties perspective, it
became clear that student responses were not consistent
with students repeatedly employing a single model in their
responses. Rather, students shifted between ideas, employing multiple ideas throughout the interview in response to
020118-4
IDENTIFYING STUDENT RESOURCES IN …
TABLE II.
PHYS. REV. ST PHYS. EDUC. RES 11, 020118 (2015)
Entropy-related resources coded and examples.
Entropy-related resources
Entropy is a form of energy
Entropy is conserved
Entropy of a system must always increase
Entropy is a measure of disorder
Entropy is related to multiplicity
Entropy describes dissipation
Entropy relates to dispersal
ΔS must be positive (or ≥0)
Entropy is related to enthalpy
TABLE III.
Example student utterance
I want to say that entropy is a form of energy, but I don’t remember exactly.
Entropy you can’t just, I can’t really remember, but I want to say, it’s like
energy, like you can’t create it or destroy it.
Yes, because even though … one is increasing in temperature, one is decreasing
in temperature, they are both moving towards the maximum number of
microstates.
It’s just weird because whenever I think of entropy I just think of more disorder.
[entropy will] increase in that it has the most possible number of, um, micro or
macrostates.
Entropy is a more general way of thinking about energy. … To, um, say a higher
potential of energy wants to dissipate to a lower potential of energy.
…like the energy’s being dispersed more evenly, and that, would yield an
increase in entropy
…the combination of the two blocks, [entropy] is increasing or remaining the
same
it’s almost like they’re interchangeable, delta H and delta S
Equilibrium-related resources coded and examples.
Equilibrium-related resources
Systems naturally move “toward” equilibrium rather
than away
Equilibrium decreases multiplicity
Application of energy can oppose equilibrium
Equilibrium corresponds to more order
Equilibrium maximizes multiplicity or probability
Which makes me think this was not possible, just because systems want to
be in a state of more equilibrium.
Temperatures going away from thermal equilibrium, I’m just thinking of
that as decreasing multiplicity.
it takes energy to keep it that way [away from equilibrium], then it
naturally goes back, spontaneously
it’s going from a state of disorder to order, to some sort of order, so that’s
why I would assume, there’s some kind of order in equilibrium
It’s that bell-ish curve, it’s that hump where the most, most macrostates?
occur. And it will always move towards that central peak.
different prompts. In many cases the ideas were conflicting
and even contradictory, even within a single response.
Students frequently recognized that they had contradicted
themselves and struggled to resolve the contradiction.
Thus, our analysis focused on the characterization of
student ideas using smaller elements of knowledge.
The appropriate grain size of these smaller knowledge
elements bears some discussion. The ideas we have
documented are typically more than diSessa’s p-prims
[18], as they are connected to specific physics concepts
and frequently span multiple sentences. These ideas might
best be described as facets, as they were not complete
concepts but rather partial concepts that were more
descriptive of aspects of a situation or problem [19]. In
any event, our goal is not to make particular claims about
the appropriate grain size, and in the subsequent discussion
we refer to these student ideas using the somewhat agnostic
term “resource” as used by Hammer [16]. In Tables II
and III we give examples of the resources that we identified
and examples of student utterances that exemplify these
resources. Some of these resembled canonically “correct”
physics ideas (“entropy is related to multiplicity”), but
others did not (“entropy is a form of energy”). The ideas as
coded are broken roughly into resources related to entropy
and those related to thermal equilibrium, though there are
examples of utterances that cross this boundary. Given the
limited data set, we do not claim this is a comprehensive set
of student resources, but it does represent the resources
coded within this interview sample.
III. CHARACTERIZATION OF
STUDENT RESPONSES
A. Entropy and multiplicity
In this sample, the connection between entropy and
multiplicity was not strong for most students. Two of the
students expressed strong preferences for multiplicity over
other ways of thinking about entropy. However, one of
these students, who used microstates to anchor her prediction about entropy, acknowledged that she often could
not apply the idea of multiplicity or distinguish between
microstates and macrostates. (In a related article, we
020118-5
MICHAEL LOVERUDE
PHYS. REV. ST PHYS. EDUC. RES 11, 020118 (2015)
describe in more detail student reasoning difficulties with
the concepts of multiplicity, microstate, and macrostate
[20].) The other students in the sample expressed either
discomfort with this connection or outright confusion.
Many of the students cited S ¼ k ln Ω, but the majority
of students did not articulate strong connections between
multiplicity and the equilibrium-seeking behavior they
were describing. For example, Jocasta seemed to connect
multiplicity (and probability) with a notion of plausibility,
rather than a quantity describing microstate:
which entropy is first introduced macroscopically, it is
impossible to say whether this is particular to the thermal
physics approach or a characteristic of students thinking
more broadly. Although many of the tasks in the interview
could be answered without referring to the Clausius
algorithm, the macroscopic analysis of the two-blocks
tasks in terms of the second law of thermodynamics
requires this step.
Jo: Not really, um, multiplicity the way I remember it, it’s
like can it happen? It’s like flipping a coin, each time
it happens, it’s random, it’s not influenced by the
other time it’s flipped. I didn’t really think about that.
On the other hand, there was one student in the sample
for whom the statistical idea of entropy as being connected
to multiplicity was very useful:
Ja: I’d never heard entropy described in a multiplicity
sense, like a statistical sense. It made so much more
sense, because every time you hear disorder, and I
hear that from people who, like, don’t know anything
about physics: “It’s just disorder.”… But I don’t
think they know what that means. I never knew what
that meant.
His unhappiness with the idea of order may have related
to an issue described further in Sec. III C 2, as he stated that
he associated the equilibrium state with more order (or, at
least, with “more orderly”) rather than more disorder:
Ja: I feel like it [equilibrium state] would be more
orderly, if anything. … Because, it feels like everything’s moving to the most likely position because,
that’s just how it is.
While it was a key part of the arguments given by Hecate
and Jason, the formulation of entropy as a statistical quantity
was not productive for most of the students. That is
particularly notable given the fact that Boltzmann formulation of statistical entropy was a major focus of the course
and the instructional approach.
B. Macroscopic entropy
One key physics idea was notable for its absence in
student responses: none of the students gave responses to
the entropy changes using the macroscopic idea that
dS ¼
dQrev
T
(sometimes referred to as the Clausius algorithm). It is
possible that the lack of references to this idea reflects
the emphasis of the thermal physics approach, in which
entropy and the second law of thermodynamics are
introduced in terms of statistical behavior. Without studies
of students in a traditional thermodynamics course, in
1. Difficulty relating entropy changes to
temperature changes
After students had completed the abstract-context task
on the back page, they were asked to describe the
usefulness of entropy in thinking about the pairs of blocks
on the first page. In most cases, the interviewer chose one of
the situations and asked specifically what would happen
to the entropy of each block and to the entropy of the twoblock system. This task was simple for some students,
but others struggled. Even students with a largely sound
understanding did not typically reason in terms of heat
transfers and temperatures. Hecate mistakenly predicted
that both blocks in part A would decrease in entropy,
without referring to the heat transfers:
H: That would mean that the entropy of both blocks in A
would be decreasing, because they are moving away
from the stable equilibrium state.
Jason, who made many correct predictions and articulated the connection between entropy and multiplicity,
struggled when asked about the entropy changes of
individual blocks:
I: What can you say about the entropy of the left block
and the right block?
Ja: The left block is increasing its entropy.
I: How about the right block.
Ja: I think it’s decreasing…?
I: But you’re not sure?
Ja: But I’m not sure. [laughs]
I: How come?
Ja: I’m trying to remember how, um, entropy is related
with temperature. So I’m not really, um, sure
about that.
The interviewer tried a second task, and asked Jason
about the changes in entropies of the two blocks in part B:
I: If I thought about the entropy of the two pieces there,
what could I say?
Ja: Individually? [laughs] I don’t really know individually. But as a whole, I would say the entropy is
decreased. But individually, I think, that’s kind of
wracking my brain right now.
Later Jason expressed confusion when asked to compare
macroscopic and statistical models, and explicitly stated
that the probabilistic interpretation was vastly preferable
to him:
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Ja: every time somebody asked me for the definition, I
always think about it in that probabilistic or multiplicity kind of way. [Darius] was doing his chemistry
homework and I was looking at it that way.
The fact that the two students most aligned with a
statistical approach had difficulty with a very simple task
involving macroscopic entropy is certain cause for reflection and further research. Whether this is idiosyncratic or
reflects a deeper conflict between statistical and macroscopic pictures of energy remains an unanswered question.
2. Entropy must increase
A key finding of Christensen’s work was that students
tended to predict that the entropy of systems must increase,
regardless of whether the system in question was isolated.
In our study this tendency was also present, but did not
appear to be as widely applied. All eight of the students at
some point stated explicitly that entropy of some system
would increase. In three of the eight cases this was stated
carefully, as a physicist might state the second law of
thermodynamics: the change in entropy of the Universe
must be greater than or equal to zero. Other students spoke
more generally about the tendency of entropy to increase,
without specifying which system(s) were being described.
Three of the eight students used conservation reasoning for
entropy at one point while also stating that the entropy of an
individual (not necessarily isolated) system must increase.
The remaining two students primarily used conservation
reasoning.
3. Conservation of entropy
As has been reported in previous studies, several student
responses were consistent with the idea that entropy is a
conserved quantity. In some cases, students explicitly
related entropy to energy. Pericles, in responding to the
prompt, stated that, “I want to say that entropy is a form of
energy, but I don’t remember exactly.” After the interview
he suggested that he had thought that entropy was another
one of the thermodynamic potentials that are taught in the
course [21]. Other students described entropy as having
units of Joules, though in at least one case the association
was with enthalpy rather than entropy:
I: So this, the 5 J, is this the entropy you’re talking about?
D: That’s just the, that’s the change in uh, that’s delta H.
But. The reason why I say it’s the entropy, cause I
think somewhere we said in our chemistry class, it’s
almost like they’re interchangeable, delta H and delta
S. … I’m struggling to remember. I think they were
interchangeable. According to them. The chemists.
Another student, Calliope, was particularly explicit in
making the connection to energy and invoking conservation:
C: You can’t just, entropy … I can’t really remember, but
I want to say, it’s like energy, like you can’t create it
or destroy it, it can’t just come out of nowhere.
In the context of an energy discussion, she spoke more
generally about the value of conservation, suggesting that
she felt it was very broadly applicable:
C: […] Things have to be conserved. That’s like the main
thing they teach us, since like middle school or high
school. Like, things just don’t appear out of nowhere,
they have to come from somewhere.
In response to the Christensen task, Jocasta used very
explicit conservation reasoning and appeared to equate
energy and entropy:
I: You said the entropy of the left box would go up, and
the entropy of the right box would go down, then you
said something about the total being, or the average,
being the same.
Jo: The sum of them would be the same.
I: And how do you know that would be the case?
Jo: [quickly] conservation of energy.
4. Entropy as a description of dispersal
As noted above, several have recently advocated teaching entropy in terms of ideas like spreading or dispersal.
The textbook, lectures, and curricular materials in this
course did not explicitly introduce the idea of entropy
using either approach, but a few of the students employed
resources of this nature nevertheless. For example, Jason
in rejecting the notion of entropy as disorder explicitly
used the phrase “dispersed more evenly” to describe the
approach to equilibrium.
Ja: The way I think about entropy is more the statistical
sense, we always talked about a measure of disorder,
but that’s sort of inaccurate, it’s more like it’s more
orderly in the sense that everything’s fitting into its
corresponding slot, like the energy’s being dispersed
more evenly, and that, would yield an increase in
entropy.
Another student, Pericles, used what might be a similar
notion, that of dissipation. Based on the way Pericles
describes it, dissipation seems to mean the same as
dispersal, i.e., spreading through space.
P: Entropy is a more general way of thinking about
energy. But it’s more broad in a sense. Because you
have exchange of energy, you have more energy that
wants to dissipate. To, um, say a higher potential of
energy wants to dissipate to a lower potential of
energy.
However, Pericles did also use what we coded as a
conservation resource when talking about entropy, and (as
noted above) described entropy as a “form of energy.” In
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the Christensen task he predicted that the total entropy of
system plus surroundings must remain the same.
While our small sample does not permit judgment that
this model is more or less useful than any other, it does
suggest that for some students at least the notion of a
quantitative measure of dispersal is a resource upon which
instructors might build. However, some caution should be
applied, as most of the students did not refer to dispersal or
spreading when describing their thinking about entropy,
and one of those that did apparently still thought of entropy
as a conserved quantity. Even in Jason’s quote, the dispersal
resource is used in connection with a discussion of whether
a system is “orderly.”
as a state, that a system could be in disorder, rather than a
semiquantitative construct that might increase or decrease:
C. Entropy and disorder
As noted above, the characterization of entropy as a
measure of disorder is a controversial one. Nevertheless, the
association of entropy with order and disorder is seemingly
inescapable. Of the students in this sample, only one did not
introduce the idea of disorder when speaking about entropy.
Two of the students who spoke about disorder did so
primarily as a disclaimer, stating that they preferred other
ways of thinking about entropy. The remainder of the
students used disorder frequently and spontaneously.
Many have criticized teaching entropy as a measure of
disorder, but research on the question is limited. Kesidou
and Duit [1] note that students use this notion readily,
but they claim that students have insufficient particulate
understanding to fully understand it. Bucy [2] came to a
similar conclusion and noted that students use the term in
disparate ways.
1. Entropy as disorder is a more subtle idea
than one might think
While the association of entropy with order and disorder
was widespread, even within this fairly small sample, the
use of disorder took a number of forms. In their responses,
students used a variety of phrasings to describe systems and
their level of order or disorder. Their phrasing suggested
that in some cases order was a quantity, but in others it was
a description of a state, or even a state itself:
disorder is a little more natural
that would be disorder
you would be changing it toward disorder
the system is getting less ordered
it would be more disordered
it’s still disordered
equilibrium state is more disorder
from a state of disorder to order
it would go toward the most disorderly state
A fairly dramatic example comes from the responses of
Gladius. In her analysis, Gladius frequently referred to
disorder. However, she used the term in at least two
different ways. Early on, she seemed to refer to disorder
G: You can’t have this go below this; That would be
disorder.
G: I don’t think that would happen, just because it’s
disorder.
Later, her language shifts subtly, to refer to “more
disorder,” suggesting disorder as a quantity rather than a
state. Note that her usage is a hybrid: the final state is
more disorder rather than has more disorder or is more
disordered.
G: The final state is more disorder.
As the interview progresses, she shifts to a more standard
usage; this may be in part due to inadvertent cuing from the
interviewer:
G: Because I said this one would be more disordered, so
this one should have more entropy.
Calliope frequently invoked the macroscopic notion of
“messiness” as a proxy for entropy, but did not speak of
more or less disorder. Rather, she referred to a room as
disordered, and referred to an association of entropy and
order or disorder, but in a way that suggests order or
disorder as states rather than measurable quantities:
…disorder is a little more natural.
If we say that disorder goes with entropy…
… unless entropy comes with order.
…if entropy comes with disorder…
Falcata used the idea of disorder and order extensively.
While he generally spoke of a state being “more disordered” or “less orderly,” he also appeared to use terms in a
manner similar to Gladius:
F: The equilibrium state is more disorder…
F: If it started off in its equilibrium state, that would be
its ordered state.
2. Some students have unexpected associations
of entropy and order
In our study, there was evidence to suggest a previously
unreported association with entropy. Four of the eight
students in this sample gave responses suggesting an
association of the equilibrium state of a system with greater
order rather than disorder, and in some cases greater
entropy.
One student (Jason) seemed to reject the relationship of
entropy to disorder based on the notion that an equilibrium
state seemed more orderly:
Ja: I think the system has to be increasing in entropy. The
way I think about entropy is more the statistical
sense, we always talked about a measure of disorder,
but that’s sort of inaccurate. It’s more like it’s more
orderly in the sense that everything’s fitting into its
corresponding slot.
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Upon probing by the interviewer, he went on to confirm
that he intended to say that the equilibrium state would be
more “orderly”:
C: It doesn’t sound right, it doesn’t sound right, that
[the blocks would] be at a high-entropy state, when
they’re in their natural state.
A fourth student stated that the initial state of two blocks
with different temperatures (case E from Fig. 1) would have
more disorder than the state in which the blocks have
reached thermal equilibrium:
Ja: I feel like it would be more orderly, if anything. And
not just because you said that! Because, it feels like
everything’s moving to the most likely position
because, that’s just how it is.
The statement “not just because you said that” gave the
interviewer pause; it was not clear whether this was a
reference to a statement earlier in the interview or a
statement in class. To be certain of the student’s intent,
the interviewer probed further, and the student emphatically
agreed:
I: And correct me if I am wrong, because I don’t want to
put words in your mouth, but it sounds like you’re
saying that the most probable state seems more
orderly to you, as opposed to more disorderly.
Ja: [student interjects] Yes!
I: Is that what you mean?
Ja: Yes, yes. Cuz that’s the way the world kind of works,
stuff tends to follow probability, that’s why they are
probabilities.
This student seemed to have integrated the relationship
between entropy, equilibrium, and order into a relatively
correct understanding by rejecting the association of
entropy with disorder. Other students did not resolve this
confusion in the same way. When asked whether entropy
would help to understand which two-block processes were
possible, Jocasta said:
Jo: It does help, seeing that it went from more entropy to
less entropy… in a state of more equilibrium.
Note that an accepted explanation of the evolution of a
system to thermal equilibrium is that the entropy is greater
in the final state. This student seems to say otherwise, and a
follow-up question both confirmed this and revealed
confusion between entropy, order, and equilibrium.
Jo: It’s just weird because whenever I think of entropy I
just think of more disorder, and so whenever I look at
this, it’s going from a state of disorder to order, to
some sort of order, so that’s why I would assume,
there’s some kind of order in equilibrium.
Another student, Calliope, brought up the idea of entropy
and her messy room, but seemed uncertain whether to
associate entropy with disorder or with order:
C: If entropy relates to disorder, then I guess a
disordered… If we say that disorder goes with
entropy, then my room would have more entropy,
unless entropy comes with order. Hm.
These student responses seem to indicate that there is a
conflict for some students between the association of
entropy and disorder and the association of the equilibrium
state with higher disorder and (or) entropy:
G: I think this one has more disorder. The initial state.
Because it’s not happy, it’s not where it wants to be.
Whereas, the final state, you know, everything is
equal, it’s kind of like a, like a stable, environment.
A similar association of equilibrium with order is
reported in the recent study by Geller.
3. Avoiding “entropy as disorder” does not prevent
students from employing the resource
The course in the study did not teach entropy in terms
of disorder, and the instructor (and study author) indeed
avoided the term entirely except when cautioning students
that disorder can be imprecise, perhaps even misleading
and ambiguous. The course textbook similarly avoided
entropy as disorder, and gave an example intended to
illustrate its ambiguity. Despite these efforts, nearly all of
the students in the interviews referred to disorder at some
point during the interviews. Only one student did not
mention disorder; two of the eight mentioned it but
primarily to state their dislike of the idea. For the other
five students, “entropy is a measure of disorder” was a
resource employed at least occasionally in solving the
interview tasks. It is clear that actively avoiding the term
disorder and cautioning students about it does not prevent
them from reasoning with it. The recent work by Geller [6]
also concludes that the prior experience of students with
notions of disorder is nearly inescapable.
At the same time, many of the students who used entropy
as disorder extensively came to a point in the interview
when they recognized some ambiguity in their usage.
Several of these examples came up in the context of the
association of order with equilibrium, above:
C: If we say that disorder goes with entropy, then my
room would have more entropy, unless entropy
comes with order. Hm.
In addition to the examples from the previous section,
Falcata recognized the ambiguity in his usage:
F: So, what I’m thinking now is that, well, ordered, um,
I’m using it very vaguely. And it would be helpful if I’d
kind of stick to something, um. I am thinking back to
an example of marbles in a box, shaking them up, if
they were organized by color, or had numbers on
them or something.
F: When I meant that that was its more ordered state, I
meant was, if you changed it. You would be changing
it toward disorder. So I guess I wasn’t really clear on
what I meant by order and disorder.
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IV. CONSISTENCY OF STUDENT RESPONSES
Previous reports on student understanding of entropy
have characterized student responses in terms of common
incorrect answers. Although there were a few strong ideas
that were used by multiple students, perhaps the most
striking feature of student responses was the lack of
consistency. Of the students in the sample, most did not
repeatedly and consistently employ a small set of the
resources described above. In contrast, we found in our
analysis that students in giving explanations would shift
between ideas similar to previously reported common
incorrect answers and canonically correct physics ideas.
For example, a student might in one statement articulate the
idea that entropy must increase, and then later state that
entropy cannot be created nor destroyed. In the discussion
below, we give three somewhat extended examples of
student responses that illustrate the extent to which students
shifted among the resources described above.
A. Example 1: Calliope
Calliope’s responses throughout the interview were
particularly striking, as she made repeated use of several
distinct, and often contradictory, resources in constructing
her responses to the questions. Her explanations bounced
between conservation, entropy as disorder, equilibrium is
ordered, and the idea that equilibrium (and entropy)
opposes energy.
Unlike many students, Calliope introduced the idea
of entropy early on, though she struggled to remember
whether the idea she was trying to express was associated
with the term entropy or enthalpy. (She had completed
the second semester of general chemistry just before the
interview and the thermal physics course a year earlier,
which may help to explain this phenomenon.) She invoked
a common metaphor for entropy, the “messy room.”
C: I can’t remember the difference between entropy and
enthalpy. I have to look through the definition. Was
entropy where everything was, wanted to go back to
where it should be?
I just remember this, I don’t know if we talked about it
in this class, just explaining to your mom why your
room is messy, and you can say it’s because of a word
from thermo, but is it entropy or enthalpy?
In response to the Christensen questions, Calliope gave
explanations consistent with the idea that entropy is a
conserved quantity.
C: This one [sum of system and surroundings] remains
the same. And I feel like if this one increases, if the
system increases, then the surroundings decrease,
and if the system decreases then the system, it’s like
the opposite. If one increases the other decreases.
This response is highly suggestive of conservation
reasoning, and indeed Calliope later explicitly related the
behavior of entropy to that of energy, and was very explicit
in invoking conservation:
C: You can’t just, entropy you can’t just, I can’t really
remember, but I want to say, it’s like energy, like you
can’t create it or destroy it, it can’t just come out of
nowhere.
From this work as well as other studies in the thermal
physics project, we believe that for physics majors in
particular the idea of conservation is a strong one.
When confronted with new ideas like entropy or multiplicity, students frequently employ a conservation resource. Calliope talked about energy conservation later,
but her language suggested a very general approach to
conservation:
C: Things have to be conserved. That’s like the main
thing they teach us, since like middle school or high
school. Like, things just don’t appear out of nowhere,
they have to come from somewhere.
In responding to the Christensen questions, Calliope first
answered that the entropy of the system in part A must
increase, but then quickly changed her mind to say it
decreased. She struggled a bit with the abstract nature of the
task and returned to her example of the messy room:
C: When I think about these things, I think about them in
a situation. I need to picture something. So I think
about my room, clean, its natural state is dirty, so
would it increase or decrease….
However, while she was certain that the natural state of
her room was to be messy, she was unsure whether this
messy state corresponded to more or less entropy.
I: Does that correspond to more entropy or less entropy,
when your room is messy?
C: [Pause] I think less. I guess I’m relating entropy to
energy, I wanna, just really want to put those two
together. I’m sure they do go together, but I don’t
really remember exactly how.
However, she also seemed to make connections between
entropy and energy that suggested that she thought of them
as opposing equilibrium and (or) one another, which may
again be related to her recent experience in a general
chemistry course:
C: Well if we think about, if I think about a gas, and its
kind of forced to be a way, and it takes energy to keep
it that way, then it naturally goes back, spontaneously, so I’m gonna say it [entropy] decreased. There
was some sort of energy to make it be there in the
first place.
Calliope did not introduce the idea of order or disorder,
but the interviewer prompted her to clarify whether her
descriptions of the messy room were related to notions of
disorder. She associated disorder with the “natural” state of
a system. However, her connection between order or
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disorder and equilibrium shifted between conflicting
resources. By “natural” she appeared to be referring to
an equilibrium state, a relaxed state of a system. For a
moment she stated that the natural state was the more
disordered state, in agreement with the widespread (albeit
controversial) notion of entropy as disorder as macroscopic
disorganization.
C: To me, disorder is a little more natural. My room just
doesn’t want to stay clean. [laughs] It takes more
work to keep it in a certain state that it doesn’t really
want to be in.
The interviewer sensed that this answer may have
contradicted previous responses on entropy, and prompted
her to reconcile them. Calliope recognized the contradiction and struggled to reconcile her conflicting intuitions,
going back and forth several times:
I: In the example of your room…earlier you said the
messy room had less entropy, is that correct?
C: Hm. Hm. I just, [pauses] I guess if I think about it that
way, it would…I’m not really making sense, because
if I’m saying, entropy, or [pause] I, If entropy relates
to disorder, then I guess a disordered… if we say
that disorder goes with entropy, then my room would
have more entropy, unless entropy comes with
order. Hm.
She continued to attempt to reconcile the notions of
entropy, disorder, and equilibrium, using both canonical
and noncanonical ideas. She changed her mind and stated
that her intuition suggested the following:
C: Now I want to say it’s increasing. Before I was almost
certain that it was decreasing. And now that we’re
saying this, if we relate entropy to the state of being
messy, then it would be higher.
Rather than describing equilibrium, she continued referring to the natural state of a system:
C: Well, if entropy comes with disorder, and it goes to its
more natural state, then my room’s entropy would
increase, the messier it would get.
I: So your natural state, or a system’s natural state, is
that a high-entropy state or a low-entropy state?
C: Hm. [pause] I want to say, low. No! Well, if they’re
inversely proportional, then yes.
…
I: The natural state is, a high-entropy state?
C: It doesn’t sound right, it doesn’t sound right, that
they’d be at a high-entropy state, when they’re in
their natural state.
She switched back and forth several more times:
C: I can’t just apply every thing to my room, because
there are things that like in the natural state, they’re
like, the chairs are stacked together, and they’re right
the way they are, it doesn’t take any energy or
whatever. Geez, I’m just confusing myself.
I want to say when, like it doesn’t take any energy to
keep it, but if they’re inversely proportional it would
be at a high entropy. No that doesn’t sound right. I’m
gonna go with low.
In the end, Calliope decided that the equilibrium state
would correspond to low entropy, adding, “You know when
I get out of here I’m going to look that up, right?”
B. Example 2: Gladius
Another student, Gladius, displayed a similar shift
between ideas. In response to the abstract-context question,
she gave a response that seemed to be consistent with the
second law of thermodynamics. However, her response
does state that the entropy of a system must increase, rather
than the entropy of the Universe, as previously identified by
Christensen:
G: If it’s spontaneous, I think, for the first one, the
entropy of the system, it’s naturally occurring, I think
[faintly]. I want to say it’s increasing.
In her analysis, Gladius frequently referred to disorder.
However, she used the term in different ways during the
course of the interview. At one point she seemed to refer to
disorder as a state, that a system could be in disorder, rather
than a semiquantitative construct that might increase or
decrease:
G: You can’t have this go below this; That would be
disorder.
G: I don’t think that would happen just because it’s
disorder.
Later, her language shifts subtly, to refer to “more
disorder,” suggesting disorder as a quantity rather than a
state. Note that her usage is a hybrid: the final state is more
disorder rather than has more disorder.
G: The final state is more disorder.
As the interview progresses, she shifts to a more standard
usage; this may be in part due to inadvertent cuing from
the interviewer (see the question posed below). In her
discussion of part E of the two-blocks question, she shifted
between three ideas. First, she gave the quote from the
previous section, indicating that the different temperatures
of the initial state corresponded to more disorder:
I: Which has more disorder overall?
G: I think this one has more disorder. The initial state.
Because it’s not happy, it’s not where it wants to be.
Whereas, the final state, you know, everything is
equal, it’s kind of like a, like a stable, environment.
Then, she said the initial and final states have the same
entropy, seemingly a conservation argument:
I: Which situation would you say has more entropy, the
initial or the final?
G: I want to say that they both have the same. Yeah.
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After the interview probed further, she gave an answer
suggesting the interpretation of entropy as a measure of
disorder, increasing as a system approaches equilibrium:
acknowledged that this might be inaccessible for introductory students:
I: Is that consistent with what you said about order or
disorder? Or should it be?
G: No it’s not.
Because I said this one would be more disordered, so
this one should have more entropy. But I wouldn’t
know how to describe that.
Note that in very short order she has used three ideas that
a physicist would identify as distinct, and contradictory,
with potentially three different conclusions about the
change in entropy of the system. But she seemed to settle
into the notion that entropy would decrease as the system
approached equilibrium:
G: Entropy would kind of equal out, that’s why I was
able to say, it would come to some kind of equilibrium. It would go from a higher entropy to a lower
entropy.
G: …but entropy does help, seeing that it went from, more
entropy to less entropy, in a state of more equilibrium.
The interviewer asked again to clarify whether the
equilibrium state would correspond to more or less entropy:
I: so is ‘more equilibrium’ associated with less entropy?
G: Yes? [laughs nervously]
G: It’s just weird because whenever I think of entropy I
just think of more disorder, and so whenever I look at
this, it’s going from a state of disorder to order, to
some sort of order, so that’s why I would assume,
there’s some kind of order in equilibrium.
In this case equilibrium is also a hybrid usage; “more
equilibrium” is suggestive of a continuous quantity rather
than a yes or no state. Such a usage may simply reflect a
shortening of the phrasing “closer to” or “further from”
equilibrium.
These two statements seem to be in conflict, but one
refers to disorder and one to entropy, so it is not clear
whether it represents a contradiction or a difficulty in
relating entropy to order or disorder. Therefore, a few
moments later, the interviewer probed for consistency, and
Gladius changed again to relate entropy to disorder, saying:
H: Disorder is generally a good way to introduce the
concept at a lower level.
I’m not gonna lie, S equals natural log of the states,
using a mathematical relationship with probability, is
not a good first introduction.
Disorder is an easy thing to, it’s something to point
us to.
Despite her stated preference for statistics, Hecate seemed
to struggle at times to reconcile statistical measures of a
system with the macroscopic questions about entropy and
the approach to equilibrium. Hecate sketched a graph of
multiplicity that was referred to frequently in class, and
correctly noted that the process would spontaneously evolve
toward the central peak, which corresponds to the classical
equilibrium state as well as a maximum of the multiplicity
of the combined system. However, she could not decide
whether this spontaneous evolution toward equilibrium was
consistent with an increase or decrease in entropy:
H: And it will always move towards that central peak.
Never away from it, as a natural spontaneous
process. What I don’t really remember is whether
that corresponds to an increase in entropy or a
decrease, just in terms of terminology.
When prompted by the interviewer what her intuition
suggested, she promptly responded that her intuition was
that the entropy would increase:
I: What’s your tummy tell you?
H: [promptly] increase. … Like that’s sort of my gut
instinct. [pauses] I can actually reason out why… I
can actually find reasons that it’s either one.
For this student, the statistical idea of entropy was an
anchoring idea that finally allowed her to resolve her
confusion, at least temporarily:
H: Increase. In that it has the most possible number of,
um, micro- or macrostates. Yeah, that makes more
sense now.
Although she appeared settled, the student apparently
continued to look for confirmation and over seven minutes
later she returned to this prediction:
G: I said this one would be more disordered, so this one
should have more entropy. But I wouldn’t know how
to describe that.
H: The entropy of the Universe has to be greater than or
equal to zero, which goes back and reasserts my…
[prior prediction of] increase.
Ultimately, she was unable to reconcile the confusion.
While the statistical idea seemed to help, she admitted
that several aspects of the statistical formulation remained
challenging to her:
C. Example 3: Hecate
Hecate stands in sharp contrast to most of the students
in the sample, as she did not choose to describe entropy
in terms of disorder, and indeed explicitly rejected the
notion. She preferred a statistical interpretation, though she
H: In class, I really struggled with, being comfortable
with, the definition of, application of, microstates
versus macrostates.
Aside from moving toward that most stable equilibrium point, I can’t really apply the multiplicities.
020118-12
IDENTIFYING STUDENT RESOURCES IN …
PHYS. REV. ST PHYS. EDUC. RES 11, 020118 (2015)
Each of these three students (Calliope, Gladius, and
Hecate) made reference to multiple resources at different
points in the interview, and none appeared to have a
completely correct understanding of all aspects of entropy.
While they were the most dramatic examples of use of
multiple resources, all of the students in the sample referred
to multiple resources.
uncommon, and even students who expressed a preference
for this idea had difficulty in using it to make certain
predictions. None of the students explicitly invoked the
Clausius algorithm for macroscopic entropy changes,
despite its emphasis in the course.
On the other hand, associations that had not been
encouraged by the instructor were quite prevalent. Some
of the responses mirrored previously published results.
Many students used conservation resources for entropy
problems, despite the fact that entropy is not generally
conserved. Many students used the notion of entropy as a
measure of disorder despite it leading to confusion on
certain predictions. In particular, students struggled to
reconcile the notion of entropy as disorder with an intuition
that thermal equilibrium is a natural, and thus ordered,
state. Students in this category either predicted that the
entropy of a system would decrease as the system
approached thermal equilibrium or shifted back and forth
between predictions.
The data suggest that efforts by instructors to banish the
idea of entropy as a measure of disorder may have only
limited success. Students were not exposed to this idea in
this course and yet many of the students in interviews
employed it repeatedly. Students used a variety of other
metaphors not explicitly introduced in the class, connecting
entropy with macroscopic or particle-level disorganization or
dispersal. Whether students are encountering these ideas in
formal classroom settings or in popular depictions of entropy
as disorganization cannot be determined from our data, but
the association of entropy with order and disorganization are
clearly widespread despite efforts to the contrary.
While this work was not directly intended to improve
instruction, we believe that a better characterization of
student thinking about entropy will help practitioners as
well as researchers.
D. Some thoughts on hidden and
suppressed resources
Subsequent to the interviews, we wondered at the fact
that the association of entropy and order in equilibrium has
not appeared in other contexts. We have posed a number of
written questions in order to probe further this connection
between entropy and order or disorder. Very few students
spontaneously gave explanations associating order with an
increase in entropy. However, it is clear even within the
interviews that students respond to cues from instructors
and take up language even if does not represent their initial
thinking. The best example comes from the responses of
Gladius, and her shifting language when talking about order;
it seems in retrospect that the interviewer, despite listening
very carefully to student language, did not recognize the
potential significance of Gladius referring to “that would be
disorder” during the interview. Later Gladius was referring to
“disordered” or “more disorder,” but it is not clear whether
this is her preferred language or represents her adoption of
the interviewer’s language. Written problems do not allow
much latitude for students to use nonstandard language,
and instructors in class might not always notice nuances in
student phrasing, or may correct the student language
without attending to the substance of the reasoning behind
it. Subsequent to these interviews, the instructor noticed at
least one student using almost this same phrasing in class,
referring to equilibrium as a more ordered state.
V. DISCUSSION
ACKNOWLEDGMENTS
The student responses to these interview prompts suggest that student thinking about entropy is more complex
than previous research might have indicated. Most of the
students in our study did not employ a small number of
strongly held ideas, but rather shifted frequently among a
larger number of ideas, most of which were at least
potentially productive.
Among the ideas employed by students, those that had
been introduced by the course instructor were not the most
prevalent. Association of entropy with multiplicity was
We thank our colleagues John Thompson and Don
Mountcastle at the University of Maine, along with former
students Trevor Smith and Brandon Bucy; David Meltzer at
Arizona State University; and Warren Christensen at North
Dakota State. We further thank colleagues at the CSUF
Catalyst Center for reading an early draft of this manuscript, particularly Sissi Li and Mary Emenike. The work
was supported by the National Science Foundation Grant
No. DUE-0817335.
020118-13
MICHAEL LOVERUDE
PHYS. REV. ST PHYS. EDUC. RES 11, 020118 (2015)
[1] S. Kesidou and R. Duit, Students’ conceptions of the
second law of thermodynamics—an interpretive study,
J. Res. Sci. Teach. 30, 85 (1988).
[2] B. R. Bucy, Ph.D. thesis, University of Maine, 2007.
[3] T. G. Amin, F. Jeppsson, J. Haglund, and H. Strömdahl,
Arrow of time: Metaphorical construals of entropy and the
second law of thermodynamics, Sci. Educ. 96, 818 (2012);
C. Brosseau and J. Viard, ’Quelques réflexions sur le
concept d’entropie issues d’un enseignement de thermodynamique‘, Enseñanza de las Cientias 10, 13 (1992).
[4] M. J. Cochran and P. R. L. Heron, Development and
assessment of research-based tutorials on heat engines
and the second law of thermodynamics, Am. J. Phys. 74,
734 (2006).
[5] A. R. Daane, S. Vokos, and R. E. Scherr, Conserving
energy in physics and society: Creating an integrated
model of energy and the second law of thermodynamics,
AIP Conf. Proc. 1513, 114 (2013).
[6] B. D. Geller, B. W. Dreyfus, J. Gouvea, V. Sawtelle, C.
Turpen, and E. F. Redish, Entropy and spontaneity in
an introductory physics course for life science students,
Am. J. Phys. 82, 394 (2014).
[7] W. M. Christensen, D. E. Meltzer, and C. A. Ogilvie,
Student ideas regarding entropy and the second law of
thermodynamics in an introductory physics course, Am. J.
Phys. 77, 907 (2009).
[8] E. Langbeheim, S. A. Safran, S. Livne, and E. Yerushalmi,
Evolution in students’ understanding of thermal physics
with increasing complexity, Phys. Rev. ST Phys. Educ.
Res. 9, 020117 (2013).
[9] D. Styer, Insight into entropy, Am. J. Phys. 68, 1090 (2000).
[10] F. L. Lambert, Disorder – A Cracked Crutch for Supporting
Entropy Discussions, J. Chem. Educ. 79, 187 (2002).
[11] H. S. Leff, Removing the mystery of entropy and thermodynamics. Part I, Phys. Teach. 50, 28 (2012); Removing
the mystery of entropy and thermodynamics. Part II, Phys.
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
020118-14
Teach. 50, 87 (2012); Removing the mystery of entropy
and thermodynamics. Part III, Phys. Teach. 50, 170 (2012);
Removing the mystery of entropy and thermodynamics.
Part IV, Phys. Teach. 50, 215 (2012); Removing the
mystery of entropy and thermodynamics. Part V, Phys.
Teach. 50, 274 (2012).
T. Moore and D. V. Schroeder, A different approach to
introducing statistical mechanics, Am. J. Phys. 65, 26
(1997).
T. A. Moore, Six Ideas That Shaped Physics (McGrawHill, New York, 2003).
D. V. Schroeder, An Introduction to Thermal Physics
(Addison-Wesley, San Francisco, 2000).
P. R. L. Heron, in Empirical Investigations of Learning, and
Teaching, Part I: Examining, and Interpreting Student
Thinking, Proceedings of the “Enrico Fermi” Summer
School on Physics Education Research, edited by E. F.
Redish and M. Vicentini (Italian Physical Society, Varenna,
Italy, 2003), pp. 341–351.
D. Hammer, Student resources for learning introductory
physics, Am. J. Phys. 68, S52 (2000).
M. E. Loverude, Student understanding of basic probability
concepts in an upper-division thermal physics course, AIP
Conf. Proc. 1179, 189 (2009); Investigating student understanding for a statistical analysis of two thermally interacting
solids, AIP Conf. Proc. 1289, 213 (2010); (unpublished).
A. A. diSessa, Towards an epistemology of physics,
Cognit. Instr. 10, 105 (1993).
J. Minstrell, in Proceedings of the Conference on Research
in Physics Learning (IPN, Kiel, Germany, 1992),
pp. 110–128.
M. Loverude, Research and curriculum development on
student understanding of the microcanonical ensemble,
Part I: Counting microstates [Phys. Rev. ST Phys. Educ.
Res (to be published)].
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Phvs Educ., Vol 19. 1984Printed In Northern Ireland
The second law
of
thermodynamics
in a historical
setting
J Strnad
Turningpoints
in physics oftenposeproblems
in teaching, as if the learning of the individual
parallels the development of physics. So itis not
surprising that the second law of thermodynamics
poses a serious didactical problem. It would even be
worth considering the possibility of not including it
in the secondary school curriculum, were it not for
the fact that theshortage of energy and relatedtopics
are nowadays of primary importance.This article
describes a particular approach to the second law,
but having alluded totheapparent
similarity of
learning andthe
development of physics it is
appropriatetostartthe
discussion with a brief
historical survey.
Historical background
It all began with a booklet entitled Reflections on
the Motive Power of Heat and on the Machines
Adapted to Develop this Power published in 1824
with limited circulation by the French engineer Sadi
Carnot (Mendoza 1960). Whilst Carnot was
interested mainly in the technical and economic
aspects of steam engines, he also was the first to
discuss underlying physical laws. At the beginning
of the 19th century heat was generally considered
to be a fluid with an unmeasurably small density,
called caloric. Such concepts wereatthattime
readily introduced to explain various effects,
including electric and magnetic ones.
As an abstraction of real steam engines Carnot
introduced the ideal heat engine containing a heat
source at temperature T2 and a heat sink at lower
J a n e Stmad teaches undergraduate physics in the
Depan?nent of Physics, University of Ljubljana,
Yugoslavia and researches in theoretical physics at
the J Stefan Institute, Ljubljana. He also writes
physics textbooks at secondary school and university
level.
94
temperature TI and operating periodically. According to him, mechanical work - W is delivered as
heat Q is transferred from the source to the sink,
the heat
engine
(figure la) being completely
analogous to the waterwheel (figure lb) with the
transferred heat Q corresponding to the mass of
water m and the temperature difference T 2 - T , to
the height difference z2-z1. We thereby adopt the
conventionthatwork,
heat,etc,are
considered
positive if received by the system, thus W < 0.
Carnot stated that motive power can be produced
everywhere where a temperature difference exists,
that it does not depend on the substance used for its
production and that its magnitude is determined by
the temperature difference of the bodies between
which the caloric is transferred. However, he also
noted that it cannot be said that it is proportional
tothe difference only. He addedthat the fall of
caloric at lowergrades of temperature gives more
motive power than at higher grades. Let us provisionally call this statement the Carnot principle.
For isothermal processes of gases he concludedas we would say today-that the amount of heat
received or rejected does not depend on chemical
composition but logarithmically on theratio
of
the final and initial volumes and it increases with
increasing temperature.
Carnot
did not use
equations, but at the end of the booklet he tried
to support his principle by numerical calculations
of the motive powerfor
cycles with air, steam
and alcohol vapour.
By analogy with the ideal waterwheel Carnot
took it forgranted thatthe ideal heatengine is
reversible in the sense that it transfers heat Q from
the sink at temperature TI to the source at temperature T2 if it receives work W. He proved that for
given temperatures T2 and TI no engine can deliver
more work than the ideal engine. If such an engine
were to exist it could be coupled to a reversed ideal
engine and deliver the work difference for nothing,
i.e. it would operate as a perpetual motion machine
(of the first kind). This was long ago accepted as
impossible. (Indeed, Sadi’s father LazareCarnot
wrote on that
subject
when studying water
machines.)
Although the idea of heat as an imponderable
fluid was widely accepted, this viewpoint was by no
means the only one. It was repeatedly stated that
the observed
phenomena
could be
understood
almost equally well if heat were associated with
molecular motion, a concept introduced in the 18th
century (Brush 1976, Friedman 1977).
Carnot’s published work was scarcely noticed by
his contemporaries. It was not reprinted until 1872
and again in 1878, this timetogether with Sadi’s
unpublished notes provided by his brother Hyppolite. These notes show that Sadi Carnot abandoned
0031-9120/84/020094+07$02 25
0 1984 The lnstltute of
Physlcs
Figure l Accordingto Carnot a, an
ideal heat engine delivers work
-W 1 Q( T2 - T,) as
caloric Q is
falling fromtemperature T2to
temperature T I in the same way as
b, an ideal waterwheel delivers
work -W = mg(z2 - zl) as the
mass m of water is falling from
height z2 to height 2,. Thomson
observed that in stationary heat
conduction
work is lost in comparison with an engine
ideal
heat
0
rz
:
b
--
--
T1 , ....r - -
*l
....
””
Q
the idea thatheat
is asubstance,
for which a
conservation lawis
valid, andconsidered
it as
motion. He anticipated the equivalence of work and
heatandthe
mechanical equivalent of heatand
proposedexperiments
that were doneindependently by J P Joule many years later.
‘. . . Heat is simple motive power, orrather
motion that haschanged form. It is amovement
among the particles of bodies. Wherever there is
destruction of motive power there is, at the same
time, production of heat in quantity exactly proportional to the quantity of motive power destroyed.
Reciprocally, wherever there is destruction of heat
there is production of motive power. . .’.
S Carnot (unpublished)
These noteswereprobablywrittenduring
the
period 1824-1826 (Wilson 1981), although there
are some indications that they may have been
In this case Carnot
written earlier (Cooper 1969).
would have purposely formulated his principle
without referring to the equivalence of work and
heat. Indeed, in Reflections he used heat (chaleur)
and caloric (calorique) differently. Although he
stated in a footnote that they have the same meaning, he never wrote ‘fall of heat’ but always ‘fall of
caloric’. Notwithstanding interesting speculations as
to the state of Carnot’s knowledge, his unpublished
work did not in any way influence subsequent
development.
Emile Clapeyron, another French engineer, took
up Carnot’s
conjecture
(1834).
Using Carnot’s
conclusions on isothermal processes for gases he
studied gas cycles extensively and provided graphical and analytical formulationsthat
are not far
away from those in present-day use. For a gas cycle
with an infinitesimally small temperature difference
between source and sink T- TI = dT, T2= T he
obtained for the work done (Klein 1974)
d T-/ dCW
( T=)Q
*
L
I
I“
(1)
with a universal function of temperature C(T). He
tried to find this function on the basis of experimental data.
AlthoughClapeyron insisted on heatconservation, his equation (1) remains valid even if the
equivalence of work and heat-i.e. the first law of
thermodynamics-is taken into account. The heat
delivered -(Q +dW) at the sink at temperature
T- d T can in this case be approximated by -Q. Is
this not another example of the greater strength of
the physicist’s intuitionthan
of his knowledge
(Wigner 1964)?
Around the middle of the 19th century the first
law of thermodynamics was established, thanks to
the work of Robert Mayer (1842, 1845), James
Prescott Joule (1843, 1847), Hermann von Helmholtz (1847) and others. Rudolf Clausius, who
becameaware
of Carnot’s work throughEmile
Clapeyron, was the first to reconcile the Carnot
principle with the first law (1850). He proved that
heat is not a thermodynamic function, i.e. that the
heat received by a system is not uniquely determined by the initial and final states. For an ideal
gas he derived cp - C V , the difference of specific
heats at constantpressureandconstant
volume,
and argued that this contradicts heat conservation.
He obtained Poisson’s equations for adiabatic
processes by putting the heat exchanged equal to zero.
For an ideal gas he found the function C(T) to be
proportional to T and deduced the dependence of
boiling point on pressure, showing by long
numerical calculations for steam that this
agreed with experimental data.
The
equation
dp/dT = q/TA(l/p)-following
fromequation
(1)
for -dW = dpAV, Q = mq and C(T)= T, where q
is the specific latent heat and AV and A( l / p ) the
changes in volume and reciprocal density, respectively, at the phase change-is thus rightly called
the Clausius-Clapeyron equation. Finally, using the
difference of specific heats for an ideal gas, Clausius
calculated the mechanical equivalent of heat.
Clausius’s principal achievement was somewhat
hidden at the beginning of the second part of his
1850 paper, when he replaced Carnot’s argument
for the maximum work delivered by an ideal heat
engine based o n the impossibility of perpetual
motion (of the first kind). He considered an engine
that would deliver more work than an ideal engine
if it received the same amount of heat or, in other
words, deliver the same work as an ideal engine on
95
receiving asmalleramount
of heat. If the first
enginewerethencoupled
with a reversed ideal
engine any amount of heat could be transferred from
a cooler to a hotter body, and this contradicts the
common experience with heat, which always shows
the tendency to balance out existing temperature
from hotter bodies into
differences by goingover
cooler ones.
The attention of William Thomson (Lord Kelvin)
was focused on Carnot’s work by Clapeyron.
Thomson derived himself equation (1) and, following a suggestion fromJoule, put C(T) = T and
defined the thermodynamic temperature by
dT/T= (dW(/Q
Being familiar with Joule’s results for heat generated by electric currents (1843) and by friction in
liquids (1847), Thomson was aware of an inconsistency. Comparingheatconduction,whereheat
is
conserved, with the action of an ideal heat engine
according to Carnot, where the fall of heat gives
rise to work, he wondered about the worklost in
heat conduction (figure IC)but he was not prepared
to abandon the idea of heat conservation in a heat
engine. The reason perhaps lay in his research into
equations of continuity in general, which had led
him to expect a quadratic dependence of work on
heat rather than Joule’s linear one (Wise 1979). It
took him about three years to accept the equivalence of work and heat.
‘. . . When “thermal agency” is thus spent in conducting heat through a solid, what becomes of the
mechanical effect which it might produce? Nothing
can be lost in the operations of nature-no energy
can be destroyed. What effect then is produced in
place of the mechanical effect which is lost? A
perfecttheory
of heat imperatively demands an
answer to this question; yet no answer can be given
in the present state of science. . .’.
W Thomson 1849 (Wise 1979)
After accepting the firstlaw Thomson clearly
recognised that two laws of nature were at stake
(1851). He called the first one Joule’s law and the
second Carnot’s and Clausius’s law. He rephrased
Clausius’s formulation: is
it
impossible forany
cyclical machine to produce no other effect than to
convey heat continuously from one body to another at
higher temperature. Clausius afterwards put it more
simply: heat cannot of itself pass from a cooler
into a hotter body?.
t P G Tait claims that in 1850 Clausius had only
implicitly stated the new law, withoutevenadding
of
itself, and ascribed the second law to Thomson,but
Thomson himself gave the credit to Clausius. This was a
part of the German-British priority debate in which in
addition to Clausius and Thomson, Rankine,Maxwell and
others were involved (Clausius 1872, Daub 1970).
96
1
P - w
Q1
Figure 2 a, According to the first law an ideal heat engine
receives heat Q 2 ,delivers work - W = Q, + Q, and
rejects heat Q,. b, In a reversed cycle, corresponding to
an ideal refrigerator or heat pump, heat Q, is absorbed,
work W received and heat - Q 2 = Q , + W rejected. Work
W, heat Q and entropy change AS are taken to be
positive if they are received by the system (arrows leading
from the surroundings into the system) or negative if they
are rejected by the system (arrows leading from the
system to the surroundings)
Thomson’s formulation of the second law (1851,
somewhat rephrased later) was as follows: a transformation whose only final result is to transform into
work heat extracted from a source which is at the
same temperature throughout is impossible. Since the
work mentioned can be converted to heat at high
temperatures Thomson’s formulation is equivalent
to Clausius’s. Nevertheless, it is significant in that it
prohibits directly the isothermal heat engine or
perpetual motion of the second kind.
Internal energy is a thermodynamic function and
its net change in a cycle is zero. Accordingly, then, to
the firstlaw the work delivered by an ideal heat
engine is
Q2> 0 is the heat received by the engine at temperature T2 at the source and Q1< O is the heat
rejected at temperature T1 at the sink (figure 2a).
A reversed engine represents the ideal heat pump
or refrigerator (figure 2b). Thomson, knowing that
the heat exchanged isothermally by an ideal gas is
proportional tothetemperature,
could adapt his
definition of the thermodynamic temperatureto
(1854):
T2/T1= QziIQlI
(3)
which is valid today. Thomson was the first to study
extensively the thermoelectric
phenomena
connected with electric currents (1851). Although he
was familiar with Clausius’s paper, he must have
developed his ideas independently.
Gradually Clausius realised that it would be
convenient to introduceasecondthermodynamic
function associated with the second law in the same
way as internal energyis with the first one. He
proved
that Q,/T,+ Q2/T2=0 for the ideal heat engine
and C i Qi/Ti = 0 or dQiT = 0 for a general rever-
sible cycle (1854). This result was also obtained by
Thomson (1854) who, however, did not introduce a
new function. Clausius introducedafunction
S
such that it is conserved in reversible processes. For
the ideal heat engine at constant temperature Tz
and TI
ASz = Q2/T2
AS1 = QJT1
(44
and
AS,,, = AS2 + AS1 = 0
Clausius’s point of departure was the recognition
that for irreversible cycles, e.g. for heat conduction,
QJT, + Q2/T2< 0 or in general $ dQ/T< 0. On the
other hand, since entropy is a thermodynamic function, in processes in which the final state coincides
with the initial one the net entropy change should
be zero. Clausius gave for the net change of the
entropy of a system, for example,
(4b)
since ASz>O as Q2>0 and ASl