We report on students’ thinking regarding entropy in an introductory calculus-based physics course. We analyzed students’ responses to a variety of questions on entropy changes of an arbitrarily defined system and its surroundings. In four offerings of the same course we found that before instruction, no more than 6% of all students could give completely correct responses to relevant questions posed in both general and concrete contexts. Nearly two-thirds of the students showed clear evidence of conservation-type reasoning regarding entropy. These outcomes were little changed even after instruction. Targeted instruction that guided students to recognize that entropy is not a conserved quantity appears to yield improved performance on qualitative questions related to this concept.

1.
S.
Kesidou
and
R.
Duit
, “
Students’ conceptions of the second law of thermodynamics–An interpretive study
,”
J. Res. Sci. Teach.
30
(
1
),
85
106
(
1993
).
2.
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
741
(
2006
).
3.
M. E.
Loverude
,
C. H.
Kautz
, and
P. R. L.
Heron
, “
Student understanding of the first law of thermodynamics: Relating work to the adiabatic compression of an ideal gas
,”
Am. J. Phys.
70
,
137
148
(
2002
).
4.
D. E.
Meltzer
, “
Investigation of students’ reasoning regarding heat, work, and the first law of thermodynamics in an introductory calculus-based general physics course
,”
Am. J. Phys.
72
,
1432
1446
(
2004
).
5.
C. H.
Kautz
,
P. R. L.
Heron
,
M. E.
Loverude
, and
L. C.
McDermott
, “
Student understanding of the ideal gas law, Part I: A macroscopic perspective
,”
Am. J. Phys.
73
,
1055
1063
(
2005
).
6.
C. H.
Kautz
,
P. R. L.
Heron
,
P. S.
Shaffer
, and
L. C.
McDermott
, “
Student understanding of the ideal gas law, Part II: A microscopic perspective
,”
Am. J. Phys.
73
,
1064
1071
(
2005
).
7.
J. R.
Thompson
,
B. R.
Bucy
, and
D. B.
Mountcastle
, “
Assessing student understanding of partial derivatives in thermodynamics
,”
Proceedings of the 2005 Physics Education Research Conference
, Salt Lake City, UT,
2005
, edited by
Paula
Heron
,
Laura
McCullough
, and
Jeffrey
Marx
J. R.
Thompson
,
B. R.
Bucy
, and
D. B.
Mountcastle
, [
AIP Conf. Proc.
818
,
77
80
(
2006
)].
8.
B. R.
Bucy
,
J. R.
Thompson
, and
D. B.
Mountcastle
, “
What is entropy? Advanced undergraduate performance comparing ideal gas processes
,”
Proceedings of the 2005 Physics Education Research Conference
, Salt Lake City, UT,
2005
, edited by
Paula
Heron
,
Laura
McCullough
, and
Jeffrey
Marx
[
AIP Conf. Proc.
818
,
81
84
(
2006
)].
9.
B. R.
Bucy
,
J. R.
Thompson
, and
D. B.
Mountcastle
, “
Student (mis)application of partial differentiation to material properties
,”
Proceedings of the 2006 Physics Education Research Conference
, Syracuse, NY,
2006
, edited by
Laura
McCullough
,
Leon
Hsu
, and
Paula
Heron
[
AIP Conf. Proc.
883
,
157
160
(
2007
)].
10.
D. B.
Mountcastle
,
B. R.
Bucy
, and
J. R.
Thompson
, “
Student estimates of probability and uncertainty in advanced laboratory and statistical physics courses
,”
Proceedings of the 2007 Physics Education Research Conference
, Greensboro, NC,
2007
, edited by
Leon
Hsu
,
Charles
Henderson
, and
Laura
McCullough
D. B.
Mountcastle
,
B. R.
Bucy
, and
J. R.
Thompson
, [
AIP Conf. Proc.
951
,
152
155
(
2007
)].
11.
E. B.
Pollock
,
J. R.
Thompson
, and
D. B.
Mountcastle
, “
Student understanding of the physics and mathematics of process variables inP-V diagrams
,”
Proceedings of the 2007 Physics Education Research Conference
, Greensboro, NC,
2007
, edited by
Leon
Hsu
,
Charles
Henderson
, and
Laura
McCullough
E. B.
Pollock
,
J. R.
Thompson
, and
D. B.
Mountcastle
, [
AIP Conf. Proc.
951
,
168
171
(
2007
)].
12.
D. E.
Meltzer
, “
Student learning in upper-level thermal physics: Comparisons and contrasts with students in introductory courses
,”
Proceedings of the 2004 Physics Education Research Conference
, Sacramento, CA,
2004
, edited by
Jeffrey
Marx
,
Paula
Heron
, and
Scott
Franklin
[
AIP Conf. Proc.
790
,
31
34
(
2005
)].
13.
D. E.
Meltzer
, “
Investigation of student learning in thermodynamics and implications for instruction in chemistry and engineering
,”
Proceedings of the 2006 Physics Education Research Conference
, Syracuse, NY,
2006
, edited by
Laura
McCullough
,
Leon
Hsu
, and
Paula
Heron
[
AIP Conf. Proc.
883
,
38
41
(
2007
)];
D. E.
Meltzer
, “
Investigating and addressing learning difficulties in thermodynamics
,”
Proceedings of the American Society for Engineering Education 2008 Annual Conference and Exposition
, July 22–25,
2008
, AC 2008–1505.
14.
See, for comparison,
R.
Ben-Zvi
, “
Non-science oriented students and the second law of thermodynamics
,”
Int. J. Sci. Educ.
21
(
12
),
1251
1267
(
1999
). Ben-Zvi reported on student use of curricular materials she developed on energy and the quality of energy. In a tenth-grade course for non-science-oriented students, Ben-Zvi found that only one-quarter of the students recognized that in processes involving energy transfer, “each transformation is accompanied by some of it being converted to heat and thus the ability to perform work decreases.”
15.
M. F.
Granville
, “
Student misconceptions in thermodynamics
,”
J. Chem. Educ.
62
,
847
848
(
1985
).
16.
P. L.
Thomas
and
R. W.
Schwenz
, “
College physical chemistry students’ conceptions of equilibrium and fundamental thermodynamics
,”
J. Res. Sci. Teach.
35
,
1151
1160
(
1998
).
17.
M.
Sözbilir
and
J. M.
Bennett
, “
A study of Turkish chemistry undergraduates’ understandings of entropy
,”
J. Chem. Educ.
84
(
7
),
1204
1208
(
2007
).
18.
See, for example,
N. J.
Tro
,
Chemistry, A Molecular Approach
(
Prentice-Hall
,
Upper Saddle River, NJ
,
2008
), pp.
783
785
.
19.
In general, we use the terms “natural,” “spontaneous,” “real,” and “naturally occurring” synonymously when modifying the word “process.”
20.
For practical purposes and to facilitate analysis, we ordinarily specify a particular system along with a separate region or reservoir referred to as the surroundings; both this system and the surroundings are then isolated from the rest of the universe. See discussion of the “concrete-context” question in Sec. III C.
21.
D.
Giancoli
,
Physics for Scientists and Engineers
, 3rd ed. (
Prentice-Hall
,
Upper Saddle River, NJ
,
2000
), p.
539
;
D.
Halliday
,
R.
Resnick
, and
J.
Walker
,
Fundamentals of Physics
, 6th ed. (
Wiley
,
New York, NY
,
2001
), p.
500
;
R. D.
Knight
,
Physics for Scientists and Engineers
(
Addison-Wesley
,
San Francisco, CA
,
2004
), p.
566
;
R. A.
Serway
,
Physics for Scientists and Engineers
, 4th ed. (
Saunders
,
Philadelphia, PA
,
1996
), pp.
629
632
.
22.
The instructor for the course in which this question was first used employed the terminology “isolated system,” so the question was written to include that language in version A. Student understanding of this terminology was not a focus of our research. We also discovered that the distracters in version A might not have fully represented students’ thinking on this question, and so we created version B. See discussion in Sec. IV B 3.
23.
See EPAPS Document No. E-AJPIAS-77-013909 for appendices. Appendix I contains detailed responses to all questions in each course. This document can be reached via a direct link in the online article's HTML reference section or via the EPAPS homepage (http://www.aip.org/pubservs/epaps.html).
24.
Almost all students (90%) who gave a “not determinable” response on part (a) (system/object) also gave a “not determinable” response on part (b) (surroundings/air) on both the general-context and concrete-context questions.
25.
After instruction in Spring 2005, this overlap proportion was almost unchanged at 41%. After use of specially designed curricular materials, this overlap proportion increased sharply; see Sec. VI A.
26.
It is conceivable that some students may confuse the word “entropy” with the word “energy.” The words are spelled similarly and sound similar, and the two concepts are closely linked. Although there may be some confusion regarding words, there is no significant evidence from their responses that students actually believe energy and entropy to be the same entity.
27.
These correspond to normalized gains of g=0.15 and 0.04, respectively, using Hake’s definition of normalized gain. See
R. R.
Hake
, “
Interactive engagement versus traditional methods: A six-thousand-student survey of mechanics test data for introductory physics courses
,”
Am. J. Phys.
66
,
64
74
(
1998
). Note, however, that correct (“not determinable”) responses regarding system entropy change on the general-context question actually declined after instruction (p=0.01).
28.
This tutorial is shown in Appendix IX on EPAPS. See Ref. 23.
29.
W. M.
Christensen
, “
An investigation of student thinking regarding calorimetry, entropy, and the second law of thermodynamics
,” Ph.D. dissertation,
Iowa State University
,
2007
, UMI No. 3274888, Chap. 5.
30.
We used a binomial-proportions test as described in
J. P.
Guilford
,
Fundamental Statistics in Psychology and Education
, 4th ed. (
McGraw-Hill
,
New York
,
1965
), pp.
185
187
. On the postinstruction “entropy of the system” general-context question (considering only those students who made a directional choice), the “increases” response is more common than the “decreases” response (p<0.001). Similarly, the postinstruction general-context response that the entropy of the surroundings increases is more popular than the response that the entropy of the surroundings decreases (p<0.01); a similar preference is expressed on the concrete-context question for the entropy of the air in the room (increases preferred over decreases, p<0.0001).
31.
Student responses are consistent with the most general form of the entropy inequality and might be considered to be partially correct. However, the questions explicitly referred to irreversible processes, or process that can actually occur, for which the total entropy always increases. The patterns of student responses we have reported appear to be independent of the specific terminology used in a particular question. That is, whether one or another term is used, the proportion of correct responses remains essentially unchanged.
32.
The complete data are provided in Appendix V on EPAPS. See Ref. 23.
33.
Other studies have explored the role of context-dependence of students’ responses, mostly in the form of different problem representations. See
D. E.
Meltzer
, “
Relation between students’ problem-solving performance and representational format
,”
Am. J. Phys.
73
,
463
478
(
2005
), and references therein.
34.
See Ref. 29, Chaps. 2 and 4.
35.
The tutorial is shown in Appendix X on EPAPS. See Ref. 23.
36.
L. C.
McDermott
, “
Bridging the gap between teaching and learning: The role of research
,”
The Changing Role of Physics Departments in Modern Universities: Proceedings of International Conference on Undergraduate Physics Education, AIP Conference Proceedings 399
, College Park, Maryland, August 1996 [AIP Conference Processings 399], edited by
E. F.
Redish
and
J. S.
Rigden
(
AIP
,
Woodbury, New York
,
1997
), pp.
139
165
.
37.
For a breakdown of the matched-sample data, see Appendix VI on EPAPS. See Ref. 23.
38.
Analogous results were found in a different context by
D. E.
Meltzer
, “
Analysis of shifts in students’ reasoning regarding electric field and potential concepts
,”
Proceedings of the 2006 Physics Education Research Conference
, Syracuse, NY,
2006
, edited by
Laura
McCullough
,
Leon
Hsu
, and
Paula
Heron
[
AIP Conf. Proc.
883
,
177
189
(
2007
)].
39.
C. H.
Crouch
and
E.
Mazur
, “
Peer Instruction: Ten years of experience and results
,”
Am. J. Phys.
69
,
970
977
(
2001
);
D. E.
Meltzer
and
K.
Manivannan
, “
Transforming the lecture-hall environment: The fully interactive physics lecture
,”
Am. J. Phys.
70
,
639
654
(
2002
);
N. W.
Reay
,
L.
Bao
,
P.
Li
,
R.
Warnakulasooriya
, and
G.
Baugh
, “
Toward the effective use of voting machines in physics lectures
,”
Am. J. Phys.
73
,
554
558
(
2005
);
I. D.
Beatty
,
W. J.
Gerace
,
W. J.
Leonard
, and
R. J.
Dufresne
, “
Designing effective questions for classroom response system teaching
,”
Am. J. Phys.
74
,
31
39
(
2006
).
40.
See Appendices VII and VIII on EPAPS. See Ref. 23.
41.
See Appendix XI on EPAPS. See Ref. 23.
42.
In contrast, we and others have found that students’ conceptual difficulties regarding processes that do not involve heat transfer, for example, the free expansion of a gas, persist to some extent. See Refs. 8 and 12.

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