Boltzman Factor Equations of State and Phase Diagrams Discussion Need help for the 6 discussion posts. Each 150 words 1- Phase transitions and phase diagr

Boltzman Factor Equations of State and Phase Diagrams Discussion Need help for the 6 discussion posts. Each 150 words

1- Phase transitions and phase diagrams

2- paper” Equations of State and Phase Diagrams” (attached below)

3- Paper”Exploring the Clapeyron Equation and the Phase Rule Using aMechanical Drawing Toy” (attached below)

4- Boltzman Factor

5- Partition Function

For item 4th and 5th use the this video:

6- Paper “Student Understanding of the Boltzmann Factor” (attached below) 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, ?, 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
+ ? 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
?
T 0.347+ c 1 – T
i
?c – 1 = c 0 1 – T
T
i=1
?
c
i+1 /3
c
where c0 = 1.9073793, c1 = 0.38225012, c2 = 0.42897885
Saturated vapor density (16 ):
2
?g
T 0.347+ d 1 – T
i
?c – 1 = d 0 1 – T
T
i=1
?
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 ?i normalized by
the total number of microstates:
?
PðMi Þ ¼ P i :
j ?j
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
decre…
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