# Development Of The 19th Century Atomic Theory -- An

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```					Development Of The 19th Century Atomic Theory -- An Outline
PART TWO – The Kinetic Theory

Contents
1. A guiding example, seen in hindsight ............................................................................ 1
2. Pre-history: Phenomenology, and the subtle fluid of caloric ....................................... 3
2.1 The virtues of the subtle fluid theory ............................................................................ 3
2.2 Empirical grounding of the theory ............................................................................... 5
2.3 Difficulties with the subtle fluid theory, favoring a rival ............................................ 6
3. Enter the kinetic theory .................................................................................................. 8
3.1 A short preview of the research program ...................................................................... 8
3.2 The early stages............................................................................................................. 9
3.3 Notes on the later difficulties ...................................................................................... 11
Appendix: Law of Dulong and Petit ................................................................................ 13

PART TWO. “The Kind of Motion We Call Heat”: Atomic
theory developed in statistical mechanics

Almost simultaneously with the advent of atomism in chemistry, physicists were
developing the kinetic theory, according to which heat and temperature are characteristics
of systems consisting of particles, and can be treated entirely in mechanical terms.
1. A guiding example, seen in hindsight
Before looking at the historic development, it is as well to think about what is
now the traditional paradigm example (a favorite in science teaching: see for instance
http://www.winbeam.com/~trebor/prelude.html
which is a very „model‟ oriented introduction). The ideal gas law, going back to Robert
Boyle in the 17th century, is that PV=rT; that is:

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the temperature of a gas is proportional to pressure and
inversely proportional to volume

To demonstrate this in rough intuitive fashion, a gas is
compressed in a cylinder by pushing down: as the volume decreases, the difficulty of
compressing increases (showing that the internal pressure is increasing). Then a flame is
placed below the cylinder: the piston begins to move up (apparently the pressure is being
released, but the volume of the gas is increasing correspondingly).
The kinetic theory explains these phenomena by identifying the temperature as
the mean kinetic energy of the molecules, and the pressure (force per unit area) as being
exerted by the impact of the moving molecules on the walls and piston. Qualitatively, the
explanation goes like this:

(1) As the temperature increases, the molecules
move faster and strike the container walls more
frequently, increasing the pressure.

(2) If, instead of increasing the concentration or
temperature, the volume of the container is
increased, the molecules must travel farther to
collide with the walls making the collisions less
frequent. Hence the pressure decreases.

(3) If the pressure of the system is kept constant and
the temperature increases, more frequent collisions will occur and the gas will
expand.

This sort of model is often called a „billiard ball model‟, since the particles are conceived

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of as perfectly hard little spheres that move fast and hit each other -- an ideal version of
billiards.

2. Pre-history: Phenomenology, and the subtle fluid of caloric
The terms still used to describe the phenomena of heat and temperature hark back
to the 18th century, when their explanation was given in terms of a „subtle fluid‟, which
flows from a hotter body to a colder one in contact. We still say precisely that: the heat
flows. Similarly, we speak of the heat capacity, in the same way that we speak of the
capacity of a rucksack or backpack or water container. Today this may seem a mere
metaphor, but at the time it was taken literally; that is how the terminology was
designed.

2.1 The virtues of the subtle fluid theory1
The thermometer was developed beginning in the 17th century, but its use was not
understood in terms in which we conceive of it now. Only in the middle of the 18th
century was the distinction between heat and temperature made precise, by Joseph Black:

5 gallons of water and 1 gallon of water may be at the same temperature (as
registered by a thermometer) but the former contains 5 times as much heat as the
latter

This difference is displayed in the phenomena if the two containers of water are used to
warm something else -- a hot-water bottle in bed on a cold night warms it better if it is
larger.
If heat is a subtle fluid that can flow from one body to another, what is
temperature? The Dutch scientist Boerhave proposed that the thermometer measures the
density of the heat in the body, so that the total heat contained is the temperature
multiplied by the volume.
That fits the example of the two containers of water well enough. But it runs into

1
In this story I will omit the important part dealing with latent heat and phase transitions, like melting and
boiling of water.

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trouble with experiments, when different substances are compared. Fahrenheit had, in an
earlier experiment, mixed different volumes of mercury and of water, and found that they
had reached a common temperature half way between their initial temperatures.
Boerhave‟s proposal would imply that this should only happen for equal volumes. At the
same time, Fahrenheit‟s results were certainly not in accord with another proposal,
namely that the heat is proportional to mass rather than volume. Neither of those two
simple ideas held.
Thus Black concluded that the amount of heat is a function of, besides
temperature, a particular attribute of the substance, its specific „heat capacity‟.   A
Swedish scientist, Wilcke, arrived at the same view, but introduced the term that lasted,
specific heat.
The experiments he then did with different substances allowed the following
conclusion about what happens to a body during a change in temperature

the heat added to a body of a given substance is proportional to the product of the
mass, the heat capacity of that substance, and the increase in temperature.

To remain with this minimal formulation, however, would be anachronistic on our part.
According to the subtle fluid theory, this addition of heat is precisely the flowing into the
body of additional caloric fluid. So the body is conceived of as a container for this fluid,
and the obvious correlate to the conclusion from the experiments for the total heat is then

the total heat contained in a body of a given substance is proportional to the
product of the mass, the heat capacity of that substance, and the temperature.

This is how Joseph Black formulated his conclusion. You see the difference: the
conclusion now goes beyond the earlier one which concerned a measure of change when
the body is heated. So stated there are some interesting implications.
Since heat can only be non-negative, with a theoretical minimum of zero,
Black‟s equation requires that the temperature is registered on a scale without negative
numbers -- unlike the scales in use, like Fahrenheit or Celsius. Thus the scale must be

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one with an absolute zero, „that than which nothing can be colder‟, so to speak -- a
conclusion drawn explicitly by Black.
The value of the theory, at this time, was precisely that it provided a way to
conceive of what is happening in such processes, and suggested novel hypotheses
concerning thermal phenomena.
Another such novel hypotheses was a conservation law: that this subtle fluid is
neither created nor destroyed, but conserved when the process is isolated. A hot body
does not cool down without heating up its surroundings or the bodies with which it is in
contact, and a body does not become warmer without being warmed, that is, being
supplied with more heat. This too appeared to be consistent with observation and
experiment, so far.

2.2 Empirical grounding of the theory
Specific heat is a theoretical quantity, its concept was introduced into the caloric
theory for theoretical reasons. If Boerhave‟s hypothesis were accepted, that the contained
amount of heat is proportional to volume as well as to temperature, what could we deduce
from the following information?2

The temperature of one cc of water was raised by 1 degree Celsius in two separate
processes:
(a) a quantity of mercury at temperature T was added to the water
(b) a quantity of alcohol at temperature T was added to the water.

Boerhave would have to say that then the initial volumes of mercury and alcohol must
have been the same. But Black‟s and other experiments did not agree. The rival
hypothesis, that the mercury and alcohol must have had the same mass, fared no better.
So to save the conservation of heat, Black introduced the compensating factor, that the
kind of substance is characterized in part by a new parameter, its specific heat.
Is this new parameter measurable? We can take as unit (as is now the standard) of

2
My example is fictional, but relevantly similar to the actual experiments, e.g. an experiment with mercury
and water that Fahrenheit had done.

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the quantity of heat the amount that must be added to one gram of water to raise its
temperature one degree Celsius -- the calorie. Then experiments of the above sort can be
done -- for example, how much heat does it take to raise the temperature of 1 gram of
mercury one degree? Or, if the same amount of heat is added (e.g. by the same constant
fire) to a gram of mercury and a gram of water, what is the difference in temperature to
which they are raised? And is it the same for all samples of mercury?
Note well what is being done here, how the reasoning leads to a value for the
specific heat of the other substance: we assume that the new hypothesis, namely that the
amount of heat added equals the product of mass, specific heat, and temperature
increment, is correct. Then we fill in the values for mass and temperature, while
asserting e.g. that the constant fire added equal amounts of heat to the two samples --
with all that given, and the hypothesis assumed, the value of the specific heat can be
deduced.
So yes, specific heat is within the theory a measurable parameter. This is a very
important point. In the terminology we will use much in our later discussion, the theory
so far is empirically grounded because each parameter is either directly measurable (like
volume) or measurable relative to the equations that express the theory‟s laws.

2.3 Difficulties with the subtle fluid theory, favoring a rival
One difficulty we will leave aside, because it was equally insuperable for the
kinetic theory: there was no accounting for radiant heat. (Heat can apparently be
conveyed in a vacuum, with the hot body not touched by anything that conducts heat.)
But there were other difficulties that were grist for the mill of those attempting to develop
a rival account.
The first was the question of weight. If the subtle fluid has mass, then heating a
body should increase its weight, and cooling it should make it lighter. The experiments
done showed, for the most part, no change in weight. Some, however, showed an
increase and some a decrease -- the latter suggesting the old Aristotelian notion of
„levity‟, negative weight as we might say, however incongruous this idea was for the
Newtonian conceptual scheme that then dominated the sciences.
Count Rumford, the American who was Minister of Defense in the German state
of Bavaria, but is now much better known for his scientific contributions, repeated some

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of the experiments with greater precision, and concluded that there was no change in
weight.
But Rumford made a greater contribution when he addressed the question whether
the law of conservation of heat is really true. He reports that when he witnessed a canon
being bored, he was amazed at how hot the drilling made the metal. If heat is a fluid,
where does it come from in this case? Could it be coming out of the metal, from some
hidden reservoir in the metal, due to the mechanical action on it? It is not as if anything
was cooling, so this was not a case of caloric fluid flowing from a hotter body to a colder
one. There was still one more possibility: what if the drilling changed the specific heat of
the metal?3
To check this, Rumford took chips of metal produced by the drilling and tested
them in comparison with other chips that were carefully and slowly made with a fine saw.
There was no difference in the specific heat. Rumford concluded that the amount of heat
was not conserved: heat was created by a mechanical process.
This was at the end of the 18th century. Not much later an English scientist, Sir
Humphrey Davy, made a careful experiment that led him to the same conclusion. He had
two pieces of ice rubbing against each other in a vacuum, and found that they were
melting, although the container as a whole was kept at a temperature below freezing.
But it was not until the middle of the 19th century that the very precise quantitative
experiments of James Joule established the mechanical equivalent of heat. His famous
1854 paper, which began by recounting the story of Rumford and Davy, ended with the
conclusion:

That the quantity of heat capable of increasing the temperature of a pound of
water … by 1 o Fahr., requires for its evolution the expenditure of a mechanical
force represented by the fall of 772 lbs through the space of one foot.4

3
Then the same equation that we saw above could be written somewhat differently, with
the specific heat changing, and the total amount of heat staying the same, to give us the
change in temperature ΔT = Q/[mΔc]
4
the omitted passage reads “(weighed in vacuo, and taken at between 55 o and 60 o)”.

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There is no easy way to conceive of this on the subtle fluid theory. Mere motion cannot
create substances out of nothing, it would seem.
There was another theory, however vague as yet, waiting in the wings all along.

3. Enter the kinetic theory

3.1 A short preview of the research program
The basic principle that would guide the elaboration of the theory and the construction of
the observable behavior and the characteristics of substances are due to the
aggregate behavior and characteristics of enormously large numbers of very small
and constantly moving elementary entities, subject to the laws of mechanics, of
which these substances consist.
This is a very general, metaphysical statement; how could it guide research? In scientific
practice it came with four methodological directives :
(i) To set up models – or define model types – introduce specific assumptions as
to the nature of the elementary entities and as to their available degrees of
freedom.
(ii) Since the motions of these entities are to be treated are 'aggregate', assume that
(although that motion is chaotic) there is for every relevant property of that
motion a mean value determined by the distribution of that property among the
molecules.
(iii) To improve the models, try to weaken or if possible eliminate the simplifying
assumptions introduced to facilitate calculation so as to simulate, as far as
possible, conditions obtaining in a real gas.
(iv) Use the specific assumptions introduced to investigate the internal properties
of gases (e.g. the viscosity) while the macroscopic (hydrodynamic)
and equilibrium properties should be derivable as limiting cases (for example, as
following in the limit as the number of elementary entities tends to infinity) .

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Thus each successive version or modification of theory, as this research program
developed, consisted in the proposal of a a particular model(-type) of a gas, constructed
in accordance with (i) and (ii), each designed to be a closer approximation to the
conditions known to obtain in a gas.
At each stage, anomalies arose; these were then attributed to the auxiliary or
simplifying assumptions constituting the model in question. For example, the shape or
structure of the entities (atoms, poly-atomic molecules) or the forces acting on them.
Thus, it was often part of the assumptions of the model that the directions of motion of
these entities were random, though more realistically, the effect of gravity could not be
absent, which would favor downward-tending motions.
There were striking successes and disheartening failures throughout the century, with
an especially dismal outlook dominating at times among the main physicists involved.
As in the story of how atomism fared in chemistry, there was constant pressure to display
ways in which the theoretical factors could be accessible to measurement, somehow, at
least in principle, relative to the theory -- or, to put it another way, to develop the theory
in directions in which this could be possible. Not that anyone asked at the time for
experiments isolating single atoms; the kinetic theory advocates insisted, after all, that it
was the aggregate behavior, rather than individual motions, that figured significantly in
how they modeled the phenomena.

3.2 The early stages5
In the middle of the 19th century we find the first working model, credited to both Joule
and Krönig. In this model the molecules are perfectly smooth, perfectly elastic spheres,
traversing rectilinear paths with the same speed, in arrays parallel to and perpendicular to
the walls of the containing vessel. Because they are perfect spheres, there is no
significance to the idea of rotation, so only translational motion is possible.
To calculate the pressure arising from the transfer of momentum at the walls of

5
I am following Peter Clark, “Atomism versus thermodynamics” and several of the historical sources in
this section.

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the container Kronig introduced a statistical hypothesis: all allowed directions of motion
are equally likely. Since only two directions were allowed (as simplification) he inferred
that on average exactly half of the of molecules within a unit volume would be moving
perpendicular to a unit area of the wall. The net transfer of momentum per unit area per
unit time could then be calculated in mechanics. Taking then the further step of
identifying the absolute temperature of the gas as proportional to the kinetic energy of the
molecules, the Boyle-Charles law for an ideal gas follows.
Much more famous than this beginning is the paper by Clausius [1857]: The
Nature of the Motion which we call Heat. Clausius improved on this simple model in
the prescribed way, by introducing more realistic assumptions about molecular motion.
He noted first of all that in the simple model he could not account for the
observed specific heat of gases. Not surprising that such a highly simplified model
would have specific deficiencies! To begin Clausius let go of Krönig's hypothesis that
the molecules were smooth elastic spheres. If they are not, and since not all molecular
collisions could be rectilinear and central, a rotational motion would ensue. Collisions
could anyway not be in general perfectly elastic, for among molecules consisting of
several atoms the impact would cause vibrations. Thus any motion of translation alone
would gradually become distributed among the other available degrees of freedom of
motion. However, once a steady state is reached, the irregularities occurring as the result
of these inelastic collisions could well be ignored. Moreover it seemed reasonable to
assume that in such a steady state, the distribution of molecular speeds would be centered
on a most frequent value, and in the calculations one could simplify by just assuming that
all the molecules have that speed. Finally, perhaps most obviously, Clausius abandoned
the idea of motion in regular arrays.
Now the theory could claim more successes. For example, by assuming
Avogadro‟s hypothesis of equal numbers of molecules in equal volumes, Clausius could
now deduce Gay-Lussac‟s law of equivalent volumes (see the discussion of Gay-Lussac
and Avogadro in the First Part) from his model assumptions. Another success related to
the still older Dalton‟s law of partial pressures: if a mixture of gases fills a volume, the
total pressure will be the sum of the pressures that each gas alone would exert in that
volume. This he deduced by assuming that the kinetic energy of translation is

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proportional to pressure and inversely proportional to volume, an assumption closely
related –via the deduction of Boyle‟s law -- to Krönig's identification of the absolute
temperature of the gas as proportional to the kinetic energy of the molecules.
Almost immediately, however (in 1858), a deficiency was found in Clausius‟
improved model. On this model, gases should instantaneously inter-diffuse, since the
collisions have almost zero duration compared to the entire duration of the rectilinear
motions of the molecules between collisions. In actual fact, gases take some time to
reach that state. So in a second paper, Clausius went still farther in bringing the model
closer to what real molecules might be like: they have finite but definite size, not
negligible with respect to its effect on dimensions of the mean free path (i.e. the average
distance a molecule moves before colliding with another molecule). Clausius had, of
course, no way of estimating the size of the molecules; he introduced the assumption that
the ratio of the mean free path to the „action sphere‟ of a molecule was 1000 to 134,
resulting in a very small mean free path length, with an estimate of such an enormous
number of collisions per cm3 that the rate of diffusion would be respectably slow. Of
course this raised to significance two other physical quantities relating to the molecules,
the size and the mean free path, with the consequent question, once again, of how these
might be accessible to at least indirect measurement, on the assumptions of the theory.

3.3 Notes on the later difficulties
.     I mentioned above that the first difficulty pointed out in the first model was that
there was no way to accommodate the specific heat there. This turned out to be a
besetting problem for the theory for decades to come – generally referred to now as the
specific heat anomaly.
In 1819 Dulong and Petit had added (see Part One) the law that atoms of every
simple body have exactly the same heat capacity. Measurement of specific heats of
various elements can then be related to the number of atoms in a unit quantity so as to
yield the atomic mass. Their hypothesis implied that the specific heat of the substance
(which is measurable) is inversely proportional to atomic weight. If indeed two of the
parameters are empirically determinable, the values of the third also admit calculation.
So that was an improvement in precisely the respect I am emphasizing. However, the

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addition does not go all the way; because only equalities are postulated and not specific
numbers, this cannot yield values beyond mass ratios.
Greater difficulty with this reasoning was actually to come, for this improvement
rested on a postulate concerning one of the most problem-beset areas for the theory:
specific heat ratios.6 The kinetic theory acquired, as a fundamental principle, the
equipartition of energy: the total energy is distributed equally between all the n
mechanical degrees of freedom. This had as consequence that the ratio of the specific
heats of a gas at constant pressure and constant volume respectively will take form
(2+n)/n. When the gas particles are assumed to be perfectly smooth and rigid spheres
having only the three degrees of translational freedom, with rotation or vibration
negligible, that ratio is 5/3, approximately 1.67, whereas experiments available by
midcentury show only 1.4. Introduction of rotational and vibratory motions did not,
however, lead to better accord with the measurement results.
In 1860 Maxwell concluded pessimistically that the discrepancy “overturns the
whole hypothesis, however satisfactory the other results may be”.7 Fifteen years later
Maxwell [1875] tells the Chemical Society “And here we are brought face to face with
the greatest difficulty which the molecular theory has yet encountered”.
The last decade of the 19th century, though it followed a long period of apparently
successful improvement, was a time of intense critique of the theory even by such
avowedly „realist‟ thinkers as Max Planck. The problem of accounting for the specific
heats of gases was first of all that the different possible models of the kinetic theory of
gases all gave values clearly different from measured values. One exception was a model
proposed by Boltzmann, but it turned out that this could not be reconciled with
mechanics! So it was derided as involving an ad hoc, inexplicable departure from basic
principles. The problem in a nutshell was that vibratory motion had to be attributed to
the molecules (as well as translational and rotational), but that the degrees of freedom of

6
For detailed discussion see especially de Regt 1996, but it is also discussed by Gardner 1979, Clark 1976:
82-88, and Nyhoff 1988. In fact, there continues to be a voluminous literature on specific heat anomalies,
for example at very low temperatures, but at present the problem they present is treated as „normal science‟.
7
In his 1860 British Association Report, quoted Nyhoff 1988: 94.

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vibration could not be allowed to play a role in the deductions concerning specific heat, if
that was to come out in accordance with the data.
To put it differently: some measurement results led, via the theory, to a one value
for this quantity, and other measurement results were incompatible with that value. In
terms introduced later, concordance failed. The determinations of its value, by
measurement, relative to the theory, were inconsistent with each other. The same was
true for the transport equations, involving the determination of a value for the mean free
path of a molecule on the way to calculating the measurable quantities pertaining to
viscosity and diffusion. The different calculations relative to the theory were not
consistent with each other (cf. Clark 1976, 82-88).        These were the failures of the kinetic
theory with respect to precisely the requirement of empirical grounding than what Planck
and others complained of at the end of the 19th century.
It is generally accepted that the situation changed radically with the work of
Perrin in the first decade of the 20th century – ironically, after the foundations of classical
physics, the framework for the kinetic theory, were just undergoing a great scientific
revolution …. This episode, and Perrin‟s work, has been the subject of a great deal of
philosophical discussion.

Appendix: Law of Dulong and Petit8

The specific heat of copper is 0.093 cal/gm K (.389 J/gm K) and that of lead is only 0.031
cal/gm K(.13 J/gm K). Why are they so different? The difference is mainly because it is
expressed as energy per unit mass; if you express it as energy per mole, they are very
similar. It is in fact that similarity of the molar specific heats of metals which is the
subject of the Law of Dulong and Petit. The similarity can be accounted for by applying
equipartition of energy to the atoms of the solids.

From just the translational degrees of freedom you get 3kT/2 of energy per atom. Energy
added to solids takes the form of atomic vibrations and that contributes three additional
degrees of freedom and a total energy per atom of 3kT. The specific heat at constant
volume should be just the rate of change with temperature (temperature derivative) of
that energy.

8
COPIED FROM http://hyperphysics.phy-astr.gsu.edu/Hbase/thermo/dulong.html#c1

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When looked at on a molar basis, the specific heats of copper and lead are quite similar:

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