Walls in Thermodynamics: Consequences to Kinetic Theory, Ideal Gas Law, Avogadro’s hypothesis, etc. And even Cosmology.
Below is a post on taking a new look at thermodynamics and the concept of work. It’s from Kent Mayhew. I don’t necessarily agree or disagree with his views, but CNPS welcomes new ideas, so please comment and let us know your thoughts.
In a previous blog (dated: Oct 15 2016), I discussed that kinetic theory should be rewritten and that by doing so the new kinetic theory will match empirical data more closely than current degrees of freedom based kinetic theory does with my new kinetic theory equation for the energy of a gas being:
E = NkT(n”+1/2)
Where n” signifies the number of atoms in each gas molecule which we shall call the polyatomic number. Note this energy can be shown to be the basis of the isometric specific heat of N gaseous molecules. T is temperature and k is Boltzmann’s constant
The net result being that my equations for both isobaric (Cp) and isometric (Cv) specific heats of gases closely follows that of accepted empirical data, while the traditional theoretical accepted equations has a completely different slope when compared to the same accepted empirical results.
Some further discussion concerning kinetic theory:
The kinematics of matter has simple explanation as to how the energy of a system seemingly relates to the Boltzmann constant (k). Basically, the three orthogonal walls of a closed system act as massive pumps, each pumping a mean energy of kT/2 onto each of the significantly smaller enclosed gas molecules. I have determined that:
1) The kinetic energy of the wall molecules equals the gas’ translational plus rotational energy. This says nothing about the relative values of rotational and translational energies of a gas, except that they are to be added and equated to the summation of the wall molecule’s mean kinetic energies along all three axes.
2) The total energy of the gas being equated to the translational plus rotational energy, plus any vibrational energy. This insight helps better explains the accepted empirical findings for the specific heats of gases, than the traditional understanding.
3) The thermal radiation of the blackbody radiation that exists in interior volumes of freespace is what allows for thermal equilibrium to exist in many systems.
4) The above stated thermal radiation is responsible for the vibrational energies of the polyatomic gases within a system i.e. polyatomic gases and condensed matter are to be treated equally.
5) The energy associated with the above stated radiations are generally infinitesimally small when compared to the kinematic energies associated with matter. Exceptions being vacuums and high temperature systems e.g. blast furnaces.
Work and Walls
In previous blogs I have discussed that most expanding useful systems must displace Earth’s atmosphere, and that this is work done onto the surrounding atmosphere, i.e. what is commonly referred to as “lost work” (W=PdV), but is so poorly understood by all those who teach, and work in thermodynamics.
Since I am discussing walls herein I am simply going to state that work is generally done thru a systems walls onto its surroundings. In traditional thermodynamics we are often wrongly taught that work goes into s system’s walls, which is completely illogical. Of course those who profess such illogical thought do not appreciate questions like
1) Why is the same amount of work required for imaginary walls as real walls, irrespective of what the walls are made of? Sure if all were elastic one might think of this as a possibly logical consequence but the fact remains rubber walls only exist around balloons, hence are not great for universally applied arguments.
2) Why is this work lost, e.g. if energy is lost into expanding walls, why does it not return when walls shrink?
A new perspective for equilibrium & the role of walls
Consider that both a given volume of gas and surrounding walls are in thermal equilibrium at temperature T. This means:
- The walls are in thermal equilibrium with the enclosed blackbody radiation e.g. both are related to the same temperature.
- The gas molecule’s translational plus rotational energy is in mechanical equilibrium with the molecular vibrations of the wall molecules.
- The gas molecule’s vibrational energies are in thermal equilibrium with the enclosed blackbody radiation e.g. both are related to the same temperature.
The combination of these three states of equilibrium causes the energies of the gas molecules and wall molecules to correlate with each other and their surrounding thermal radiation, i.e. if isothermal walls did not surround the gas then true equilibrium between the gas molecules and the surrounding thermal radiation may not exist. A fact overlooked by traditional analysis!
In order to better understand reconsider an ideal gas in thermal equilibrium with its surrounding walls, as illustrated in Fig 1.9.5. Not only does this mean that the gas molecules obtain as much molecular energy from the walls as it gives, but it also means that both the gas molecules, and the walls absorb as much blackbody radiation as they radiate.
Remember monatomic gases lack intramolecular vibrations therefore they neither absorb, nor radiate thermal radiation. Conversely, polyatomic gases have intramolecular vibrations therefore they absorb, and emit, thermal radiation.
Consider a dilute monatomic gas wherein the blackbody radiation moves freely until it strikes the walls. The mean speed (magnitude of the mean velocity) of each gaseous molecule will be a function of the system’s temperature, and this has to do with the gaseous molecules hitting the container’s walls, and bouncing back into the fray. Extrapolating this further, consider that the vibrational energy of the container wall’s molecules is a function of the wall’s temperature. Therefore, the mean energy impeded upon each gaseous molecule during its last wall collision is a function of the wall’s temperature! The distribution of the magnitude of molecular velocities about their mean value is commonly referred to as; Maxwell’s velocity distribution, for that gas.
Now consider a polyatomic gas. The above still applies in so far as molecular energies are concerned, except now the gas molecules also absorb and radiate radiation, such that their vibrational energies remain related to the system’s temperature. The blackbody radiation density, vibrational energy and the translational energy within the gas’s volume, are independent of each other. However, the fact that they are all dependant upon the wall’s temperature means that correlations exist in systems, which are contained by walls i.e. bounded by condensed matter. We now begin to understand what Maxwell & all who have followed failed to understand.
True Adiabatic Expansion
Reconsider the gaseous molecules and blackbody radiation, except now imagine that both the walls and surroundings are magically removed, as is depicted on the L.H.S. of Fig 1.9.6. Without walls, the gas would disperse, expanding outwardly, as is depicted on the R.H.S. of Fig 1.9.6. Without any surroundings, then the expanding gas does no work.
Understandably, without the influx of thermal radiation from/through the walls, the blackbody radiation density within the system must decrease over time, as the system expands. As the blackbody radiation density decreases, the rate at which polyatomic gaseous molecules absorbs thermal radiation must decrease with time. Hence their vibrational energy must decrease. Of course monatomic gases will not be affected in the same way.
For both a polyatomic and a monatomic gases. If a thermometer is placed into that expanding wall-less gas, the thermometer should read a lower temperature due to the decrease in blackbody radiation density. One might ponder to what extent will the thermometer’s reading drop? That should depend upon the kinetic energy density of gas molecules, in comparison to the energy density of the blackbody radiation, i.e. its current associated temperature.
Furthermore, as the system’s temperature decreases, our expectation becomes: The mean frequency of thermal radiation emitted by the polyatomic gaseous molecules will decrease as the gas molecules become colder although their mean velocity remains constant. Ultimately, the gas molecules get colder, as the density of surrounding radiation decreases although their velocities are constant.
Kinetic Energy & Wall-less Expansion
How about the kinetic energy of the gas molecules? Imagine that the gaseous molecules experience no external forces e.g. gravity, or EM, then conservation of momentum implies that the gaseous molecules will forever continue upon the trajectories that they attained after their last collision. Ultimately, the gaseous molecules will travel outwards towards infinity, as is illustrated in the R.H.S. of Fig 1.9.6. Since the gaseous molecule’s momentum would be conserved, then so too is their kinetic energy.
If the total kinetic energy of the gas molecules within our system is constant, then for a truly adiabatically expanding ideal gas, Boyle’s law holds (PV=constant). For our truly adiabatically expanding gas, the system’s temperature should be decreasing because the blackbody radiation density is decreasing. If the temperature is decreasing then the ideal gas law cannot apply, even though Boyles law holds.
To some, it may seem mind-boggling that Boyle’s law applies to the expansion of a gas irrelevant of whether its expansion is isothermal in a system surrounded by walls, or, non-isothermal in a wall-less true adiabatic system. Now reconsider the ideal gas law. It must only apply to systems wherein the blackbody radiation is at the same temperature as the matter contained within that system, that being in systems with actual walls.
Blackbody Density, Kinetic Energy, Temperature and Walls
Generally, without walls the mean velocity, hence the mean kinetic energy, and consequentially the pressure of a gas can decouple from the blackbody radiation density that resides in the surrounding freespace. In other words, equilibrium no longer exists between matter and its contiguous blackbody radiation. Does this mean that the magnitude of gas molecule’s velocity is no longer a function of the system’s temperature? Probably! Therefore, temperature does not necessarily hold the same meaning for all systems without walls, as it does for systems with! Of course such conceptualization is foreign to traditional theory wherein temperature is wrongly limited to molecular motions.
Can you think of a truly naturally occurring adiabatically expanding system? Perhaps, the only case of a true adiabatically expanding system is the expansion of our universe, as was envisioned by Hubble: This of course assume that Hubble was right which may or may not be the case. Now this statement assumes that our universe is actually expanding and that no energy is exchanged between our universe and its surroundings. This may become a subject for debate, if our new perspective is adopted.
So far as an adiabatically expanding universe is concerned, the traditional consideration that work (PdV) being done onto the expanding system’s walls7,8,14 no longer applies. Interestingly, in commenting upon this traditional consideration: “Enrico Fermi once suggested”… “the work done, which is equal to the energy expended by the expanding gas, goes “into the hands of god”, which is another way of saying that we do not know”20. Our new perspective that work involves the movement of a mass against a gravitational force, renders Fermi’s commentary as nonsensical, e.g., if our universe is surrounded by nothingness then no work can be done onto such surroundings.
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 1810)7. The implication becomes, when comparing two gases, and then those two gases will have the same mean kinetic energy per unit volume.
One of the concerns of Avogadro’s hypothesis is that it seemingly confounds the conservation of momentum. From classical mechanics we know that when two dissimilar objects, which possess different masses namely M1 and M2, collide then their momentum is conserved. The net result becomes the more massive object attains a lower velocity than the less massive object, after their collision. Specifically, if they both start with the same magnitude of velocity then the relation becomes:
V2=(M1/M2)V2 eqn 1
Where V is velocity and M is mass of colliding particles
Based upon the above equation (eqn 1), one might then assume, that two interacting gases should not have the same mean kinetic energy after their collision. A corollary would become they should not tend to occupy the same mean molecular volume hence the same total volume for the same number of molecules, at a given pressure. In other words, we have a conundrum, wherein Avogadro’s hypothesis seemingly goes against the principle of conservation of momentum. This may explain why Avogadro’s hypothesis was not appreciated when it was first formulated, i.e. many researchers such as John Dalton, were against it. Anyhow by circa 1860, facts seemed to indirectly confirm the validity of Avogadro’s hypothesis. Why?
Explaining the above conundrum. Imagine a gas molecule traveling along the positive x-axis strikes a wall. As was previously discussed in Section 1.5, 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 wall acts 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. The net result becomes that all gas molecules will possess a mean kinetic energy of kT/2, along each orthogonal direction. Thus explaining 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 above
In conclusion, it must be that the walls (and/or, other similar surfaces of condensed matter) within a system are what causes Avogadro’s hypothesis to be valid21, at least when talking about gases near room temperature and 1 atm pressure. It turns out that if we were to perform an analysis of the mean free path of such gas molecules, then it would be determined that gas molecules tend to bounce of the walls much, much more than they collide with each other. This implication being that we may not always be able to directly apply our findings obtained in a laboratory, wherein walls exist, to other systems specifically those without walls21.
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 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.
A more likely scenario occurs in high-density gases, wherein the mean free path becomes too short, meaning gas molecules become more likely to collide with each other rather then the surrounding walls. Therefore, for high-density gases, the velocities of gas molecules will tend to obey eqn 1, rather than Maxwell’s velocity distribution, which is a result of kinetic theory. And herein Avogadro’s hypothesis falters.
Taking this a step further and realizing that since both the ideal gas constant, and the ideal gas law are based upon 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 of the polytropic equation, which is often used to explain what we witness in stellar bodies like stars.
We are taught that collisions between gaseous molecules are elastic, i.e. energy is conserved. Imagine, when gaseous molecules do collide that heat is given off, hence the collision is not perfectly elastic. If inelastic collisions occur within a closed system, then the other gas molecules and/or the surrounding walls should absorb any collisional 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.
Now we 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 inelastic21,22,23,24. This also raises the question to what degree are collisions between thermal radiation and condensed matter elastic?
Implications to Cosmology and sciences in General
Well I have previously discussed that entropy does not belong in cosmology. After reading this those whose interest is in cosmology must realize that traditional thermodynamics often cannot be applied to cosmology because what we are taught really concerns systems with walls. And unless our universe has walls, then one must question what we are doing.
Of course when considering thermodynamic here on Earth, we tend to compartmentalize systems by giving them walls, hence separation. Not only is this generally done in experiments, but it is also the standard for most industrial applications.
The science of thermodynamics that we are taught is based upon systems with walls, e.g. most experiments being performed in walled closed system. If we remove such walls then the science as a whole can fall apart.
Kinetic theory, the ideal gas law, Avogadro’s hypothesis, Maxwell’s distributions etc all require that the:
1) temperature associated with a system’s Blackbody radiation
2) temperature associated with the kinematics the gaseous molecules, and/or any other matter with said system
3) temperature of the walls themselves
all are one and the same.
In other words, all are in thermal equilibrium. What is not taught is that it is the system’s walls themselves that actually make all this possible. Specifically kinetic theory, the ideal gas law, Avogadro’s hypothesis, Maxwell’s distributions all are based upon systems with walls. Remove the walls, as is often the case in cosmology, and the science can be turned upside down.
Now there will be those who will argue that temperature is associated with the kinematics of matter. As I discussed in previous blogs on this website, I would argue that a vacuum has a temperature that being the temperature associated with the blackbody radiation within!
When all is said and done the way that thermodynamics is traditionally taught and therefore envisioned requires a rethink.
References used in this blog.
- “Fundamentals of Statistical and Thermal Physics”, F. Reif, McGraw-Hill, New York, 1965
- “Statistical Physics”, F. Reif, McGraw-Hill, New York, 1967
- Statistical Thermodynamics and Microscale Thermophysics”, V. Carey, Cambridge U 1999
- “Genesis of the Big Bang”, R.A. Alpher and R.C.Herman, Oxford University Press, UK 2001
- K. Mayhew, Phys. Essays 19, vol 4, 604(2013)
- P. Marmet IEEE Transactions on Plasma Science vol 18 issue 1 (1990)
23 . P. Marmet, Phys. Essays, vol 1, pp 24-32 (1998)
24 . M. Jauch and F. Rohlich, The theory of Photons and Electrons. Cambridge, MA:Addison-Wesley,195