Logic dictates that a gas molecule obtaining kinetic energy from vibrating wall molecules should behave in a similar fashion to a tennis ball striking a racquet. No wonder Maxwell was surprised that the rotational and translational energies have the same value in traditional theory! Although they certainly can be, the vast majority of the time, the translational and rotational energies due to given impacts will not be the same! Herein, we depart from traditionally accepted degree of freedom based arguments and employ common sense. Obviously Maxwell was right to be surprised at the traditionally accepted result, and should have followed his gut feeling rather than profess classical traditionally accepted kinetic theory.
Accepting that a monatomic gas nolecukle will be have similarly to a tennis ball means that a monatomic gas will have both kinetic and rotational energy whose total mean value will be determined by the energetics of the wall's molecules. Thus in terms of total kinetic plus rotational energy [Etotal], would now be better written:
The total mean energy (Etotal) for an N molecule diatomic gas must now equal the addition of its mean translational and rotational energy [eqn 17)], plus its mean vibrational energy (Evibrational =kT) i.e.:
Etotal=E(kinetic+rotational) +Evibrational (19 a)
Etotal = 3kT/2+kT = 5kT/2 (19 b)
If a polyatomic gas molecule absorbs or emits thermal photons in a manner similar to diatomic molecules, then an n”-molecule polyatomic gas molecule should have a mean vibrational energy of:
Evibrational = (n”-1)kT (22)
where n” signifies the number of atoms in each gas molecule which we shall call the polyatomic number. Therefore, as an approximation the mean total energy [Etotal] for a polyatomic gas molecule should be the addition of its mean translational and rotational energy as defined by equation (17) plus its mean vibrational energy as defined by equation (22), giving:
Collecting the terms gives:
And for an N gas molecules becomes:
In deriving equation (25), we treated the system as if there were limited exchanges of vibrational energies between the vibrating polyatomic gases and vibrating walls. This is most likely not the case. However, if we assume that on average a polyatomic gas gives as much vibrational energy as they extract with each wall collision, then there is very limited or no net exchange of vibrational energy. Of course we may expect that as the polyatomic gas gets larger (more atoms) there may be some substantial net energy exchanges with each wall collision and this may take extensive modeling.
Those indoctrinated in traditional thermodynamics may not appreciate what is theorized herein. Certainly, our altered perspective has no dependence upon the traditional conceptualization of degree of freedoms whereby all the kinetic energy is treated as translational for monatomic molecules. One may care to say in our analysis that the degrees of freedom still apply, however, we now have both the gas’ rotational and translational energies along each axis exchanged with the wall molecules kinetic energy along that axis.
What is most important is that we no longer require those accepted exceptions. For example there is no longer a reason to believe that a monatomic gas has no rotational energy. Or, a diatomic gas has no real vibrational energy and only possesses two degrees of rotational freedom, thus making theory match empirical data. Or, why other gases have no real vibrational energy when condensed matter must at the same temperature. Or, why large polyatomic gases have (3n"-6) vibrational degrees of freedom while diatomic molecules do not have any vibrational energy at room temperature. Or, why large polyatomic molecules have vibrational energy but no potential energy associated with the bonds as condensed matter does.
This may have consequences to traditional fundaments such as Maxwell’s velocity distributions for gases (If you consider Maxwell's distribution in terms of energy then now the energy is both rotational and translational). It must be stressed that in our analysis we do not define the magnitude of translational energy compared to rotational energy, except to say that they add up to and equal the summation of the wall molecule’s kinetic energies along three axis. It is more than likely that the translational energy is significantly greater than the rotational energy for most gases
Beside: See Graph 1 showing that the the theory presented herein fits empirical data better than the traditional theory illogically based upon degrees of freedom can.
Note this blog page was based upon my July 2017 paper in Journal Progress in Physics. An expanded interpretation is in my April 2018 paper in the same journal, which clearly shows:
1) why collisions are inelastic
2) why empirical findings for large polyatomic gases do not match theory (mine nor traditional), the reason being what this author calls "flatlining". see paper
Copyright Kent W. Mayhew
Copyright Kent W. Mayhew