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The Derivation of the Distribution of the Accelerations of Charged Particles in a Gas as a Function of Temperature and Its Relationship to Blackbody Radiation |
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Human perception of the heat and radiation has evolved slowly over time. First came the recognition that perceptions of hotness and coldness could be quantified and measured as temperature. Then came the realization that heat could be identified in terms of the random motion of the molecules of matter and temperature with the average kinetic energy of those molecules in random motion.
Along with those perceptions of heat and temperature there was the observation that metal heated sufficiently hot glowed red and sufficiently further glowed white. The spectra of glowing objects was measured and it was found that objects not hot enough to glow visible light also emitted longer wavelength radiation. The source of such thermal radiation was a bit of a puzzle.
In the 1890's the proposition was developed that when electrical charges experience acceleration or deceleration they radiate electromagnetic waves. Acceleration or deceleration can be a change in the direction of motion as well as a change in the magnitude of its velocity. The sharp changes in the direction of movement of molecules involved in collisions were taken to the source of thermal radiation. It is a quite plausible explanation but technical details need to be examined. This is the topic of this webpage.
One of the great successes of mathematical physics in the 19th century was the development of the kinetic theory of gases. This theory is now based upon Statistical Mechanics. In Statistical Mechanics the canonical distribution of molecules in various energy states is a negative exponential; i.e., it is of the form
where p(E) is the probability or frequency density and α and β are parameters.
The parameter α is a normalizing constant. This means that the aggregation (integral or summation) of p(E) over the allowable set of values of E is equal to unity. From this condition the value of α is derived.
For example, if the allowable range of E is 0 to ∞
The parameter β is a characteristic of the statistical ensemble.
A major element of the success of the analysis of the kinetics of gas molecules was the Maxwell-Boltzmann distribution of non-interacting molecules. The frequency of molecules in an energy state of magnitude E is a negative exponential function of absolute temperature T; i.e.,
where p(E) is the probability density of a molecule being having the energy E, α is the normalizing constant, T is absolute temperature and k is the Boltzmann constant.
The integral of p(E) over a possible range of E from 0 to ∞ for the kinetic energies of molecules in a gas, is equal to unity. From the previous section it is known that α is equal to 1/kT.
The above distribution is for kinetic energy, but from it the distribution of other variables which are functions of kinetic energy can be derived.
These distributions apply to gases composed of fermions and bosons, respectively. Fermions obey the Pauli Exclusion Principle; bosons do not.
If the gas molecules are monoatomic their energy state is entirely due to their linear kinetic energy, ½mv², where m is the molecular mass and v is its velocity. The probability density function for molecular speed (absolute value of velocity) is
It is widely believed that space and time are not infinitely divisible. The ancient Greeks imagined cutting bits of matter into smaller and smaller pieces. They felt on a purely philosophical basis that that process had to terminate and a piece that could be cut into smaller pieces was called an atom from the Greek terms a and tom for not and cut. Now the term quantum is used for the minimum possible unit of some quantity.
Let ε and δ be the quanta of length and time, respectively. It is presumed that ε=cδ, where c is the speed of light. The quantum of volume is then ε³.
When a molecule impinges upon a wall of the gas container the component of velocity which is perpendicular to the wall surface is reversed. Let θ be the angle between the velocity vector and the normal to the surface of the wall. The component of velocity perpendicular to the wall surface is then v·cos(θ) so the change in velocity is −2v·cos(θ)
Let this on average change in velocity be denoted as γv, where γ is some quantity less than or equal to 2. If the velocity is perpendicular to the wall then γ is equal to ±2.
If the collision and rebound of the molecule from the container wall takes place in time period δ then the acceleration a is
So a² is simply a multiple of the kinetic energy and hence its distribution is a negative exponential. Thus
where β=(γ/δ)²(2/mkT).
The Relativistically correct formula for such radiation generation is the Liénard-Weichert formula but for velocities small compared to the speed of light that formula reduces to the Larmor formula
were R is the rate of radiant energy generation, a is acceleration, q is charge and c is the speed of light.
When the charge is spatially distributed over M pieces the Larmor formula becomes
When the charge is distributed over a volume V and the quantum of length is ε then
The energy E_{col} that goes to create a photon is the energy that is generated over a quantum time interval δ; i.e., Rδ. Also it is presumed that ε=cδ. Thus
Note that the c³ term has been eliminated.
The acceleration arises from the rebounding of a molecule off a wall of the container. This produces a change in velocity of γv occurring in a quantum of time; i.e.,
and hence
This means that the energy of the generated photon is proportional to the kinetic energy of the molecule creating it. This also means that the distribution of the energy of the photons is of the same form as the distribution of the kinetic energies of the molecules, namely a negative exponential. This is a problem because the distribution of photon energies for blackbody radiation is not simply a negative exponential distribution.
A pinnacle of success of mathematical physics was Max Planck's discovery and subsequent derivation of the formula for the frequency distribution of blackbody radiation. It is of the form
where ν is wave frequency, h is Planck's constant, c is the speed of light, k is Boltzmann's constant and T is absolute temperature.
But photon energy is hν so the above formula may be expressed as
This is quite different from a simple negative exponential function.
Thus the generation of radiant energy from the acceleration/deceleration of molecules collisions cannot be the source of the distribution of thermal blackbody radiation. The difference in the two distributions is all that is needed to establish the proposition that the accelerations and decelerations of thermal collisions are not the source of thermal radiation. There is however no doubt that thermal motion is the source of thermal radiation; it just isn't by way of the accelerations and decelerations of the Larmor or related formula.
To further establish that accelerations due to molecular collisions cannot be the source of blackbody thermal radiation some computations are carried out below to show that there is a quantitative discrepancy between the energies of the photons in thermal radiation and the energies of photons that would be generated by molecular collisions.
Wien's Displacement Law says that the most frequent wavelength λ_{max} of radiation for a blackbody at absolute temperature (°K) of T is
where b is equal to 0.0029 mK. Thus for a temperature of 300° K the most frequent wavelength of thermal radiation is about 10^{-5} meters. The energy of a photon of this radiation is
The charge diameter of a proton is approximately 1.68 fermi = 1.68×10^{-15} meters. This makes the quantum of time equal to 5.6×10^{-24} seconds.
The average velocity of an H_{2} molecule at room temperature (300deg; K) is about 2000 m/s. In a head-on collision with a container wall the direction is reversed. If this takes place in one quantum of time the magnitude of the acceleration is (4×10^{3}/5.6×10^{-24})=7.14×10^{26}. This quantity squared is 5.1×10^{53. }
The rate of energy generation according to the Larmor formula is
where a is acceleration, q is charge and c is the speed of light. The charge of the proton is 1.6×10^{-19} coulombs. This squared is 2.56×10^{-38}. The value of R is the rate of energy generation; the quantity of energy generated in one quantum of time is then
The energy generated at room temperature by a head-on collision of an H_{2} molecule with a container wall of 6.74×10^{-33} joules is a far cry from the energy of 1.99×10^{-21} joules. It is about a third of a trillion times smaller.
Max Planck formulated some units of length and time that some suggest might be the quanta of length and time. The Planck values are 1.616×10^{-35} meters and 5.4×10^{-44} seconds.
If a velocity of 2000 m/s is reversed by a head-on collision with the wall of a container in the Planck unit of time its acceleration is (4000/5.4×10^{-44} is 7.4×10^{46} and this squared is 5.5×10^{93}.
The unit of volume for the Planck length is 4.22×10^{-105} m³ The charge of a proton is distributed over a volume of 2.48×10^{-45} m³. This means the charge is distributed in a number of pieces equal to
The Larmor formula with a distributed charge is R = (2/3)a²q²/(Mc³)
The energy generated in a quantum of time is then
E_{col} = δR = ( 5.4×10^{-44})[(2/3)( 5.5×10^{93})( 2.56×10^{-38})/( 5.9×10^{59})(2.7×10^{25})]
E_{col} = 3.18×10^{-73} joules
This is effectively infinitesimal compared with the energy of 1.99×10^{-21} joules of the thermal radiation for the same temperature.
Although it was found that for a quantum of length equal to the Planck value and for one equal to the diameter of a proton that the energy generated by molecular collisions is far less than the energy of the photons involved in blackbody radiation there might be the possibility that at some quantum of length between those extreme values thermal collisions could generate photons of the energy of blackbody radiation. The following shows that this is not true.
The formula derived above is
As stated previously, the average velocity of a hydrogen molecule at room temperature (300° K) is about 2000 m/s. The maximum acceleration that an average H_{2} molecule can experience is the complete reversal of direction from a head-on collision with the wall of the container. Thus the parameter γ will be taken equal to −2. .
The charge radius of a proton is approximately 0.84 fm = 0.84×10^{-15} m. This makes its volume V equal to 2.48×10^{-45} m³.
The value of E_{col} is set equal to 1.99×10^{-21} joules and δ solved for; i.e.,
This makes the quantum of length ε equal to 1.276×10^{-9} meters. This supposedly would be the minimum possible length of anything physical. This cannot be correct since the diameter of a proton is 1.68×10^{-15} meters. Therefore it is impossible to find a quantization of length and time that derives the thermal radiation for 300° K from the accelerations generated by collisions in a gas at that temperature.
Gustav Kirchhoff, in the mid-19th century, established the principle that there is a close relationship between the emission of radiation from a material and the absorption of radiation by that material. This principle further discredits the notion that thermal radiation is created by the accelerations and decelerations of charges in molecular collisions. There is no way that thermal radiation can directly create such accelerations and decelerations.
The way that is accpted as the mechanism by which radiation is absorbed in a gas and turned into thermal motion is through the gas' content of triatomic molecules such as H_{2}O and CO_{2}. The absorption spectra of such molecules are not continuous but the emission at discrete frequencies is spread through such things as the Doppler effect.
Although dipole oscillators are cited any multipole oscillator would have the same effect.
Contrary to what is widely presumed thermal radiation having Planck's blackbody distribution cannot be generated from the accelerations and decelerations produced by collisions resulting from thermal motion. This is due to the two phenomena having different forms as functions of temperature. It is also found that for the allowable values of the quanta of length and time it is impossible to derive the typical photon energy of blackbody radiation at 300° K from the accelerations experienced by the molecules of H_{2} gas at that temperature.
The tentative alternative to the transformation of thermal motion in a gas into thermal radiation via the accelerations/decelerations of collisions due to motion is through the multipole oscillators in the gas.
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