3 posts. Each 150 words. Do not use formulas or equations in the discussions1-Equilibrium entropy conditions2- The Macroscopic Definition of Entropy3-Attached paper; “Identifying Student Resources in Reasoning About Entropy and the Approach to Thermal Equilibrium”

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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
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.
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
Published by the American Physical Society
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.
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.
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
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
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
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
Time since completing course
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.
300 K 300 K
320 K 280 K
360 K 300 K
320 K 340 K
360 K 300 K
340 K 320 K
340 K 280 K
300 K 300 K
275 K 325 K
300 K 300 K
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
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
PHYS. REV. ST PHYS. EDUC. RES 11, 020118 (2015)
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