Taken from my 2015 book but not my 2018 book

By Kent W Mayhew Mar 29 2017
Avogadros’ hypothesis and kinetic
theory: Where and why they falter
The following is from my revised book. I thought it might be of interest. I thought of posting
it in forums on kinetic theory but I find getting equations subscripts to read a real headache. I am also looking for some people
who may be interested to read through my manuscript and let me know what you think. Let me know if anyone is interested. And the ideal
persons are NOT necessarily a fully indoctrinated individual in traditional thermodynamics, and that is simply because of the nature
of humans.
Avogadro’s Hypothesis/Conundrum
A fundamental principle that allows us to readily deal with gases was devised by Amedeo
Avogadro (17761856); Avogadro’s hypothesis states that: “Equal volumes of different gases measured at the same temperature and pressure
contain an equal number of molecules” (circa 1811)^{7}. An implication becomes, when comparing two gases in thermal equilibrium
then those two gases will have the same mean kinetic energy per unit volume. This is based upon all gas molecules at a given temperature
possess the same kinetic energy, as defined by kinetic theory.
A forgotten concern of Avogadro’s hypothesis becomes that it seemingly
confounds the conservation of momentum. Imagine two gases in thermal equilibrium with different masses and in mechanical equilibrium
i.e. exert the same pressure. If they exert the same pressure and occupy the same mean molecular volume, then they have the same momentum.
Hence from classical mechanics we know that when two dissimilar objects, i.e. masses M_{1} and M_{2}, collide then their total momentum
is conserved, therefore in terms of their velocities (v_{1} and v_{2}):
v_{2}=(M_{1}/M_{2})v_{1} Or v_{1}=(M2/M1)v_{2} 1.10.36
If one thinks about it, the conceptualization that different gases in thermal equilibrium will have the same mean pressure,
and same mean kinetic energy per volume seems at odds with logic. In order to better understand, realize that the pressure exerted
by a gas molecule is directly related to that gas molecule’s momentum. Remember conservation of momentum means that the total momentum
is conserved, and not that each size of molecule has the same mean momentum hence can apply the same mean pressure. Now ask; how can
all gas molecules at a given temperature exert the same pressure, and possess the same mean kinetic per unit volume, when they have
different masses? It is improbable and may even be impossible!
Another perspective; based upon eqn 1.10.36, the net result of
a collision becomes the more massive object attains a lower velocity than the less massive object. One can readily envision that over
a period of time, that this may somehow allow the different gas molecules to all attain the same momentum hence same pressure because
they occupy the same mean molecular volume. Does this not lend itself to question how different gas molecules (differing mass) maintain
the same mean kinetic energy?
Do we have a conundrum? Avogadro’s hypothesis seemingly goes against the principle of conservation
of momentum, all in order to maintain the bias of conservation of kinetic energy. Perhaps this helps to explain why Avogadro’s
hypothesis was not appreciated when it was first formulated, i.e. researchers such as John Dalton, were against it. It is accepted
that our understanding of molecules and elements was naďve during this period. Anyhow by circa 1860, kinetic theory and other facts
seemed to confirm the validity of Avogadro’s hypothesis. Why?
Resolving Avogadro’s Conundrum
Explaining the above conundrum. Imagine
a gas molecule traveling along the positive xaxis strikes a wall. As was previously discussed in kinetic theory, that gas molecule
exchanges a mean energy with the wall of kT/2. If we now consider the wall as being a massive immovable object, then the gas molecule
does not simply bounces off of the wall. Rather, the wall pumps a mean kinetic energy of kT/2, onto the gas molecule with each and
every collision. Ultimately, the walls act like an engine, continually exchanging/pumping a mean kinetic energy of kT/2, along each
axis that is perpendicular to that wall, onto each and every gas molecule, directed along the xaxis.. The net result being that all
gas molecules will possess a mean kinetic energy of kT/2, along each orthogonal direction. This explains why the kinetic theory of
gases is actually valid! Furthermore, when a gas molecule collides with another gas molecule, then we expect that it will adhere to
the conservation of momentum, i.e. eqn 1.10.36, and not necessarily conservation of kinetic energy.
Could it be that the walls (and/or,
other similar surfaces of condensed matter) within a system are what causes Avogadro’s hypothesis to be valid, at least when talking
about dilute gases near room temperature? A requirement would be that gas molecules tend to bounce of the walls much more frequently
than they collide with each other. Hence, the gas must be sufficiently dilute. An implication being that laboratory findings, where
walls exist, cannot always be simply applied systems without walls.
Mean Free Path and Sufficiently Dilute
The concept of sufficiently
dilute gases was previously used in various sections (like 1.5) without providing any clarity. Now, a sufficiently dilute gas can
be understood as a gas in a container whose dimensions are smaller than, or at least not significantly greater than, the gas’s mean
free path.
The mean free path (l) of a gas molecule with a high velocity relative to an ensemble of similar gas molecules at
random locations is defined in terms of the gas molecule’s crosssectional diameter (d) and number of molecules per unit volume (n)
by:
l=1/(pie)ddn 1.10.37
Interestingly, the mean free path does not directly depend upon the gas molecule’s velocity,
however in reality it does, because at a given pressure gas molecules with higher mean momentum will tend to occupy a greater
mean molecular volume, hence a lower number of molecules per unit volume (n). Furthermore eqn 1.10.37 remains an approximation as
molecules have attractions at large distances and repulsion at short distances i.e. LennardJones potential.
Elastic vs Inelastic
Collisions
Traditional kinetic theory considers that all collisions between gaseous molecules are elastic, i.e. energy is conserved.
For any elastic collision, the relative velocity before of the two colliding masses equals the minus of the relative velocity after
the collision. The following solution for a elastic collision is derived in Appendix B.8 of my revised book to be out soon (hopefully).
M_{2}/M_{1}=1[2v_{1f}/(v_{2f}v_{2i})]
A
traditionalist could argue that a gas molecule behaves like a superball when it collides with a wall, i.e. v_{1f}= v_{1i}. Of course this
fits with an elastic collision and relative velocities, but not necessarily with eqn 1.10.38 because the walls mass is infinite compared
to the gas molecule. The awkwardness goes beyond this, because no real mechanism for both the walls and gas molecules being related
to kT/2 is given. If you accept this author’s assertion, that massive walls pump a given energy onto the gas molecules, then
the relationship is understood without the requirement of elastic collisions.
Furthermore, collisions between differing
gases that obey eqn 1.10.38 are not readily envisioned. So although a plausible solution for elastic collisions exists, the assertion
remains questionable. The more logical useful solution remains that intermolecular collisions are not elastic, and that kinetic theory
retains a certain validity simply because the gas is sufficiently dilute that the predominate collision is that between the walls
and the gas molecules.
Now imagine, when gaseous molecules do collide that heat is given off, hence such collisions are not
perfectly elastic, yet energy is conserved. If inelastic collisions occur within a closed system, then the other gas molecules and/or
the surrounding walls should absorb any collision derived heat that is given off. Accordingly, such heat would become part of the
equilibrium state between molecular collisions and vibrations, along with the emission and absorption of thermal radiation.
Furthermore
we now also have the basis for an explanation for viscous dissipation, and/or the natural PT system relationship i.e. molecular collisions
are not necessarily elastic, therefore heat is generally given off. The implication being that intermolecular collisions even in the
condensed matter states are not necessarily elastic, as was previously envisioned.
This also may seemingly fit with our knowledge from
quantum mechanics that even collisions between photons and electrons are inelastic^{21,22,23,24}. This also raises the question to what
degree are collisions between thermal radiation and condensed matter elastic?
Mathematical Argument for Avogadro’s Hypothesis
Start
off with the ideal gas law, i.e. eqn 1.1.21: PV=NkT. Which can be rewritten:
The mean kinetic energy of N gaseous molecule is given by eqn 1.5.12: E_{k}=3NkT/2 . Eqn 1.5.12 can be rewritten as:
Substituting eqn 1.10.40 into eqn 1.10.39, gives:
The point of eqn 1.10.41 is that the
volume (V) occupied by N molecules does not depend upon the molecule’s mass. Rather it depends upon the gas molecule’s kinetic energy,
which was attained from its collisions with the walls, as was determined in kinetic theory of gases. Hence, Avogadro’s hypothesis
is confirmed.
In reality the above mathematical analysis is a circular argument, because we started with the ideal gas
law, whose validity must be questioned when walls do not exist. Interestingly, a whole gambit of thermodynamic relations may falter
without walls.
Avogadro’s Hypothesis Limitations
There must exist conditions, wherein Avogadro’s hypothesis becomes invalid. If the
radius of gaseous atoms were large enough, then the scattering crosssection of gaseous molecules within a system would become significant.
In which case eqn 1.10.36 would dominate, and Avogadro’s hypothesis would falter. However this is unlikely since the freespace associated
with most gases is so much greater than the molecule’s radius.
More likely scenarios occur in highdensity gases, wherein the mean
free path becomes too short, or a dilute gas in a large container, or even our atmosphere where walls do not exist. In such situations
the gas molecules are more likely to collide with each other rather then the surrounding walls. Therefore, the velocities of gas molecules
will tend to obey eqn 1.10.36, rather than obey Maxwell’s velocity distribution, which is based upon kinetic theory, which now falters.
And herein Avogadro’s hypothesis also falters.
Taking this a step further and realizing that since both the ideal gas constant, and
the ideal gas law are also based upon sufficiently dilute gases as is Avogadro’s hypothesis. Then their validity must decline as a
gas’s density increases. It is accepted that the ideal gas law is not valid for highly dense gases^{14}, and now we begin to understand
why that is so. This also helps explain the need for the polytropic equation when dealing with stars, as was previously discussed
in Section 1.9. An unrelenting traditionalist, may adhere to elastic collisions, but then they cannot explain why the polytropic is
required, unless they argue that elastic collisions are limited to dilute gases, but then give no explanation as to why dilute gases
are elastic and dense gases are not.
Avogadro’s Hypothesis at Low Temperatures
If the gases obeyed Avogadro’s hypothesis
for all temperatures, then the: “Number of moles per m^{3}” would be a decreasing linear function of temperature. We can see that it
is approximately so from 100^{ }K to 320 K. However, as the gas’s temperature approaches absolute zero, the number of mols/m^{3} increases
in an exponentialtype fashion.
Copyright Kent W. Mayhew