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 (1776-1856); 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 M₁ and M₂, collide then their total momentum is conserved, therefore in terms of their velocities (v₁ and v₂):              

v₂=(M₁/M₂)v₁       Or     v₁=(M2/M1)v₂                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 naïve 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 x-axis 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 x-axis.. 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 cross-sectional 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. Lennard-Jones 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).                              

M₂/M₁=1-[2v_1f/(v_2f-v_2i)]                                 1.10.38

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 P-T 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:

  V=NkT/P                                        1.10.39

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:

  NkT=2E_k/3                                         1.10.40

Substituting eqn 1.10.40 into eqn 1.10.39, gives:

V=2E_k/3P                                                   1.10.41

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 cross-section 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 high-density 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 gases¹⁴, 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

Interestingly, Avogadro’s hypothesis also falters at low temperatures. Graph 1.10.2 is the plot for the “Number of Moles per Cubic Meter versus Temperature”, for five elements: Hydrogen, helium, argon, nitrogen and oxygen. The graph for each element starts near their boiling point, and continues until 320 K. The data⁵ for Graph 1.10.2 is located in Table 1.10.5, at the end of this section. 

If the gases obeyed Avogadro’s hypothesis for all temperatures, then the: “Number of moles per m³” 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/m³ increases in an exponential-type fashion.

 Low Temperatures Gases

A simple traditional argument for the behavior of gases at low temperatures is that the gas molecules have less kinetic energy, therefore other forces, e.g. the electromagnetic forces (EM), become prevalent. However, this argument is troublesome because as we just witnessed in Graph 1.10.2, Avogadro’s hypothesis did not hold for the noble gas, helium wherein the EM interaction between molecules should be a minimal. Perhaps then, it could be a case of other forces such as gravity affecting what we see.

There are also quantum mechanical arguments to help explain the behavior of gases at low temperature, e.g. the Bose-Einstein condensation (BEC). Basically it is concludes that equipartition falters at low temperatures. The reasoning is based upon quantum effects becoming significant; “when thermal energy kT is smaller than the quantum energy spacing in a particular degree of freedom, the average energy and heat capacity of this degree of freedom are less than the values predicted by equipartition. Such a degree of freedom is said to be frozen out when the thermal energy is smaller than the spacing. For example: The heat capacity of a solid decreases at low temperatures as various types of motion become frozen out, rather than remain constant as predicted by equipartition”¹⁸

The above quantum argument is rather complex and out of the scope of this text. So, can we find a simpler argument? This author cannot help but ponder if the simplest explanation for Avogadro’s hypothesis faltering at low temperatures is simply because the thermal radiation density would no longer simply be directly proportional to temperature i.e. eqn 1.4.7: @=a’T [Rayleigh-Jeans approximation (@ is energy density, a' is a constant)] is no longer valid. Of course the fact that a gas’s energetics is no longer proportional to T means that kinetic theory would no longer be valid; more provocative food for thought.

 For a better understanding see my papers on kinetic theory and or my book.

 Copyright Kent W. Mayhew