Research: Science and Education
Equations of State and Phase Diagrams
L. Glasser
Molecular Sciences Institute, School of Chemistry, University of the Witwatersrand, P. O. Wits 2050, South Africa;
glasser@aurum.wits.ac.za
An equation of state (EoS) is a mathematical summary
of the (usually thermodynamic) equilibrium properties of a
material (1). At the very minimum (in the absence of external
constraints, such as fixed temperature), such an equation relates
the mechanical, thermal, and volumetric properties of a single
phase of the material; other properties might be needed to fully
describe its behavior if the material exists in a particular
environment, such as in a strong electric or magnetic field,
or localized in a surface.
Among the simplest possible EoS’s is that of the ideal gas
pVm = RT
(1)
where p (pressure) represents the mechanical state of the
system, T (absolute temperature) the thermal state, and Vm
(molar volume) the volumetric state. Clearly, there are only
two independent variables (which may be arbitrarily selected
from among the three) while the remaining variable takes on
a value dependent on the values assigned to the other two,
through the EoS. EoS’s may be expressed with other variables than p, T, and Vm; for example density, ρ, may replace
Vm, but the three- (or more)-variable structure remains.
A more elaborate EoS, such as van der Waals equation
p=
RT – a
V m – b V m2
(2)
retains the same structure of variables, though differently
expressed. Furthermore, the van der Waals equation and most
EoS’s for the fluid state generally receive a further thermodynamic interpretation. Such cubic (as well as some other)
functions display an S-shaped structure with loops between
the liquid and vapor regions. The regions are divided into
three using Maxwell’s construction (2) to guide the division
into gas/vapor, liquid, and phase gap, by recognizing that the
liquid and vapor are in equilibrium at a given temperature,
so that their molar Gibbs functions are equal. Maxwell’s
construction then requires equal areas above and below the
tie line, which connects liquid and vapor in equilibrium,
passing through the loops of the EoS function.
Thus, the full representation of a simple EoS is a threedimensional image, encompassing the three variables. Threedimensional models (3–6 ) and diagrams (7 ) of single-phase
systems are not uncommonly found (Fig. 1). However, in
graphical representation of multiphase systems, partly because
of the difficulty of drawing three-dimensional images (and partly
because of the lack of the required data), it is traditional to
use projections of the three dimensions onto two-dimensional
planes in preference to the full three-dimensional representation (e.g., pV diagrams [1, 2, 8–10] in place of pVT diagrams),
while three-dimensional descriptions (3, 11, 12) are usually
presented schematically, not to scale. (Petrucci [13] has presented a three-dimensional diagram of p, V, and composition,
but for a hypothetical binary system, at a fixed temperature.)
Equilibria between two phases are represented by a (two874
dimensional) curve within the three-dimensional diagram, and
equilibrium among three phases is represented by a straight line
situated along the constant temperature–pressure equilibrium
condition. Figure 2 shows that the triple “point” of a pT
projection is actually a misnomer for a tie line in full phase
diagram space; Sandler (11) terms this the “triple-point line”.
Thus we see that, although the common practice of
depicting only projections suffices for many purposes, it
also hides some of the features of the relations between the
EoS’s of the different phases. This paper illustrates the phase
relations in a unary (single-component) system with a full
three-dimensional diagram using authentic data. Such a diagram
does not seem to have appeared before.
To provide this authentic representation of a unary system,
we chose carbon dioxide (Fig. 3), for which much (but not
all) of the required data is available in equation form. Where
the equations required could not be found, they were fitted
to experimental data or closely approximated by standard
relations, as described in the Appendix. To encompass the
wide range of values encountered, it is necessary to represent
the pressure and volume data on logarithmic axes.
The data required are equilibrium temperatures and pressures as well as data for
Sublimation curve: solid and vapor densities (or molar
volumes)
Melting curve: solid and liquid densities
Saturation curve: liquid and vapor densities
Solid state phase changes: these data are largely uncertain or
unknown (10), and so are omitted from the diagram
The Appendix lists the equations (whether fitted or theoretical)
for carbon dioxide equilibria, or the experimental data used
Figure 1. Photograph of a model of the ideal gas pVT surface,
constructed using templates kindly provided by D. B. Hilton (5).
Journal of Chemical Education • Vol. 79 No. 7 July 2002 • JChemEd.chem.wisc.edu
Research: Science and Education
to generate the curves. These data were extracted from Gmelin
(14) and Landolt-Börnstein (15) for transitions involving the
solid state, and from the International Thermodynamic Tables
of the Fluid State for carbon dioxide (16 ). The calculated results
were prepared on an Excel spreadsheet.
For ease of access, a recent EoS for the single-phase fluid
region is also listed in the Appendix (17), but was not utilized
in preparation of Figure 3. Much thermophysical (and other)
data for many materials, including carbon dioxide, is available
on the WebBook of the National Institute of Standards and
Technology (18).
Literature Cited
Figure 2. Orthographic (isometric) three-dimensional pVT diagram.
Adapted from ref 11: Sandler, S. I. Chemical and Engineering
Thermodynamics, 2nd ed.; Wiley, New York, 1989; p 217; copyright © 1966 by Blaisdell Publishing Co. (John Wiley & Sons, Inc.).
Reprinted by permission of John Wiley & Sons, Inc.
Figure 3. An orthographic pVT diagram for carbon dioxide, with
projections onto the pT, pV, and VT planes. To accommodate the full
range of data, the logarithms of the pressure and molar volume
axes are used. The horizontal lines (constant pT ) are tie lines connecting phases in mechanical and thermal equilibrium across the
phase gaps. The critical point condition and the triple-“point” tie-line
are labeled. The dot at the end of the liquid–vapor line in the pT
projection represents its termination at the critical point. Volumes in
the diagram (corresponding to areas in the projections) are labeled
solid, liquid, vapor and (above the critical point) gas. The space
curves depicted are A: solid sublimation, in equilibrium with vapor;
B: vapor condensation, in equilibrium with solid; C: liquid saturation,
in equilibrium with vapor; D: vapor saturation, in equilibrium with
liquid; E: solid–liquid melt equilibrium—on the scale of the diagram
the two separate curves nearly overlie one another. The small break
barely discernible at the junction of curves A and C arises from the
difference in molar volumes of solid and liquid at the triple “point”
(cf. Fig. 2).
1. Atkins, P. W. Physical Chemistry, 6th ed.; Oxford University
Press: Oxford, 1998.
2. Wisniak, J.; Golden, M. J. Chem. Educ. 1998, 75, 200.
3. Petrucci, R. H. J. Chem. Educ. 1965, 42, 323.
4. Peretti, E. A. J. Chem. Educ. 1966, 43, 253.
5. Hilton, D. B. J. Chem. Educ. 1991, 68, 496.
6. Coch Frugoni, J. A.; Zepka, M.; Rocha Figueira, R.; Coretti, M.
J. Chem. Educ. 1984, 61, 1048.
7. Remark, J. F. J. Chem. Educ. 1975, 52, 61.
8. Halpern, A. M.; Lin, M.-F. J. Chem. Educ. 1986, 63, 38.
9. Lieu, V. T. J. Chem. Educ. 1996, 73, 837.
10. Gramsch, S. A. J. Chem. Educ. 2000, 77, 718.
11. Sandler, S. I. Chemical and Engineering Thermodynamics, 2nd ed.;
Wiley: New York, 1989; p 217.
12. Logo for 14th Russian Conference on Chemical Thermodynamics, St. Petersburg, Jun 30–Jul 5, 2002; http://rcct2002.
nonel.pu.ru/ (accessed Apr 2002).
13. Petrucci, R. H. J. Chem. Educ. 1970, 47, 825.
14. Gmelins Handbuch der Anorgorganischen Chemie, 8th ed.; Section C: Part 1, Carbon Dioxide; von Backzo, C., Ed.; Verlag
Chemie: Weinheim, 1970.
15. Landolt-Börnstein Zahlenwerte und Funktionen, Vol. 2, 6th ed.;
Schäfer, K.; Beggerow, G., Eds.; Springer: Heidelberg, 1971;
Part 1, p 723.
16. International Thermodynamic Tables of the Fluid State—3:
Carbon Dioxide; Angus, S.; Armstrong, B.; de Reuck, K. M., Eds.;
Pergamon: Oxford, 1976.
17. Mäder, U. K.; Berman, R. G. Am. Mineral. 1991, 76, 1547.
18. National Institute of Standards and Technology; NIST
WebBook; http://webbook.nist.gov (accessed Mar 2002); at
present free of charge.
Appendix (from ref 16 )
Critical point:
Tc = 304.21 K; pc = 73.825 bar; Vm,c = 94.428 cm3 mol᎑1
Triple point:
T3 = 216.58 K; p3 = 5.18 bar; Vm,3(s) = 29.09 cm3 mol᎑1;
Vm,3() = 37.338 cm3 mol᎑1; Vm,3(g) = 3133.79 cm3 mol᎑1
Saturated vapor pressure/temperature (16 ):
p
ln p = a 0 1 – T
Tc
c
1.935
4
+ Σ ai
i=1
Tc
–1
T
i
where
a0 = 11.377371, a1 = ᎑6.8849249, a2 = ᎑9.5924263, a3 = 13.679755,
JChemEd.chem.wisc.edu • Vol. 79 No. 7 July 2002 • Journal of Chemical Education
875
Research: Science and Education
a4 = ᎑8.6056439
T/K
Saturated liquid density (16 ):
2
ρ
T 0.347+ c 1 – T
i
ρc – 1 = c 0 1 – T
T
i=1
Σ
c
i+1 /3
c
where c0 = 1.9073793, c1 = 0.38225012, c2 = 0.42897885
Saturated vapor density (16 ):
2
ρg
T 0.347+ d 1 – T
i
ρc – 1 = d 0 1 – T
T
i=1
Σ
c
c
p – p3
T
p 3 + 648.13886 = 3 ln T + ln 648.13886
3
Melting: Volume Changes (refs 14, 15)
P / bar
∆Vm,melt /(cm3 mol᎑1)
267.65
2,942.0
4.71
281.65
3,922.7
4.31
294.55
4,903.4
3.94
306.25
5,884.0
3.62
317.35
6,864.7
3.32
328.35
7,845.4
3.07
338.95
8,826.0
2.83
348.55
9,806.7
2.65
357.75
10,787
2.48
368.45
11,768
2.34
220
5.185
166.91
We have fitted the above data to the following quadratic polynomial (given to 10-decimal places—this precision is not justified by
the underlying data, but is required to yield calculated values of
sufficient accuracy):
—
37.34
—
6.38
36.77
29.82
225
412.57
6.19
36.07
29.69
230
669.39
6.01
35.49
29.57
937.63
5.82
35.02
29.45
5.75
34.85
29.41
237
1,048.2
267.65
2,942.0
4.72
33.80
28.86
281.65
3,922.7
4.30
33.40
28.68
i+1 /3
Melting curve (16 ):
T/K
216.58
235
where d0 = ᎑1.7988929, d1 = ᎑0.71728276, d2 =1.7739244
ln
∆Vm,melt(quad) Vm,/cm3 mol᎑1 Vm,s(calcd)
P/bar
Selected Values from Table 8, Reference 16
References 14 and 15
294.55
4,903.4
3.93
33.03
28.57
306.25
5,884.0
3.62
32.71
28.50
317.35
6,864.7
3.34
32.41
28.48
328.35
7,845.4
3.07
32.16
28.48
338.95
8,826.0
2.84
31.92
28.51
348.55
9,806.7
2.64
31.74
28.56
357.75
10,787
2.46
31.57
28.64
368.45
11,768
2.27
31.43
28.75
Sublimation curve (applicable down to 90 K) (16 ):
T
p
ln p = 14.57893 1 – 3 – 14.48067 ln T +
T
T3
3
2
3
65.35685 T – 1 – 47.14593 T – 1 + 14.53922 T – 1
T3
T3
T3
Sublimation Data (Using the Ideal Gas Equation)
T/K
Vm,v/(cm3 mol᎑1)
P/bar
90
6.63 × 10
᎑9
1.13 × 1012
100
2.15 × 10᎑7
3.87 × 1010
110
3.79 × 10᎑6
2.41 × 109
120
4.21 × 10
᎑5
2.37 × 108
᎑4
3.34 × 107
∆Vm,melt = 6.9542 × 10᎑5T 2 – 0.0686277044T + 18.1107164247
130
3.24 × 10
140
0.00186
6.26 × 106
Melting results:
We have fitted the liquid Vm, versus T data below to the following
cubic polynomial (given to 10 decimal places—this precision is not
justified by the underlying data, but is required to yield calculated
values of sufficient accuracy):
Vm, = ᎑3.0331 × 10᎑6T 3 + 0.0028688133T 2 –
150
0.00843
1.48 × 106
160
0.0314
424,000
170
0.0995
142,000
180
0.276
54,300
190
0.684
23,100
200
1.55
10,700
0.9200958384T + 132.4799710302
∆Vm,melt(quad) is the calculated value for the volume change at the
corresponding melting temperature, using the quadratic fit above.
Vm,s(calcd) (= Vm, – ∆Vm,melt) is the corresponding calculated molar
volume of solid CO2 using the difference of the above two equations:
∆Vm,melt(quad) = ᎑3.0331 × 10᎑6T 3 + 2.799268 × 10᎑3T 2 –
210
3.27
5,340
216.58
5.19
3,470
0.85146813560T + 114.3692546
NOTE: The van der Waals equation, using a = 3.688 bar cm6 mol᎑2,
b = 42.67 cm3 mol᎑1 (1), gives essentially the same values.
Fluid equation of state (17 ):
p=
RT
V – B 1 + B 2T –
B3
–
A1
TV
2
+
A2
V4
3
V +C
with C = B3/(B1 + B2T ) and B1 = 28.06474, B2 = 1.728712 × 10᎑4,
B3 = 8.365341 × 104, A1 = 1.094802 × 109, A2 = 3.374749 × 109
in units of cm3 mol᎑1, K, and bar.
876
Journal of Chemical Education • Vol. 79 No. 7 July 2002 • JChemEd.chem.wisc.edu
PHYSICAL REVIEW SPECIAL TOPICS—PHYSICS EDUCATION RESEARCH 11, 020123 (2015)
Student understanding of the Boltzmann factor
Trevor I. Smith,1 Donald B. Mountcastle,2 and John R. Thompson2,3
1
Department of Physics and Astronomy and Department of STEAM Education,
Rowan University, Glassboro, New Jersey 08028, USA
2
Department of Physics and Astronomy, University of Maine, Orono, Maine 04469, USA
3
Maine Center for Research in STEM Education, University of Maine, Orono, Maine 04469, USA
(Received 29 September 2014; published 23 September 2015)
[This paper is part of the Focused Collection on Upper Division Physics Courses.] We present results of
our investigation into student understanding of the physical significance and utility of the Boltzmann factor
in several simple models. We identify various justifications, both correct and incorrect, that students use
when answering written questions that require application of the Boltzmann factor. Results from written
data as well as teaching interviews suggest that many students can neither recognize situations in which
the Boltzmann factor is applicable nor articulate the physical significance of the Boltzmann factor as an
expression for multiplicity, a fundamental quantity of statistical mechanics. The specific student difficulties
seen in the written data led us to develop a guided-inquiry tutorial activity, centered around the derivation of
the Boltzmann factor, for use in undergraduate statistical mechanics courses. We report on the development
process of our tutorial, including data from teaching interviews and classroom observations of student
discussions about the Boltzmann factor and its derivation during the tutorial development process. This
additional information informed modifications that improved students’ abilities to complete the tutorial
during the allowed class time without sacrificing the effectiveness as we have measured it. These data also
show an increase in students’ appreciation of the origin and significance of the Boltzmann factor during the
student discussions. Our findings provide evidence that working in groups to better understand the physical
origins of the canonical probability distribution helps students gain a better understanding of when the
Boltzmann factor is applicable and how to use it appropriately in answering relevant questions.
DOI: 10.1103/PhysRevSTPER.11.020123
PACS numbers: 01.40.Fk, 01.40.gb, 05.20.-y, 05.70.-a
I. INTRODUCTION
The study of student understanding of advanced topics
is becoming increasingly prevalent in physics education
research [1–18]. Investigating upper-division undergraduate students provides a snapshot of the intellectual journey
from novice introductory student to expert physicist that
may reveal key components of this transition [19].
Moreover, the National Research Council has recently
emphasized the need for more study of advanced undergraduate education in many science disciplines [20]. As
part of a broader study on student learning in thermal
physics, we have investigated student understanding of the
Boltzmann factor with the goal of developing instructional
strategies to improve that understanding.
Statistical mechanics provides a mechanism for understanding the emergence of macroscopic phenomena from
the collective properties of individual microscopic systems;
as such, it is a cornerstone of contemporary physics.
However, due to its complexity and sophistication, students
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)=020123(17)
do not typically encounter statistical mechanics until late in
their undergraduate (or even graduate) studies, and comparatively little research has been done to document student
difficulties and successes in this field [13,15,17,18,21,22].
This work showed that even after instruction students
often struggle to distinguish microstates of a system from
macrostates and to appropriately relate the two. The
fundamental assumption of statistical mechanics states that
all accessible microstates of a system (microscopic arrangements of a system’s particles in phase space) are equally
probable [23]. Microstates that share common macroscopic
properties (system volume, internal energy, etc.) may be
grouped into measurable macrostates. The probability of
finding the system in a particular macrostate, Mi , is
determined by the number of microstates corresponding
to that macrostate, i.e., the multiplicity ωi normalized by
the total number of microstates:
ω
PðMi Þ ¼ P i :
j ωj
ð1Þ
Much of the intellectual effort of statistical mechanics is
spent defining the relevant properties of the microstates and
macrostates and determining the multiplicity given the
macroscopic properties of the system [24].
020123-1
Published by the American Physical Society
SMITH, MOUNTCASTLE, AND THOMPSON
PHYS. REV. ST PHYS. EDUC. RES 11, 020123 (2015)
Loverude reports that many students have difficulty
distinguishing microstates and macrostates in the context
of binary systems [17]. In one question he asked students,
after flipping six coins, if the probability of getting five
heads was more than, less than, or equal to the probability
of getting six heads. About 20% of the students incorrectly
stated that the probabilities were the same, often claiming,
“all probabilities have equal occurrences,” which is true for
microstates but not for macrostates (see Ref. [17], p. 190).
In another question, students had to compare the probabilities of a six-child family having two different sequences
of boys and girls (GBGBBG vs BGBBBB). Over one-third
of students incorrectly stated that the second sequence was
less probable because families are more likely to have equal
numbers of boys and girls rather than only one girl out of
six, thus connecting the probabilities of a macrostate (the
relative number of boys and girls) to an individual microstate (a specific birth sequence).
Loverude also provides evidence that students struggle to
distinguish microstates from macrostates, especially in the
context of interacting systems. In the context of Einstein’s
model for a solid lattice structure, Loverude asked students
to determine the most likely energy distribution between
two lattices of different sizes [18]. About 40% of students
incorrectly stated that the most probable macrostate is the
one in which each solid has the same amount of energy and
disregarded the number of oscillators within each lattice.
Loverude also reports that students often add the multiplicities of interacting Einstein solids to determine the total
multiplicity rather than appropriately multiplying them [18].
A key aspect of equilibrium statistical mechanics is that,
when dealing with large systems (∼1023 particles), the most
likely state of the system is overwhelmingly the most
probable. This result is due to the fact that the statistical
spread of the macrostate probability distribution tends to
decrease as σ ∝ N −1=2 , where σ is the standard deviation
and N is the number of particles in the system. When N is
large, nearly 100% of all microstates exist within a range of
macrostates that are virtually indistinguishable from each
other, i.e., within the limits of measurable uncertainty. This
single most likely “system state” is the equilibrium state
(with microscopic fluctuations) of the macroscopic thermodynamic system [25].
Mountcastle, Bucy, and Thompson studied students’
understanding of probability distributions by asking them
to determine the most probable number of “heads” when
flipping N coins as well the uncertainty in this value
(reported as a Δa) [22]. About a third of students
incorrectly indicated that the relative uncertainty remains
constant as N increases, e.g., Δa=a ¼ 15% for all cases,
and about 20% stated that the uncertainty covers the entire
range of possible values (the most probable result is
N=2 N=2). However, students readily recognized that
performing additional measurements would reduce the
uncertainty of the mean, e.g., using more rain gauges to
measure amount of rainfall [22]. Further investigation
showed that students have difficulty reconciling the “overwhelmingly probable” equilibrium state with calculations
and graphs showing that the probability of the single most
likely macrostate actually decreases with increasing N:
Pmax ¼ N!=½2N ðN=2Þ!ðN=2Þ! for the binomial distribution. Some students took this idea to the extreme on the
coin toss question by stating that the most probable result
of flipping 6 × 1023 coins is 3 × 1023 1 heads. The
distinction between a single discrete macrostate and an
equilibrium thermodynamic “state” (consisting of a range
of virtually indistinguishable macrostates) is subtle and
requires careful attention by both students and instructors.
These results, along with Loverude’s [18], provide the
foundation for studies into students’ understanding of the
statistical treatment of thermodynamic systems where
states are defined by continuous (rather than discrete)
quantities.
The canonical probability distribution defined by the
Boltzmann factor has been described as “the quintessential
expression of the statistical mechanical approach”
(Ref. [26], p. 109) and “the most powerful tool in all of
statistical mechanics” (Ref. [23], p. 200). By knowing the
possible microscopic energy eigenstates, one may deduce
the thermodynamic equilibrium properties of any system
at constant temperature, including average internal energy,
free energy, entropy, pressure, heat capacity, etc. This
connection between microscopic and macroscopic properties is known to be difficult in multiple contexts in physics
[17,18,27–30] as well as chemistry [31]. As the core of
statistical mechanics is this micro-macro connection, this
topic is an optimal context for an investigation of this
nature. Our investigation of student understanding of the
Boltzmann factor provides additional information about
difficulties students have with this connection; these results
have implications for studies of more complex systems and
topics.
In this paper we present results of our investigation into
student understanding of the physical significance and
utility of the Boltzmann factor in several simple models.
We identify various justifications, both correct and incorrect, that students use when answering written questions
that require application of the Boltzmann factor. Results
from written data as well as teaching interviews suggest
that many students can neither recognize situations in
which the Boltzmann factor is applicable nor articulate
the physical significance of the Boltzmann factor as an
expression for multiplicity, a fundamental quantity of
statistical mechanics. The specific student difficulties seen
in the written data led us to develop a guided-inquiry
tutorial activity, centered around the derivation of the
Boltzmann factor, for use in an undergraduate statistical
mechanics course. We report on the development process of
our tutorial, including data from teaching interviews and
classroom observations on student discussions about the
020123-2
STUDENT UNDERSTANDING OF THE …
PHYS. REV. ST PHYS. EDUC. RES 11, 020123 (2015)
Boltzmann factor and its derivation during the tutorial
development process. This additional information informed
modifications that improved students’ abilities to complete
the tutorial during the allowed class time without sacrificing the effectiveness as we have measured it. Our findings
provide evidence that working in groups to better understand the physical origins of the canonical probability
distribution helps students gain a better understanding of
when the Boltzmann factor is applicable and how to use it
appropriately in answering relevant questions.
The canonical partition function (Z) is the result of the
normalization constraint that the sum of probabilities
[Pðψ j Þ] over all j must be unity:
II. PHYSICS OF THE BOLTZMANN FACTOR
Before discussing our research on student understanding
of the Boltzmann factor, it is useful to provide an overview
of the physics (and mathematics) of the Boltzmann factor
and the canonical partition function. The particular derivation of the Boltzmann factor and the canonical partition
function through which students are guided in the tutorial is
included in the Appendix.
The underlying assumption of the canonical ensemble is
that the thermodynamic system has a fixed equilibrium
temperature, a fixed number of particles, and may exchange
energy with its surroundings. A standard model for the
canonical ensemble is a very small system in equilibrium
with a large thermal energy reservoir (free to exchange
energy but not particles, see Fig. 1). The Boltzmann factor
is a mathematical expression for the probability that a
system in equilibrium at a fixed temperature is in a
particular energy state,
Pðψ j Þ ∝ e−Ej =kT ;
ð2Þ
where ψ j denotes the microstate with a particular energy
Ej , k is Boltzmann’s constant, and T is the temperature of
the system [32]. The decaying exponential form of the
Boltzmann factor results from an expression of the multiplicity of the reservoir derived from Boltzmann’s equation,
S ¼ k lnðωÞ:
ð3Þ
As the energy of the system decreases, the energy (and
multiplicity) of the reservoir increases in such a way that
the total probability increases (see the Appendix for full
details).
FIG. 1. Sample system for the Boltzmann factor instructional
sequence. An isolated container of an ideal gas is separated into a
small system (C) and a large reservoir (R). The label “C” is used
to avoid confusion with entropy.
X e−Ej =kT
X
Pðψ j Þ ¼
¼ 1;
Z
j
j
Z¼
X
e−Ei =kT ;
ð4Þ
ð5Þ
i
where Z depends on temperature but is independent of the
energy value Ej [33]. One may also express the partition
function in terms of the energy of a macrostate E:
Z
Z¼
DðEÞe−E=kT dE;
ð6Þ
all E
where the density of states function DðEÞ accounts for the
degeneracy (or multiplicity) of the macrostate. In this way
the canonical partition function is equally valid for systems
with discrete energy microstates, as in Eq. (5), and those
with continuous energy distributions, as in Eq. (6).
The canonical partition function can be used to express
equilibrium (macroscopic) thermodynamic quantities. For
example, the Helmholtz free energy of a system may be
written as a function of Z,
F ¼ −kT lnðZÞ;
ð7Þ
derivatives of F yield information about the system’s
entropy, pressure, magnetization, and many other thermodynamic variables. Moreover, the average energy of a
system hEi may be expressed as a derivative of the natural
logarithm of Z. Because of these connections, Schroeder
refers to the canonical probability distribution as “the most
useful formula in all of statistical mechanics” (Ref. [23],
p. 223). The canonical partition function and the
Boltzmann factor are cornerstones of statistical mechanics,
and a thorough understanding of when and how they are
useful (when examining an equilibrium system at constant
temperature) is essential for study in the field.
III. STUDENT UNDERSTANDING OF THE ORIGIN
AND UTILITY OF THE BOLTZMANN FACTOR
We gathered data in several different forms to study
student understanding of the Boltzmann factor from multiple perspectives. In one investigation, we gave students
an ungraded written survey to determine whether or not
they use the Boltzmann factor in an appropriate context.
Additionally, we conducted teaching experiments with
several students as well as classroom observations to assess
their understanding of the physical origin and significance
of the mathematical expression of the Boltzmann factor.
Our results indicate that students often do not use the
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SMITH, MOUNTCASTLE, AND THOMPSON
PHYS. REV. ST PHYS. EDUC. RES 11, 020123 (2015)
Boltzmann factor in appropriate contexts, instead using
vague notions of lower energies having higher probabilities
to make conclusions about the ratios of these probabilities.
Moreover, we find that students may not recognize the
physical significance of the Boltzmann factor, even after
having memorized its mathematical derivation.
(school 1); data were collected from seven successive
classes (N ¼ 50). Students at school 1 are typically senior
undergraduates who have competed studies in classical
mechanics, electrodynamics, and quantum mechanics.
The PRQ was also administered once in a single-semester
upper-division thermal physics course at a comprehensive
public university in the western United States (school 2,
N ¼ 32). Students at school 2 are typically junior undergraduates who have completed studies in modern physics
and classical mechanics. The PRQ was administered at
both schools immediately before students participated in
guided-inquiry activities regarding the Boltzmann factor
and the canonical partition function (our Boltzmann Factor
tutorial, see Sec. IV). At school 1 the activities were used
after lecture instruction, and the PRQ was given after
lectures. At school 2 the activities were used in place of
lecture instruction, and the PRQ was given before instruction to establish a baseline for students’ understanding
before the tutorial.
Student responses to the PRQ were coded in two ways:
first by the response given (equal to, greater than, less than,
or other), second by whether or not the Boltzmann factor
was used. Figure 3(a) shows the response frequencies for
the entire seven-year data corpus from school 1, and
Fig. 3(b) shows the response frequencies from school 2.
Green diagonal stripes indicate the students who used the
Boltzmann factor or stated that the energy of the ground
state was irrelevant (in part B) to obtain their chosen
answers; these students are considered to have used correct
explanations regardless of which answer they chose [35].
We used a grounded theory approach to analyze students’ explanations for their responses; the entire data
corpus was examined for common trends, yielding categories defined by the data, and all data were reexamined to
group them into those defined categories [36,37]. One goal
of our analysis was to focus on describing rather than
interpreting students’ explanations while defining the
categories. In this way our analysis stays as true to the
A. Student use of the Boltzmann factor in
appropriate contexts
One desired result of teaching students about the
Boltzmann factor is that they will recognize applicable
situations and use it appropriately to make claims about
probabilities of the occupation of specific energy states.
The probability ratios question (PRQ, shown in Fig. 2)
probes their ability to do this. The correct solution to the
PRQ requires students to recognize three pieces of
information:
• The probability of a single particle being in each of
three energy states is proportional to the Boltzmann
factor for each state,
• a ratio of exponential functions is the exponential of
the difference of their exponents,
• the differences in energies between adjacent states
are the same for each particle (ΔEn;n−1 ¼ 0.05 eV).
The first two items indicate that each ratio of probabilities is
an exponential function of the energy difference between
the two states. The third item reveals that both pairs of
ratios in the PRQ are equal [34]. Students were also
considered to have given a correct explanation to part B
of the PRQ if they stated that the two ratios were equal
because the only difference between the two particles is the
energy of the ground state.
1. Recognizing the need for the Boltzmann factor
The PRQ was administered to students in an upperdivision statistical mechanics course at a land-grant
research university in the northeastern United States
FIG. 2.
Probability ratios question (PRQ) given as an ungraded survey before tutorial instruction.
020123-4
STUDENT UNDERSTANDING OF THE …
40
30
School 1
Part A: Within
System
Part B: Shifted
Ground Level
Used Boltzmann factor or
Correct explanation
90%
80%
70%
60%
50%
20
40%
30%
10
32
100%
20%
Number of Students
Number of Students
50
PHYS. REV. ST PHYS. EDUC. RES 11, 020123 (2015)
School 2
Part A: Within
System
Part B: Shifted
Ground Level
24
80%
60%
16
50%
40%
30%
8
20%
10%
Other
0%
Less than
Greater than
Equal to
Other
Less than
Greater than
Other
Equal to
0
0%
Less than
Greater than
Equal to
Other
Less than
Greater than
Equal to
90%
70%
Used Boltzmann factor or
Correct explanation
10%
0
100%
FIG. 3. PRQ pretutorial results. (a) School 1, after lecture instruction on the Boltzmann factor over seven years (N ¼ 50); (b) school 2,
before any instruction on the Boltzmann factor in one year (N ¼ 32). The green diagonal stripes indicate the students who used the
Boltzmann factor or stated that the energy of the ground state was irrelevant (in part B) to obtain their chosen answers. Students in the
“Other” column often provided no explicit answer or stated that there was not enough information to determine the answer. Only 24
students from school 1 and four students from school 2 used the Boltzmann factor on both parts.
data as possible by limiting researcher biases and interpretations. This is consistent with Heron’s identification of
specific difficulties [38].
The data represented in Fig. 3 suggest two questions.
(1) What is the prevalence of invocation of the Boltzmann
factor, regardless of the correctness of the response?
(2) How do students justify their answers if they do not
apply the Boltzmann factor? To answer the first of these
questions, the data show four categories of responses:
• correct response (equal to) using the Boltzmann factor
(or stating that the energy of the ground state was
irrelevant in part B),
• correct response without using the Boltzmann factor,
• incorrect response using the Boltzmann factor,
• incorrect response without using the Boltzmann
factor.
This coding scheme enables highlighting of the number
of students who are and are not invoking the Boltzmann
factor to answer the PRQ. A natural question associated
with this coding scheme is, how might someone invoke the
Boltzmann factor but arrive at an incorrect response? One
route is to make a computational error. On the other hand,
one could compare the wrong ratios, but do so correctly
using the Boltzmann factor. Data also indicate that some
students imposed degeneracy terms when using the
Boltzmann factor to answer the PRQ. In coding responses,
a student who wrote that probability is related to a decaying
exponential of the energy was coded as using the
Boltzmann factor independent of the final answer obtained.
Using the Boltzmann factor and stating that the energy of
the ground state is irrelevant were grouped together because
both are correct physical justifications for concluding
that the ratios of probabilities in part B of the PRQ
are equal.
Table I shows the percentages of students who occupy
each of the four response categories at each school for both
parts of the PRQ. From the data shown in Fig. 3 and Table I,
it is clear that the distribution of responses is different at the
two schools. A Fisher’s exact test showed this to be true
(p ¼ 0.008 for part A, p < 0.001 for part B) [39–41].
TABLE I. Results from the PRQ pretest at both school 1 (N ¼ 50) and school 2 (N ¼ 32). Students are grouped
by whether or not they gave the correct answer (“equal to”) and whether or not they used the Boltzmann factor (BF)
(or stated that the energy of the ground state was irrelevant in part B).
Part A
School 1
School 2
Used BF
No BF
Used BF
No BF
Part B
Correct
Incorrect
Total
Correct
Incorrect
Total
34%
6%
9%
13%
14%
46%
3%
75%
48%
52%
12%
88%
62%
6%
16%
22%
6%
26%
0%
63%
68%
32%
16%
84%
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SMITH, MOUNTCASTLE, AND THOMPSON
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Another Fisher’s exact test showed that students at school 1
are using the Boltzmann factor on the PRQ pretest more
than students at school 2 (p < 0.001 on both parts) [42].
This is not surprising given that students at school 1 had
received lecture instruction on the Boltzmann factor, while
students at school 2 had not. On the other hand, only 48%
of students at school 1 used the Boltzmann factor on part A
and only 68% did so on part B, indicating that lecture
instruction alone was not sufficient for all students to gain a
robust understanding of when and how to use the
Boltzmann factor.
The most common incorrect response at school 1 for
both parts of the PRQ is that Pð0.10 eVÞ=Pð0.05 eVÞ <
Pð0.05 eVÞ=Pð0.00 eVÞ (“less than” for part A and
“greater than” for part B, see Table II). These answers
are considered consistent because the second and third
energy levels in particle B have the same numerical values
as the first and second energy levels in particle A,
respectively. A Fisher’s exact test shows the distribution
of “less than” and “greater than” responses from school 1 to
be significantly different for part A as compared to part B
(p ¼ 0.035). However, the data from school 2 show the
exact opposite trend: more students answer “greater than”
for part A and “less than” for part B [see Fig. 3(b) and
Table II]. A Fisher’s exact test shows that this difference at
school 2 approaches significance (p ¼ 0.061). Additional
tests show that the results from school 2 are significantly
different from those at school 1 (p ¼ 0.038 for part A and
p ¼ 0.036 for part B).
Figure 3(a) also shows that students at school 1 are more
likely to answer part B correctly (which discusses an
effective shift in the ground state energy of a system) than
part A (comparing two different sets of probabilities for
states within the same system), with only 34% using the
Boltzmann factor to obtain the correct response on part A
compared to 64% providing a correct explanation on part B
(statistically significant, p ¼ 0.035). One student at school
1 justified his response for part B in stating that, “… it does
not matter what the ‘baseline’ is, just the amount of energy
added.” This higher performance on part B could be a result
of our coding scheme in that explanations involving comments about the arbitrariness of the ground state energy
were considered correct for part B regardless of the
student’s response to part A. This phenomenon is not
significantly observed at school 2 [see Fig. 3(b), Fisher’s
exact test yields p ¼ 0.45].
TABLE II. Pretest response comparison: “greater than” versus
“less than.” Numbers shown indicate the percentage of incorrect
responses at each of the two schools. This is necessary because
significantly more students answered the PRQ correctly at school
1 than at school 2. Only by looking at the percentages of incorrect
responses can meaningful comparisons be made.
Part A
School 1
School 2
Part B
Greater than
Less than
Greater than
Less than
23%
54%
38%
25%
44%
26%
19%
44%
2. Incorrect reasoning about probability ratios
The justifications students used to support their final
answers were sorted into several categories developed
using a grounded theory approach. At school 1, 24 students
(out of 50) used the Boltzmann factor within their explanation of their answers on the PRQ; only four out of 32
students at school 2 used the Boltzmann factor. Of the
remaining students at each school, roughly half (15 out of
26 at school 1 and 13 out of 28 at school 2) used a ranking
of probabilities as their primary justification; e.g.,
PA ð1Þ > PA ð2Þ > PA ð3Þ. An additional five students at
school 2 stated that the lowest energy is most probable but
did not make claims about the relative probabilities of
energy states 2 and 3. Using probability ranking, either
explicit or implied, is the most common incorrect justification at both schools, and no students provided a physical
explanation for why the probabilities of the various energy
levels would be ranked as they claimed.
Of the students who ranked the probabilities to
justify their answers, eight students at school 1 and seven
at school 2 made claims about the relative difference in
probability between states 1 and 2 and between states 2
and 3. Some claims were made in sentence form, e.g., “… it
is more likely that the system will have less energy so
the difference between [states] 3 & 2 is less than
[between states] 2 and 1” (student’s emphasis); other
claims took the form of a mathematical expression, e.g.,
“PA ð1Þ − PA ð2Þ > PA ð2Þ − PA ð3Þ.” Both of these statements imply the idea that PA ð1Þ ≫ PA ð2Þ > PA ð3Þ. All
seven students at school 2 used this idea to claim that
PA ð3Þ=PA ð2Þ > PA ð2Þ=PA ð1Þ. However, the students at
school 1 used similar reasoning to come to three different
conclusions:
PA ð1Þ ≫ PA ð2Þ > PA ð3Þ →
PA ð3Þ PA ð2Þ
>
;
PA ð2Þ PA ð1Þ
ð8Þ
PA ð1Þ > PA ð2Þ ≫ PA ð3Þ →
PA ð3Þ PA ð2Þ
<
;
PA ð2Þ PA ð1Þ
ð9Þ
PA ð1Þ ≫ PA ð2Þ ≫ PA ð3Þ →
PA ð3Þ PA ð2Þ
¼
:
PA ð2Þ PA ð1Þ
ð10Þ
Interestingly, this third case was used to justify a correct
response.
In each of these cases, students seem to be considering
the probabilities in pairs and using the relative difference
between each pair to compare the ratios of the pairs. This is
consistent with a strategy for comparing fractions that
Smith refers to as compare numerator-denominator
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STUDENT UNDERSTANDING OF THE …
PHYS. REV. ST PHYS. EDUC. RES 11, 020123 (2015)
differences (NDD) [43]. The NDD strategy is categorized
by students using the within-fraction difference between
the denominator and the numerator as a comparative
measure. Examples of the NDD strategy in Smith’s study
include students determining that 3=5 ¼ 5=7 (because
5 − 3 ¼ 7 − 5 ¼ 2) and that 14=24 > 7=12 (because
24 − 14 > 12 − 7) [43]. In our case, students are explicitly
or implicitly using the differences between the probabilities
as justification for comparing their ratios. Arons cites
difficulties interpreting ratios as one of the most prevalent
cognitive gaps for students at the secondary and undergraduate levels (Ref. [44], pp. 4–9).
Students who ranked the probabilities as simply PA ð1Þ >
PA ð2Þ > PA ð3Þ also made claims that are consistent with
some of Smith’s other classifications. Using this ranking to
claim that PA ð3Þ=PA ð2Þ > PA ð2Þ=PA ð1Þ (“greater than” on
part A) is consistent with Smith’s denominator principle
(fractions with larger denominators are smaller than fractions with smaller denominators), and using this ranking to
claim that PA ð3Þ=PA ð2Þ < PA ð2Þ=PA ð1Þ (“less than” on
part A), is consistent with both the numerator principle
(fractions with larger numerators are larger) and larger
components (fractions with larger numerators and denominators are larger) [43]. However, since no student admitted
to exclusively using either the numerator or the denominator of each ratio to compare the two, we cannot be certain
that students used these strategies, only that the students’
final responses are consistent with their use.
Students who used probability rankings to justify their
answers were categorized as being consistent with one (or
more) of Smith’s strategies. The reliability of the categories
for our classification was checked by an independent
classification of the data from school 2. There was initial
agreement for 72% of the student responses; after discussion and negotiation, agreement of 91% and at least
partial agreement of 97% of students was obtained (one
analysis placed some students simultaneously in two
categories, while the other only agreed on one of the
categories for this group).
At both school 1 and school 2 more student responses
were aligned with the NDD strategy than either the
denominator principle or the numerator principle and larger
components, and no significant differences were found
between the two student populations in terms of their use of
these strategies. In many cases it is unclear precisely why a
student chose the response they did based on the ranking
provided, but it is interesting to note the similarities
between their claims and those made by the adolescent
students in Smith’s study.
The key difficulty identified so far is that many students
do not apply the Boltzmann factor when it is appropriate to
do so, even after lecture instruction. Instead, these students
provide responses that are consistent with using novicelike
reasoning strategies for comparing ratios. Most students
recognized that lower energies are more probable, but they
offered no physical justification for why this is so and could
not use this information alone to make conclusions about
the probability ratios in question.
B. Recitation of a mathematical derivation without
physical understanding
In an effort to probe student understanding of the
Boltzmann factor more deeply, we conducted individual
interviews with four students at school 1 after classroom
instruction in the first year of tutorial implementation to
determine their familiarity with the Boltzmann factor, its
applications, and its origin. Two interview participants had
participated in the first half of the Boltzmann Factor tutorial
during class (in which they discussed the definitions of
macrostates, microstates, and multiplicity for the microcanonical and canonical ensembles; see Sec. IV), while the
other two had not seen the tutorial. The interviews were
conducted in the style of a teaching experiment [15,45,46]
and consisted of asking students to complete a guidedinquiry activity that started with asking them to consider
how probability relates to multiplicity in the divided
container (C-R) scenario (see Fig. 1) and culminated with
the derivation of the Boltzmann factor [47].
The teaching experiment is a unique form of interview
as “it is an acceptable outcome … for students to modify
their thinking” during the course of the interview [46].
According to Steffe and Thompson, “a teaching experiment
involves a sequence of teaching episodes … a teaching
agent, one or more students, a witness of the teaching
episodes, and a method of recording what transpires during
the episode” [45]. For our purposes the interviewer alternated roles as both teaching agent and witness during each
interview. In a sense, the activities used during the interview may also be seen as a teaching agent as they included
tasks for students to complete and students interacted with
the document in an intellectual manner. Our goal for the
interviews was not to simply determine students’ understanding of the Boltzmann factor, but rather to examine
how well they could complete instructional tasks based
on previous knowledge related to the Boltzmann factor.
Students worked on their own; the interviewer solicited
explanations for their work and gave assistance when
required. Field notes were taken during the interviews
and students’ written work was collected afterward.
Results from the teaching interviews provide further
evidence of the need for the Boltzmann Factor tutorial,
especially with regard to the origin of the Boltzmann
factor itself. None of the interview participants found the
tasks to be trivial, and none correctly articulated how the
Boltzmann factor as an expression of probability relates to
multiplicity prior to the interview. A major finding during
these interviews was the identification of students’ difficulties in executing the Taylor series expansion as part of
the derivation of the Boltzmann factor; we reported these
difficulties previously [15].
020123-7
SMITH, MOUNTCASTLE, AND THOMPSON
PHYS. REV. ST PHYS. EDUC. RES 11, 020123 (2015)
One episode during one of the student interviews was of
particular interest. One student (Joel [48], who had participated in portions of the Boltzmann Factor tutorial in
class) was very familiar with the applications of the
Boltzmann factor and seemed to be just as familiar with
its origin. In one portion of the activity, students were given
a table of multiplicities for various discrete system energy
levels and asked to determine the most probable macrostate
(see Table III). The desired result was for students to
conclude that the macrostate with the greatest reservoir
multiplicity would be the most probable. Joel wanted to use
the Boltzmann factor rather than thinking about multiplicities, even though no information had been given about
the relative energy values [49]. The interviewer asked Joel
to show where the Boltzmann factor came from before
applying it to this situation, at which point Joel quoted the
textbook derivation of the Boltzmann factor practically
verbatim. The final portion of Baierlein’s mathematical
derivation is as follows [[26,50], p. 92],
Boltzmann factor he had implicitly written that it was
proportional to ωR [connecting Eqs. (11) and (14)], but
without explicit help from the interviewer, Joel could not
recognize that the multiplicity of the reservoir when it has
energy, Etot − Ej [rhs of Eq. (11)], is proportional to the
exponential function, exp ð−Ej =kTÞ [rhs of Eq. (14)].
Furthermore, Joel had great difficulty relating the physical
example used in the textbook (a “bit of cerium magnesium
nitrate … in good thermal contact with a relatively large
copper disc” (Ref. [26], p. 91) to the ideal gas example used
during our interview. He was unable to recognize and
articulate the important physical characteristics of each
scenario that make the Boltzmann factor applicable, i.e., a
system with fixed temperature and variable energy. Joel’s
failure to make these connections suggests an incomplete
understanding of the physical reasoning used to derive the
Boltzmann factor, even after memorizing the textbook
derivation.
Results from the teaching interviews and the PRQ
suggest that many students can neither recognize situations
in which the Boltzmann factor is applicable nor articulate
the physical significance of the Boltzmann factor as an
expression for multiplicity, one of the fundamental quantities of statistical mechanics. These difficulties prompted
our development of the Boltzmann Factor tutorial to help
students better understand the physical origin of the
Boltzmann factor and how it may be applied in various
contexts.
Pðψ j Þ ¼ const ×
multiplicityof reservoir when
ithasenergyEtot − Ej
;
ð11Þ
1
ð12Þ
Pðψ j Þ ¼ const × exp SR ðEtot − Ej Þ ;
k
1
∂SR
Pðψ j Þ ¼ const × exp SR ðEtot Þ þ
ð−Ej Þ ; ð13Þ
k
∂ER Etot
Pðψ j Þ ¼ ðnew constantÞ × exp ð−Ej =kTÞ:
ð14Þ
This derivation exploits the fact that the combined energy
of the system and reservoir (Etot ) is a fixed quantity in order
to write the energy of the reservoir (ER ) in terms of the
energy of the system (Ej ).
Joel’s ability to reproduce the derivation might suggest
an understanding of the physical significance of the
Boltzmann factor. However, when asked how the multiplicity of the reservoir relates to the Boltzmann factor, Joel
was at a loss. During his replication of the derivation of the
TABLE III. Sample energy and multiplicity values for the “toy
model” system (C) and reservoir (R); see Fig. 1. This table was
presented to students during the teaching interviews and is also
used in the Boltzmann Factor tutorial. A key element of this
situation is that the combined energy of C and R is a fixed
value, Etot .
EC
ωC
ER
ωR
E1
E2
E3
E4
E5
1
1
1
1
1
Etot − E1
Etot − E2
Etot − E3
Etot − E4
Etot − E5
3 × 1018
5 × 1019
4 × 1017
1 × 1020
7 × 1018
IV. DESIGN AND IMPLEMENTATION OF THE
BOLTZMANN FACTOR TUTORIAL
Given students’ apparent lack of recognition of when
to apply the Boltzmann factor to a physical scenario, we
designed a guided-inquiry tutorial activity to lead students
through its derivation and encourage deep cognitive connections between the physical quantities involved. The
derivation chosen for use in the Boltzmann Factor tutorial
is included in the Appendix and may be found in many
widely used textbooks, including the one used at the
primary research site [26].
Our Boltzmann Factor tutorial gives students the opportunity to productively struggle with the connections
between the mathematical formalism and the physical
interpretations within the derivation of the Boltzmann
factor [15]. Fostering physics-mathematics connections,
such as gaining facility with taking limits and making
approximations, as well as knowing when to take these
steps, is an important and nontrivial component of upperdivision courses as students transition from novices to
experts in the field [19].
The desired student outcomes during the tutorial are
consistent with the concept of productive disciplinary
engagement (PDE) [51]. The small group setting, with
explicit instructions to discuss responses and reasoning
with group mates, fosters engagement, which is evident
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STUDENT UNDERSTANDING OF THE …
PHYS. REV. ST PHYS. EDUC. RES 11, 020123 (2015)
through student-student discourse. Because disciplinary
content is the core of the tutorial, most engagement in a
tutorial constitutes disciplinary engagement. Engle and
Conant define productive disciplinary engagement as
episodes where students are making progress in their
engagement with the content [51]. Evidence of this
productivity includes students recognizing their confusion
about a concept or making a new connection as a result of
the interaction. Indeed, the tutorial pedagogy and the
materials themselves provide a setting designed to foster
PDE. The pedagogy used in the canonical set of tutorials,
Tutorials in Introductory Physics, to address specific
student difficulties, fully matches the parameters of PDE
(Ref. [52] and see also Ref. [53], p. iii). Our primary goal is
that students discuss topics in a way that helps them
progress through the tutorial tasks while gaining a better
understanding of those topics (discussing relevant concepts, synthesizing information, engaging with the connections between the mathematics and the physics, etc.).
While in some cases other, less time-consuming pedagogical approaches may also foster PDE and help students with
specific difficulties, in this case the depth of the content and
the difficulty of the sequence of steps in the derivation of
the Boltzmann factor suggested that a tutorial would be the
most effective way to achieve this. Given that a lecture on
this topic typically uses an entire class period, we expected
the tutorial would occupy a full class period as well as some
time outside of class.
system microstate in which the macrostate of R has the
largest multiplicity (ωR ) is the most probable (E4 in
Table III) because all reservoir microstates are equally
likely. Careful consideration of the relative probabilities of
each macrostate leads to the proportionality between the
probability of the jth microstate of C and the multiplicity of
the reservoir: Pðψ j Þ ∝ ωR ðψ j Þ.
The final section of the Boltzmann Factor tutorial is the
derivation of the Boltzmann factor itself. The core of this
derivation is a Taylor series expansion of SR ðER Þ about the
value ER ¼ Etot to obtain the expression for SR as a linear
function of EC given in Eq. (A4) [15]. The students are
explicitly asked to consider the physical significance of
each term in the expansion and to determine the final linear
expression on their own. Then, using the relationship
between entropy and multiplicity in Eq. (3), they are
guided to derive an expression for ωR :
A. Boltzmann Factor tutorial
The Boltzmann Factor tutorial begins by asking students
to consider an isolated container of an ideal gas. They are
guided to recognize that the container has a fixed internal
energy (Etot ) and that all accessible microstates are equally
probable.
Once the properties of the contents of the isolated
container have been established, the students are presented
with a scenario in which the container of ideal gas is
separated into relatively small and large sections (see
Fig. 1). The small system of interest (C) is said to be in
thermal equilibrium with the large reservoir (R), and the
students are asked to compare the values of various
thermodynamic properties of C to those of R to highlight
the fact that the intensive properties (temperature, pressure)
will have the same value for both C and R, while the values
of the extensive properties (volume, number of particles,
internal energy) of C are much smaller than those of R.
The third section of the tutorial uses the fact that the
multiplicities of C and R are so different (ωC ≪ ωR ) to
justify a single-particle “toy model” in which ωC ¼ 1
(ωtot ¼ ωC ωR ¼ ωR ), and the energy of C can only take
on a handful of discrete values, EC ∈ fEj g ¼ fE1 ; E2 ; …g
(see Table III). The students are asked to determine which
system microstate (j ¼ 1, 2, 3, 4, or 5) is most probable and
which is least probable. The desired solution is that the
ωR ¼ eSR =k ¼ eSR ðEtot Þ=k−EC =kT ;
ð15Þ
and because SR ðEtot Þ is a constant,
ωR ∝ e−EC =kT ;
ð16Þ
i.e., the Boltzmann factor. Students find that Pðψ j Þ ∝
ωR ðψ j Þ and that ωR ðψ j Þ ∝ e−Ej =kT , leading to the proportionality in Eq. (2). Finally, they obtain the expression
for the canonical partition function Z by normalizing the
probability.
The post-tutorial homework assignment is an application
of the Boltzmann factor to a three-state system with
unevenly spaced energy levels. Students are asked various
questions about the ratios of probabilities of the system
being in a particular state. These questions are similar to the
PRQ, but the students are given specific values for T and N
and asked to determine numerical values for the probability
ratio rather than compare two different ratios. They are also
asked to determine an expression for the generic ratio
between the probabilities of any two energy levels. This
homework assignment was used as a continuation of the
tutorial, not as an assessment or research tool.
B. Tutorial implementation
At school 1, the Boltzmann Factor tutorial was implemented after all lecture instruction on the Boltzmann factor.
Students were given one 50-min class period to complete
the tutorial. The course instructor and one additional
facilitator were available during the tutorial session as
observers and facilitators [54]. No course credit is offered
for participation in the tutorial itself, but the course grade
does include a component for class participation. Several
groups were videotaped during tutorial sessions (in three
years of classes) to monitor tutorial progress and document
student reasoning regarding the Boltzmann factor and
related topics.
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The Boltzmann Factor tutorial was implemented at
school 2 once in place of lecture instruction. Students
were given one 50-min class period and an additional
20 min during the next period to complete the tutorial. As
described in Sec. III A, the PRQ was administered as a
pretest at both institutions before tutorial instruction, and a
similar question was used on course examinations.
through in the tutorial improve both students’ ability to
apply the Boltzmann factor appropriately and their understanding of the physical basis for the Boltzmann factor,
including the connections between some of the mathematical steps and the physical scenario.
V. FINDINGS DURING IMPLEMENTATION OF
THE BOLTZMANN FACTOR TUTORIAL
In order to probe the effect of tutorial instruction on
students’ tendency to invoke the Boltzmann factor in an
appropriate situation, we administered written post-tests
at both schools on midterm examinations. The PRQ was
given on a course examination after the Boltzmann Factor
tutorial in two years at school 1. A similar question,
referred to as the PRQ Analog (shown in Fig. 4), was
developed by the instructor at school 2 and asked on a
course exam in one year at both institutions. The PRQ
Analog requires students to apply the same knowledge as is
used to correctly answer the PRQ: that the probability of a
particle being in one of the energy states is proportional
to the Boltzmann factor, and that a ratio of probabilities
would be equivalent to a ratio of Boltzmann factors, which
depends only on the difference between the energy levels.
However, the PRQ Analog adds some complexity by using
systems with energy levels that are not evenly spaced and
requiring students to recognize that the number of particles
occupying each energy level will be proportional to the
probability of a single particle having that energy. Despite
this added complexity, Fig. 5 shows that students’ use of
the Boltzmann factor was very similar on both the PRQ and
PRQ Analog exam questions.
From the three implementations at school 1 there are 19
sets of matched (pre- and post-tutorial) data of students
who participated in the Boltzmann Factor tutorial. There
are 29 sets of matched data from school 2. Figure 5 shows
the exam data from all students broken down by question
An interesting question is whether recognizing when to
use the Boltzmann factor serves as a direct proxy for an
understanding of the physical significance and meaning of
the expression—its origin and why it describes the relative
occupation of states. Instructors typically assume that this is
the outcome of presenting the derivation of such functions
to students: that the clear description of the steps of the
derivation, including the explicit connections between the
mathematical steps and the physical constraints, assumptions, etc. that drive the mathematics, provides students
with the intended insight. Thus, the subsequent assumption
is that the proper invocation of the Boltzmann factor
implies an understanding of its meaning and significance.
However, in the process of pilot testing the Boltzmann
Factor tutorial, we observed that this is not necessarily the
case. We have evidence from students working through
sections of the tutorial either in class or during teaching
interviews that suggest that (a) the students do not have a
sense of the physical basis for the Boltzmann factor before
the tutorial and (b) the sequencing of the tutorial provides
the students the opportunity to gain an understanding and
appreciation for this physical foundation.
Below, we describe the findings from the data collected
during and after tutorial implementation. These data—
collected in written and video form—provide evidence to
support the claim that the activities the students work
A. Improving student use of the Boltzmann factor
in appropriate contexts
FIG. 4. PRQ Analog developed by instructor at school 2. Administered on a course exam once at school 1 and at school 2.
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FIG. 5. Post-tutorial results from PRQ and PRQ Analog,
administered during course examinations. The green diagonal
stripes indicate the students who used the Boltzmann factor or
stated that the energy of the ground state was irrelevant (in part B)
to obtain their chosen answer(s). For the PRQ Analog, “Equal to”
corresponds to choice II, “Greater than” to choice I, “Less than”
to choice III, and “Other” to choice IV. Data are shown for
students who completed the PRQ pretest, participated in the
Boltzmann Factor tutorial, and completed either the PRQ or the
PRQ Analog exam question.
and school. These data provide evidence that the Boltzmann
Factor tutorial helps students recognize the utility of the
Boltzmann factor and how to apply it properly in the
context of these questions.
The most striking feature of Fig. 5 is that all 13 students
at school 1 used appropriate Boltzmann factor reasoning on
both parts of the PRQ [55]. Moreover, all but one student at
school 1 (about 83%) used the Boltzmann factor correctly
to answer the PRQ Analog after tutorial instruction. This is
a marked improvement over lecture instruction alone (only
about half consistently used the Boltzmann factor on the
PRQ pretest). Similarly, all but two students at school 2
(almost 95%) used the Boltzmann factor to answer the PRQ
Analog exam question.
In order to perform statistical analyses to compare the
exam results with the pretest results, data were grouped
into the four categories discussed in Sec. III A 1. This
reduced coding scheme is necessary because we essentially asked three questions at various times (PRQ parts A
and B and the PRQ Analog): the specific responses to the
various questions, e.g., “greater than,” cannot necessarily
be considered the same response. As such, the only
categories available for grouping responses are either the
correct response or one of the incorrect responses; along
with this we have the dimension of whether or not a
student used the Boltzmann factor appropriately to justify
his or her response, consistent with the four categories
above. These general categories do not allow claims to be
made about how reasoning patterns differ within the
incorrect responses, but they do allow comparisons of the
frequency with which students use the correct Boltzmann
factor reasoning and whether or not it yielded a correct
response. Using these categories, a Fisher’s exact test
showed that all exam data are statistically similar
(p ¼ 0.125). Additionally, a Fisher’s exact test comparing
the data at school 1 showed that the results from the
exams are statistically significantly better than the results
on the PRQ pretest on both parts A (p ¼ 0.019) and B
(p ¼ 0.012) [56]. A Fisher’s exact test also shows that
the exam results at school 2 are significantly better than
the pretest data (p < 0.001 for both parts).
The written pretest results suggest that students were not
aware of the contexts in which the Boltzmann factor is
applicable, even after lecture instruction. The written posttest results demonstrate a marked improvement in the
correct use of the Boltzmann factor in these situations.
These results suggest that the Boltzmann Factor tutorial
helps improve student understanding of how and when to
use the Boltzmann factor when it is used either as a standalone activity (school 2) or as a supplement to lecture
instruction (school 1).
B. Improving student understanding of the physical
basis for the Boltzmann factor
As mentioned above, in addition to improving appropriate student use of the Boltzmann factor, the other
major goal for the Boltzmann Factor tutorial was for
students to gain an appreciation for the physical basis of
the Boltzmann factor, which did not occur based on
lecture instruction, even when a student was able to recite
the derivation exactly, as shown in Sec. III B. However,
we anticipated that students would gain this appreciation
by working through the Boltzmann factor derivation in
small groups while emphasizing the physical justifications for each step therein. Documenting the acquisition
of this appreciation or understanding is not possible using
written data of the sort typically gathered. So, in order to
monitor student progress and success in achieving this
instructional goal for the Boltzmann Factor tutorial, we
videotaped several groups of students while they completed the tutorial during the first three years of implementation at school 1.
Segments from these classroom episodes were selected
for transcription and further analysis based on the content
of student discussions. Given our focus on investigating
students’ understanding of particular topics, our methods
of gathering video data align with Erickson’s description
of manifest content approaches, in which particular
classroom sessions are selected to be videotaped based
on the content being discussed [57]. We chose to
videotape classroom sessions in which students were
engaging in our tutorial because we are primarily
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interested in their ideas regarding the conceptual and
mathematical content of our tutorial and students’ ability
to negotiate tutorial prompts in an efficient and productive manner. We have already explained that our use of
“productive” follows its use in productive disciplinary
engagement [51]; we classify “efficient” interactions as
those enabling the students to complete the tutorial within
the intended 50-min class period. In some respects this
categorization of student interactions is done with an eye
toward the end justifying the means: an interaction
cannot necessarily be considered productive or efficient
without knowing the conversations that take place after
that interaction.
Over three years of tutorial implementation we videotaped a total of four groups containing 13 students. To
analyze video data, we watched each video in its entirety
and made note of conversations that seemed interesting; we
later watched these segments many times and recorded both
what was discussed and why we thought it was interesting.
Quotations included in this section were often selected for
their uniqueness. Several students made comments and
statements that indicated difficulties that were not expected
and have not been previously documented. Data do not
exist to verify the pervasiveness of these difficulties, but we
feel their existence is noteworthy. In cases where more than
one student displayed a similar difficulty, we have included
multiple quotes to allow the reader to evaluate the similarities and differences between the data.
Video data from the second tutorial implementation at
school 1 provide evidence that students gain an appreciation for the origin of the Boltzmann factor while participating in the Boltzmann Factor tutorial. Two students (Sam
and Bill, who worked in a group on their own) participated
in several conversations throughout the tutorial session that
indicate their contemplation of relevant physical ideas.
During the Boltzmann Factor tutorial they discussed which
macrostate (from Table III) is most probable:
physical quantities involved, and relating their expression
for multiplicity to the Taylor series of entropy, Sam and Bill
had a realization [58]:
Bill
Sam
Bill
Sam
—Probably the one with more microstates
—Yeah… the one with the highest
multiplicity
…
—“Give a general expression for the
probability of the system”… so
probably just use omega R (ωR ),
so we’d say omega R j (ωRj ) over
the sum of all of them.
—Yeah, that’s what we said: omega R j
over P
the sum of omega R j
ðωRj = ωRj Þ.
Later in the tutorial, after completing the Taylor series
expansion (with instructor intervention), interpreting the
Sam
Bill
Sam
Bill
Sam
—That’s cool. Look, see, you get the
Boltzmann factor. You solve for
omega (ω): e to the minus E over
k T (e−E=kT ).
…
—I guess that’s where it comes from.
—’Cause we didn’t know where it
came from.
—I had no idea.
—I was just like, “OK.”
These excerpts indicate that Sam and Bill are discussing
relevant physical quantities and principles and gaining an
appreciation for the origin of the Boltzmann factor as a
result of the Boltzmann Factor tutorial. In particular, they
are correctly relating the Boltzmann factor of the system
with the multiplicity of the reservoir as an indicator of
probability. It should be noted that before tutorial instruction, Sam answered both parts of the PRQ correctly using
correct reasoning, and Bill used the Boltzmann factor
correctly but made errors in his calculations. These data
indicate that students who are able to successfully use the
Boltzmann factor after lecture instruction may not have a
complete understanding of the conceptual meaning behind
the mathematics they are using.
During the third year of tutorial implementation one
group struggled to interpret the derivative of entropy (with
respect to energy) obtained from completing the Taylor
series as the inverse of the temperature T −1 (see Ref. [15] for
details). However, once they had written an expression for
the entropy of the reservoir, one student had a particularly
expressive realization upon solving for the multiplicity,
Actually wait, ohhh, heyyy, because then that becomes
the partition [function]… and there’s your Boltzmann
factor.
Similar statements were made by Jake (who had participated in the first three sections of the tutorial in class) and
others during the teaching interviews (see Sec. III B),
indicating that they had not developed a robust understanding of the physical significance of the Boltzmann
factor after lecture instruction alone. All observation and
interview data indicate that these same students can gain an
appreciation for the physical significance of the Boltzmann
factor while participating in the Boltzmann Factor tutorial.
C. Revising the tutorial
The development process for instructional materials is
iterative; modifications are typically made to improve the
instructional experience based on earlier implementation(s).
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For this reason data are collected during the tutorial
implementations to ascertain the impact the materials are
having on students’ abilities to interact with the tutorial
activities, including the extent to which (a) students interpret
the instructions as the developers intended, (b) the tasks and
questions in the tutorial generate productive discussions
among the students, elicit specific difficulties targeted by
the developers, and guide students to the desired outcomes, and (c) students are able to complete the tutorial
tasks in the allotted time. The video data from school 1
serve this purpose, as does detailed feedback from the
instructor at school 2 regarding students’ abilities to
perform tutorial tasks as well as specific places where
they had particular difficulty. Additional data came from
the teaching interviews described in Sec. III B, which
were conducted after the initial implementation at school
1 (when many students did not complete the tutorial). In
this authentic instructional setting, we find evidence of
additional specific difficulties that written data would
not elicit, as well as examples of student discussions
prompted by the materials that inform the development of
the tutorial and further research.
All of these data were used to inform tutorial revisions
and modifications. Some revisions were minor, such as
wording changes to improve clarity for the students. Other
changes were more extensive, and included removing
sections entirely or moving activities and tasks to be
completed as either pretutorial homework [the initial steps
in the Taylor series expansion of SR ðER Þ [15]] or posttutorial homework (obtaining the expression for the canonical partition function). It should be noted that data do not
exist to determine the precise effect that each individual
tutorial modification has on student learning and understanding of the Boltzmann factor. However, the data do
suggest that the collective modifications have led to
increased student efficiency in completing tutorial tasks
during later implementations, allowing students to complete more of the tutorial in the time allotted. Increased
efficiency benefits students by giving them the opportunity
to arrive at the “punchline” of the Boltzmann Factor
tutorial: the derivation of the Boltzmann factor itself.
During the first tutorial implementation at school 1
several unanticipated difficulties were observed. The first
occurred while students completed the first page of the
tutorial on which it asked them to “estimate (to order of
magnitude) how many microstates (molecular configurations) exist such that the total energy of the gas [in the
isolated container] is Etot .” This language cued the students
to attempt to find a formula for calculating the multiplicity
of the gas based on its energy [59]. The intent of the task,
however, was for the students to recognize that there would
be many molecular configurations that would have a total
energy of Etot and to just write down any appropriately large
number. Students spent four minutes on this task before
asking the instructor for help. (This was not expected to take
very long; a rigorous calculation was neither intended nor
possible, and thus it should only have taken about a minute.)
The wording of the question was altered in subsequent
implementations to ask the students, “How many microstates (molecular configurations) would you estimate exist
such that the total energy of the gas is Etot : 1, 1000, 10N ?”
Data from the second tutorial implementation at school 1
indicate that students found this order-of-magnitude estimate much easier than the year before.
One observation noted during the teaching interviews
was that some students focused strongly on a relationship
between multiplicity and energy ðω ∝ V N E3N=2þ1 Þ that
was given in an introductory paragraph of the interview
(and the tutorial section). The intent of the statement was to
connect the Boltzmann Factor tutorial to the density of
states function ½DðEÞ ≡ dω=dE ∝ V N E3N=2 , which they
had recently learned about, and to motivate the notion
that ωC ≪ ωR (given that V C ≪ V R and EC ≪ ER ).
However, students tried to use this expression to relate
the multiplicities given in Table III to the energies.
One student (Jake, see last paragraph in Sec. V B) even
stated that since the EC ¼ E3 microstate has the lowest
reservoir multiplicity (ωR ¼ 4 × 1017 , rightmost column in
Table III), E3 must be the lowest energy (of C) and,
therefore, be the most probable. What he failed to consider
is that the multiplicity of the reservoir is the lowest, making
ER the lowest, and E3 the highest value (by conservation
of energy). Jake’s reasoning, in fact, reached the exact
opposite conclusion of what was intended.
The intent of the energy and multiplicity table (Table III)
and related questions is to motivate the connection between
multiplicity of the reservoir and probability of the system
being in the corresponding microstate. The students were
meant to realize that the EC ¼ E4 microstate is the most
probable since it has the largest corresponding multiplicity
for the reservoir, leading them to conclude that E4 must be
the lowest energy of the system because ER must be at its
highest value. Two other interview participants displayed
this tendency to latch onto the given expression relating
multiplicity to energy; it was also observed during the inclass tutorial session to a lesser extent. The statement
reminding students about the connection between multiplicity and energy was removed from later implementations
of the Boltzmann Factor tutorial along with most of the
original introductory paragraph.
Other in-class observations indicated that students did
not always refer to their own work from previous sections
of the tutorial when answering more difficult questions
later. In particular, when answering questions about multiplicity concerning the divided container (see Fig. 1),
students did not necessarily refer to the conclusions they
had made about the original undivided container. Specific
references to previous tutorial sections were added to
encourage students to make these connections and build
on knowledge they had previously constructed.
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The most consistent observation from the first two
implementations of the Boltzmann Factor tutorial at school
1 and the implementation at school 2 is that students could
not complete the tutorial in one 50-min class session. The
students at school 1 during the first year were only able to
complete the first three sections of the tutorial, ending in an
expression indicating Pðψ j Þ ∝ ωR ðψ j Þ. They did not have
the opportunity to even begin the Taylor series expansion
that would lead to the derivation of the Boltzmann factor
(the portion of the tutorial that we expected to be the most
difficult). After revising the tutorial to address the specific
difficulties discussed above, students at school 1 were
able to successfully complete the first four sections of
the tutorial (culminating with the derivation of the
Boltzmann factor) within one 50-min class during
the second year, but they were not able to complete the
normalization of probability to determine an expression
for the canonical partition function. A similar result was
reported at school 2 in that six out of seven groups of
students (≈4 students per group) were able to derive the
Boltzmann factor after the entire 70 min allotted by the
instructor, but only 1 or 2 groups had enough time to derive
Z as well. The students who did work through that portion
of the tutorial, both those in class at school 2 and those in
the teaching interviews at school 1, had little trouble
normalizing their expression for probability to get Z.
Based on the overwhelming majority of students not
completing the entire tutorial, even after modifications, we
removed the fifth section of the tutorial, in which students
derive the canonical partition function from the in-class
activities, and added it as the first question in the posttutorial homework assignment. The in-class portion of the
tutorial now ends with the derivation of the Boltzmann
factor as well as a comment on the term “Boltzmann factor”
and a reference to the homework assignment in which
students will determine an exact expression for the probability rather than just a proportionality. Classroom observations from subsequent implementations at school 1
indicate that these revisions have improved the efficiency
of the tutorial, and that most students are able to complete
the derivation of the Boltzmann factor during a single
50-min class period [60].
students’ failure to appropriately apply the Boltzmann
factor, we developed the Boltzmann Factor tutorial to
improve their understanding of situations in which the
Boltzmann factor is appropriate by guiding them through a
derivation of the Boltzmann factor, one that is particularly
rich in connecting the physics to the progression through
the derivation. Modifications were made to the tutorial
based on teaching interviews and in-class observations in
order to optimize student productive disciplinary engagement during class time. Results from several tutorial
implementations indicate that students are far more likely
to use the Boltzmann factor properly after tutorial instruction than after lecture instruction alone (results are statistically significant at the p < 0.05 level). The Boltzmann
Factor tutorial can be an effective supplement to (as at
school 1) or replacement for (as at school 2) lecture
instruction.
We anticipated that guiding students through this particular derivation of the Boltzmann factor would provide
them with the opportunity to engage in the physical
reasoning behind the derivation of the Boltzmann factor,
which our data suggested was not an outcome of lecture
instruction—even for a student who invests the effort to
memorize the textbook derivation. We have shown that
participating in tutorial instruction on this derivation helps
students gain an appreciation of the physical implications
and meaning of the mathematical formalism behind the
formula that had previously eluded them, e.g., Sam and Bill.
We have previously reported two related studies on
students’ understanding of Taylor series expansions [15]
and the relationship between the Boltzmann factor and
the density of states as expressions of multiplicity [61].
These results support our current claim that deriving the
Boltzmann factor is subtle and complex, and a robust
understanding of its physical meaning is not trivial.
One major avenue for future research is a study on the
pervasiveness of student understanding of the Boltzmann
factor after tutorial instruction. Do students use the
Boltzmann factor appropriately in situations that do not
involve probability ratios of discrete, nondegenerate energy
states? Do they recognize situations in which the
Boltzmann factor is and is not applicable? Our original
study has been necessarily focused on helping students
understand a basic application of the Boltzmann factor.
However, the Boltzmann factor is considered “the most
powerful tool in all of statistical mechanics” [23]. Do
students understand this tool well enough to use it to
maximum potential? Studying how students use the
Boltzmann factor and the canonical partition function to
derive other physical quantities and investigating how well
students understand the physical significance behind the
relevant mathematical procedures could help answer this
question and provide better insight into what students do
and do not understand about the “quintessential expression
of the statistical mechanical approach” [26].
VI. SUMMARY AND IMPLICATIONS
FOR FUTURE WORK
Our results show that students often do not use the
Boltzmann factor when answering questions related to
probability in applicable physical situations after lecture
instruction alone. These results have been replicated over
several years. Students instead tend to use statements about
a ranking of the relative probabilities to make novicelike
claims about probability ratios, consistent with literature in
mathematics education. This is a common error among
students regardless of whether or not they had received
lecture instruction on the Boltzmann factor. To address
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ACKNOWLEDGMENTS
We thank members of the University of Maine Physics
Education Research Laboratory, and our other colleagues,
Warren Christensen, Michael Loverude, and David
Meltzer, for their continued collaboration and feedback
on this work. We especially thank Jessica Clark for her
assistance with data analysis. We are deeply indebted to
the instructors of the courses in which data were collected.
This material is based on work supported by the National
Science Foundation under Grants No. PHY-0406764 and
No. DUE-0817282.
APPENDIX: DERIVATION OF THE
BOLTZMANN FACTOR
To understand the mathematical form of the Boltzmann
factor, consider the interactions between the system under
investigation (we call this C to avoid confusion with
entropy S) and the thermal reservoir (R); see Fig. 1 [62].
The probability of finding the system in a particular state
will depend on the total multiplicity of the system-reservoir
combination [PðEC Þ ∝ ωtot ], which is the product of the
individual multiplicities of the system and the reservoir
(ωtot ¼ ωC ωR ). In fact, if one considers a small enough
system (perhaps a single particle), the energy of the system
may only occupy a handful of discrete energy levels
(EC ∈ fEj g ¼ fE1 ; E2 ; …g). If these energy levels are
nondegenerate, the system would have a constant multiplicity, ωC ¼ 1 [65]. The total multiplicity of the systemreservoir combination will then be exactly equal to the
multiplicity of the reservoir:
ωtot ¼ ωR ωC ¼ ωR :
ðA2Þ
remains constant. The energy of the system, however, may
fluctuate about some average value,
EC ¼ hEC i δE:
∂SR
E þ
SR ðER Þ ¼ SR ðEtot Þ −
∂ER Etot C
¼ SR ðEtot Þ −
ðA3Þ
The magnitude of these energy fluctuations (δE) may be
relatively large compared to hEC i, but insignificant compared to hER i; thus, we are justified in considering R a
reservoir as its energy does not change appreciably.
EC
;
T
ðA4Þ
where ð∂S=∂EÞV;N ¼ T −1 from the fundamental thermodynamic relation (dE ¼ TdS − PdV þ μdN) and ER ¼
Etot − EC . The equality in the second line is valid because
the temperature of the system (and reservoir) is fixed:
higher-order derivatives of entropy are derivatives of
temperature and thus vanish. In this manner one obtains
an expression for SR as a function of EC and constants.
Revisiting Eq. (3) one obtains
ðA1Þ
The challenge now is to determine an expression for ωR in
terms of EC (the defining parameter of the macrostate). To
accomplish this, one must first relate EC to the properties of
the reservoir.
It is reasonable to assume that the system-reservoir
combination is isolated from the rest of the Universe such
that its total energy,
Etot ¼ EC þ ER ;
Qualitatively, by conservation of energy, as the energy
of the system decreases, the energy of the reservoir must
increase, increasing ωR and ωtot , yielding a higher probability; therefore, lower energy states for the system (C) are
more probable than higher energy states.
One must now be concerned with the precise mathematical form of multiplicity as it relates to energy, but
while energy is an extensive variable, multiplicity is neither
extensive nor intensive. This dilemma is solved by relating
multiplicity to the extensive quantity entropy via Eq. (3).
Given that entropy is an extensive variable, it may also be
written as a function of other extensive variables, e.g., as
SR ðER Þ. Because the reservoir is so much larger than the
system, ER ¼ Etot − EC ≈ Etot , and a Taylor series expansion is appropriate to approximate SR ðER Þ about the point
ER ¼ Etot ,
ωR ∝ e−EC =kT ∴ PðEC Þ ∝ e−EC =kT ;
ðA5Þ
giving the desired result of PðEC Þ, from Eq. (2).
It should be noted that the above method is not the only
way to derive the Boltzmann factor. Schroeder, for example, uses an approximation of the fundamental thermodynamic relation rather than a Taylor series expansion to
determine an expression for SR in terms of EC [23]. Carter,
on the other hand, uses the method of Lagrange multipliers
to maximize lnðωÞ with the constraints that the average
energy and number of particles in the system are both fixed;
this derivation does not require the assumption of a large
thermal reservoir, as the multiplicity of the reservoir is
never used [66]. The derivation presented in this section
was chosen for use within our Boltzmann Factor tutorial as
it is presented in the textbook used at the primary research
site [26] as well as several other commonly used texts
(see Refs. [63,64]) and because the physical significance of
the Boltzmann factor (the multiplicity of the reservoir or
surroundings) is emphasized throughout.
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Ei are unique; duplicate terms appear in the summation to
account for all degenerate macrostates.
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[47] Teaching interviews were not audio or video recorded so as
to provide a more informal atmosphere. Analysis is based
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[48] All names are pseudonyms.
[49] Joel had also provided this reasoning during the in-class
tutorial session.
[50] Equation (13) is not explicitly shown in Ref. [26], but Joel
wrote it during his interview.
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