Disorder Metaphor Reading Reflection Write a reading reflection on the attached papers on entropy discussing the disorder metaphor. It should address one o

Disorder Metaphor Reading Reflection Write a reading reflection on the attached papers on entropy discussing the disorder metaphor. It should address one or more of the reading reflection prompts. Make sure it is focused and coherent (there is a a unifying theme to your reflection). The reading reflection should be around 250 words long.Reading reflection prompts: what are the author’s purpose in writing this article? What can you take from it that can be applied in your own classroom? How does the article impact your own physics content and pedagogical knowledge? In the Classroom
Disorder—A Cracked Crutch for Supporting Entropy Discussions
Frank L. Lambert†
2834 Lewis Dr., La Verne, CA 91750; flambert@att.net
This article decries the use of “disorder” in teaching
beginning students about thermodynamic entropy. It is
cautionary rather than proscriptive about “disorder” being used
warily as a device for assessing entropy change in advanced
work or among professionals.1
Overview
To help students visualize an increase in entropy, many
elementary chemistry texts use artists’ before-and-after drawings
of groups of “orderly” molecules that become “disorderly”. This
has been an anachronism ever since the ideas of quantized
energy levels were introduced in elementary chemistry. “Orderly–
disorderly” seems to be an easy visual support, but it can be
so grievously misleading as to be characterized as a failureprone crutch rather than a truly reliable, sturdy aid.2
After mentioning the origin of this visual device in the
late 1800s and listing some errors in its use in modern texts,
I will build on a recent article by Daniel F. Styer. It succinctly
summarizes objections from statistical mechanics to characterizing higher entropy conditions as disorderly (1). Then,
after citing many failures of “disorder” as a criterion for evaluating entropy—all educationally unsettling, a few serious, I
urge the abandonment of order–disorder in introducing
entropy to beginning students. Although it seems plausible,
it is vague and potentially misleading, a non-fundamental
description that does not point toward calculation or elaboration in elementary chemistry, and an anachronism since the
introduction of portions of quantum mechanics in first-year
textbooks.3
Entropy’s nature is better taught by first describing
entropy’s dependence on the dispersion of energy (in classic
thermodynamics) and the distribution of energy among a large
number of molecular motions relatable to quantized states,
microstates (in molecular thermodynamics). 4 Increased
amounts of energy dispersed among molecules result in
increased entropy that can be interpreted as molecular occupancy of more microstates. (High-level first-year texts could
go further to a page or so of molecular thermodynamic
entropy as described by the Boltzmann equation.)
The History and Use of “Disorder” to Characterize
Entropy
As is well known, in 1865 Clausius gave the name “entropy” to a unique quotient for the process of a reversible
change in thermal energy divided by the absolute temperature
(2). He could properly focus only on the behavior of chemical
systems as macro units because in that era there was considerable doubt even about the reality of atoms. Thus, the behavior
of molecules or molecular groups within a macro system was
totally a matter of conjecture (as Rankine unfortunately
demonstrated in postulating “molecular vortices”) (3). Later in
the 19th century, but still prior to the development of quantum
mechanics, the greater “disorder” of a gas at high temperature
compared to its distribution of velocities at a lower temperature

Professor Emeritus, Occidental College, Los Angeles, CA 90041.
was chosen by Boltzmann to describe its higher entropy (4).
However, “disorder” was a crutch; that is, it was a contrived
support for visualization rather than a fundamental physical or
theoretical cause for a higher entropy value. Others followed
Boltzmann’s lead; Helmholtz in 1882 called entropy “Unordnung” (disorder) (5), and Gibbs Americanized that description
with “entropy as mixed-up-ness”, a phrase found posthumously
in his writings (6 ) and subsequently used by many authors.
Most general chemistry texts today still lean on this conceptual crutch of order–disorder either slightly with a few
examples or as a major support that too often fails by leading to
extreme statements and overextrapolation. In the past century,
the most egregious errors of associating entropy with disorder
occurred simply because disorder is a common language word
with nonscientific connotations. Whatever Boltzmann meant
by it, there is no evidence that he used disorder in any sense
other than strict application to molecular energetics. But over
the years, popular authors have learned that scientists talked
about entropy in terms of disorder, and thereby entropy has
become a code word for the “scientific” interpretation of
everything disorderly from drunken parties to dysfunctional
personal relationships,5 and even the decline of society.6
Of course, chemistry instructors and authors would
disclaim any responsibility for such absurdities. They would
insist that they never have so misapplied entropy, that they
used disorder only as a visual or conceptual aid for their
students in understanding the spontaneous behavior of atoms
and molecules, entropy-increasing events.
But it was not a social scientist or a novelist—it was a
chemist—who discussed entropy in his textbook with “things
move spontaneously [toward] chaos or disorder”.7 Another
wrote, “Desktops illustrate the principle [of ] a spontaneous
tendency toward disorder in the universe”.7 It is nonsense to
describe the “spontaneous behavior” of macro objects in this
way: things like sheets of paper, immobile as they are, behave
like molecules despite the fact that objects’ actual movement
is non-spontaneous and is due to external agents such as
people, wind, and earthquake. That error has been adequately
dismissed (7 ). The important point here is that this kind of
mistake is fundamentally due to a focus on disorder rather than on
the correct cause of entropy change, energy flow toward dispersal.
Such a misdirected focus leads to the kind of hyperbole one
might expect from a science-disadvantaged writer, “Entropy
must therefore be a measure of chaos”, but this quote is from
an internationally distinguished chemist and author.7,8
Entropy is not disorder. Entropy is not a measure of
disorder or chaos. Entropy is not a driving force. Energy’s
diffusion, dissipation, or dispersion in a final state compared
to an initial state is the driving force in chemistry. Entropy is
the index of that dispersal within a system and between the
system and its surroundings.4 In thermodynamics, entropy
change is a quotient that measures the quantity of the unidirectional flow of thermal energy by dS ≥ dq/T. An appropriate paraphrase would be “entropy change measures energy’s
dispersion at a stated temperature”. This concept of energy
dispersal is not limited to thermal energy transfer between
JChemEd.chem.wisc.edu • Vol. 79 No. 2 February 2002 • Journal of Chemical Education
187
In the Classroom
system and surroundings. It includes redistribution of the
same amount of energy in a system—for example, when a
gas is allowed to expand into a vacuum container, resulting
in a larger volume. In such a process where dq is zero, the
total energy of the system has become diffused over a larger
volume and thus an increase in entropy is predictable. (Some
call this an increase in configurational entropy.)
From a molecular viewpoint, the entropy of a system
depends on the number of distinct microscopic quantum
states, microstates, that are consistent with the system’s
macroscopic state. (The expansion of a gas into an evacuated
chamber mentioned above is found, by quantum mechanics,
to be an increase in entropy that is due to more microstates
being accessible because the spacing of energy levels decreases
in the larger volume.) The general statement about entropy in
molecular thermodynamics can be: “Entropy measures the
dispersal of energy among molecules in microstates. An entropy
increase in a system involves energy dispersal among more
microstates in the system’s final state than in its initial state.”
It is the basic sentence to describe entropy increase in gas
expansion, mixing, crystalline substances dissolving, phase
changes, and the host of other phenomena now inadequately
described by “disorder” increase.
In the next section the molecular basis for thermodynamics is briefly stated. Following it are ten examples to
illustrate the confusion that can be engendered by using
“disorder” as a crutch to describe entropy in chemical systems.
The Molecular Basis of Thermodynamics
The four paragraphs to follow include a paraphrase of
Styer’s article “Insight into Entropy” in the American Journal
of Physics (1).9
In statistical mechanics, many microstates usually correspond to any single macrostate. (That number is taken to
be one for a perfect crystal at absolute zero.) A macrostate
is measured by its temperature, volume, and number of
molecules; a group of molecules in microstates (“molecular
configurations”, a microcanonical ensemble) by their energy,
volume, and number of molecules.
In a microcanonical ensemble the entropy is found simply
by counting: one counts the number W of microstates that
correspond to the given macrostate10 and computes the entropy
of that macrostate by Boltzmann’s relationship, S = kB ln W,
where kB is Boltzmann’s constant.11
Clearly, S is high for a macrostate when many microstates
correspond to that macrostate, whereas it is low when few
microstates correspond to the macrostate. In other words, the
entropy of a macrostate measures the number of ways in which
a system can be different microscopically (i.e., molecules be
very different in their energetic distribution) and yet still be
a member of the same macroscopic state.
To put it mildly, considerable skill and wise interpretation
are required to translate this verbal definition into quantitative
expressions for specific situations. (Styer’s article describes
some conditions for such evaluations and calculations.)
Nevertheless, the straightforward and thoroughly established
conclusion is that the entropy of a chemical system is a
function of the multiplicity of molecular energetics. From this,
it is equally straightforward that an increase in entropy is due
to an increase in the number of microstates in the final
macrostate. This modern description of a specifiable increase
188
in the number of microstates (or better, groups of microstates)
contrasts greatly with any common definition of disorder,
even though disorder was the best Boltzmann could envision
in his time for the increase in gas velocity distribution.
There is no need today to confuse students with 19th
century ad hoc ideas of disorder or randomness and from these
to create pictures illustrating “molecular disorder”. Any valid
depiction of a spontaneous entropy change must be related
to energy dispersal on a macro scale or to an increase in the
number of accessible microstates on a molecular scale.
Examples of “Disorder” as a Broken Crutch
for Supporting Illustrations of Entropy
1. Entropy Change in a Metastable Solid–Liquid
Mixture ( 1)
This example, a trivial non-issue to chemists who see
phenomena from a molecular standpoint and always in terms
of system plus surroundings, can be confusing to naive adults
or beginning chemistry students who have heard that “entropy
is disorder”. It is mentioned only to illustrate the danger of
using the common language word disorder.
An ordinary glass bowl containing water that has cracked
ice floating in it portrays macro disorder, irregular pieces of
a solid and a liquid. Yet the spontaneous change in the bowl
contents is toward an apparent order: in a few hours there will
be only a homogeneous transparent liquid. Of course, the
dispersal of energy from the warmer room surroundings to
the ice in the system is the cause of its melting. However, to the
types of individuals mentioned who have little knowledge of
molecular behavior and no habit pattern of evaluating possible
energy interchange between a system and its surroundings, this
ordinary life experience can be an obstacle to understanding. It
will be especially so if disorder as visible non-homogeneity or
mixed-up-ness is fixed in their thinking as signs of spontaneity
and entropy increase. Thus, in some cases, with some groups of
people, this weak crutch can be more harmful than helpful.
A comparable dilemma (to those who have heard only
that “entropy is disorder” and that it spontaneously increases
over time) is presented when a vegetable oil is shaken with
water to make a disorderly emulsion of oil in water (8b).
However (in the absence of an emulsifier), this metastable
mixture will soon separate into two “orderly” layers. Order
to disorder? Disorder to order? These are not fundamental
criteria or driving forces. It is the chemical and thermodynamic properties of oil and of water that determine such
phase separation.
The following examples constitute significantly greater
challenges than do the foregoing to the continued use of disorder in teaching about entropy.
2. Expansion of a Gas into a Vacuum (9)
When this spontaneous process is portrayed in texts with
little dots representing molecules as in Figure 1, the use of
disorder as an explanation to students for an entropy increase
becomes either laughable or an exercise in tortuous rationalization. Today’s students may instantly visualize a disorderly
mob crowded into a group before downtown police lines.
How is it that the mob becomes more disorderly if its individuals spread all over the city? Professors who respond with
their definition must realize that they are particularizing a
common word that has multiple meanings and even more
Journal of Chemical Education • Vol. 79 No. 2 February 2002 • JChemEd.chem.wisc.edu
In the Classroom
1
1
2
2
Vacuum
SA
SA
∆S 2-1 > 0
2SA
∆S 2-1 = 0
Figure 1. Expansion of a gas into a vacuum.
Figure 2. More disorderly?
implications. As was well stated, “We cannot therefore always
say that entropy is a measure of disorder without at times so
broadening the definition of ‘disorder’ as to make the statement
true by [our] definition only” (10).
Furthermore, the naive student who has been led to focus
on disorder increase as an indicator of entropy increase and is
told that ∆S is positive in Figure 1 could easily be confused
in several ways. For example, there has been no change in the
number of particles (or the temperature or q), so the student
may conclude that entropy increase is intensive (besides the
Clausius equation’s being “erroneous”, with a q = 0). The
molecules are more spread out, so entropy increase looks as
if it is related to a decrease in concentration. Disorder as a
criterion of entropy change in this example is even worse than
a double-edged sword.
How much clearer it is to say simply that if molecules can
move in a larger volume, this allows them to disperse their
original energy more widely in that larger volume and thus
their entropy increases. Alternatively, from a molecular viewpoint, in the larger volume there are more closely spaced—
and therefore more accessible—microstates for translation
without any change in temperature.
In texts or classes where the quantum mechanical behavior of a particle in a box has been treated, the expansion
of a gas with N particles can be described in terms of microenergetics. Far simpler for other classes is the example of a
particle of mass m in a one-dimensional box of length L
(where n is an integer, the quantum number, and h is Planck’s
constant): E = (n2h2)/(8mL 2). If L is increased, the possible
energies of the single particle get closer together. As a consequence, if there were many molecules rather than one, the
density of the states available to them would increase with
increasing L. This result holds true in three dimensions,
the microstates become closer together, more accessible to
molecules within a given range of energy.
beginning students can be seen to be broken—not just weak.
(Generally, as in this example, entropy is extensive. However, its additivity is not true for all systems [11a].)
3. Doubling the Amount of a Gas or Liquid,
in Terms of Disorder
Does any text that uses disorder in describing entropy
change dare to put dots representing ideal gas molecules in a
square, call that molecular representation disorderly, attach
it to another similar square while eliminating the barrier lines,
and call the result more disorderly, as in Figure 2? Certainly the
density of the dots is unchanged in the new rectangle, so how
is the picture more “disorderly”? In the preceding example,
if the instructor used a diagram involving molecular-dot
arrangements, an implication any student could draw was that
entropy change was like a chemical concentration change;
entropy was therefore an intensive property. However in this
example, the disorder description of entropy must be changed
to the opposite, to be extensive! With just these two simple
examples, the crutch of disorder for categorizing entropy to
4. Monatomic Gases: Massive versus Light Atoms
( 1, but with Helium Atoms)
Helium atoms move much more rapidly than do atoms
of krypton at the same temperature. Therefore, any student
who has been told about disorder and entropy would predict
immediately that a mole of helium would have a higher entropy
than a mole of krypton because the helium atoms are so much
more wildly ricocheting around in their container. That of course
is wrong. Again, disorder proves to be a broken crutch to
support deductions about entropy. Helium has a standard-state
entropy of 126 J K᎑1 mol᎑1, whereas krypton has the greater
S °, 164 J K᎑1 mol᎑1.
The molecular thermodynamic explanation is not obvious
but it fits with energetic considerations, whereas “disorder” does
not. The heavier krypton actually does move more slowly than
helium. However, krypton’s greater mass, and greater range
of momenta, results in closer spacing of energy levels and
thus more microstates for dispersing energy than in helium.
5. The Crystallization of Supercooled Water,
a Metastable System
NOTE: In this example and the one that follows, students
are confused about associating entropy with order arising
in a system only if they fail to consider what is happening
in the surroundings (and that this includes the solution
in which a crystalline solid is precipitating, prior to any
transfer to the environment). Thus they should be repeatedly reminded to think about any observation as part
of the whole, the system plus its surroundings. When orderly crystals form spontaneously in these two examples,
focusing on entropy change as energy dispersal to or from
a system and its surroundings is clearly a superior view
to one that depends on a superficiality like disorder in
the system (even plus the surroundings). Example 7 is
introduced only as a visual illustration of the failure of
order–disorder as a reliable indicator of entropy change
in a complex system.
Students who believe that spontaneous processes always
yield greater disorder could be somewhat surprised when
shown a demonstration of supercooled liquid water at many
degrees below 0 °C. The students have been taught that liquid
water is disorderly compared to solid ice. When a seed of ice
or a speck of dust is added, crystallization of some of the
liquid is immediate. Orderly solid ice has spontaneously
formed from the disorderly liquid.
Of course, thermal energy is evolved in the process of
this thermodynamically metastable state changing to one that
is stable. Energy is dispersed from the crystals, as they form,
JChemEd.chem.wisc.edu • Vol. 79 No. 2 February 2002 • Journal of Chemical Education
189
In the Classroom
to the solution and thus the final temperature of the crystals
of ice and liquid water is higher than originally. This, the
instructor ordinarily would point out as a system–surroundings
energy transfer. However, the dramatic visible result of this
spontaneous process is in conflict with what the student has
learned about the trend toward disorder as a test of spontaneity.
Such a picture might not take a th…
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