A New Thermodynamics

Blog: Kinetic theory: Part 3: New Conceptualization

By Kent W. Mayhew


Kinetic Theory: A new conceptualization 


Consider that you hit an incoming tennis ball with a suitable racquet. If the ball hits the racquet at a 90 degree angle then the ball will have significant translational energy in comparison to any rotational energy. Conversely, if the ball hits the racquet at a different angle (obtuse or acute), although the same force is imparted onto that ball, the ball’s rotational energy may become significant when compared to the ball’s translational energy. The point becomes, in real life both the translational and rotational energy are due to the same impact exchange.

    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:


Etotal=3NkT/2                                                             (17)


    All the considerations in formulating eqn 17 in tradiational kinetic theory apply, except we now have the realization that the wall’s kinetic energy is transformed in a combination of the gaseous molecule’s kinetic and rotational energy. This applies to all molecules from monatomic thru polyatomic, which is to say that we do no longer require the fudge factoring that tradiational theory has that being the irrational concept that monatomic molecules have no rotational energy.


  The isometric molar heat capacity for a monatomic gas is still:


          Cv=3R/2                                                                      (18)


  Now consider a diatomic gas


  What about the vibration energy of the diatomic gas molecules? As is the case for condensed matter, this would be related to the absorption and emission of the blackbody/thermal radiation surrounding the diatomic gas molecules.


   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)


  Unlike traditional kinetic theory wherein the vibrational energy of a diatomic molecule is considered to be zero thus enabling the theory to match empirical data, our new conceptualization is now a perfect match  to the accepted value for isometric molar heat capacity (Cv) of a diatomic gas that being:


          Cv=5R/2                                                                       (20)


   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:


               Etotal=(n”-1)kT+3kT/2                                       (23)                             


    Collecting the terms gives:


             Etotal=(n”+1/2)kT                                                 (24)                             


   And  for an N gas molecules becomes:


           Etotal=NkT(n”+1/2)                                               (25)        


    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                                

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In the above Graph 1 you can see how the traditional kinetic theory, (which is wrongly based upon degrees of freedom) skyrockets away from empirical data, while our new perspective adheres much closer to the same data.
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