A New Thermodynamics

Blog: Kinetic theory: Part 2: Broken Traditional Concepts

By Kent W. Mayhew

www.newthermodynamics.com

Note: Equation numbers simply are those from my paper (2017) thus do not start at (1).

 

Traditional Kinetic Theory is presented along with some commentary concerning it's illogical application.

 

The law of equipartition states that the total energy of a gas is equally distributed among all of its degrees of freedom. Equipartition theorem is based upon the energy of a system being defined by a set of f generalized coordinates (q1….qf) and their corresponding generalized momentum.

 This leads to equipartition theorem stateing that the mean energy for each independent quadratic term is  ½ kT. For an ideal gas, the translational energy of each gaseous molecule can be envisioned as being purely translational kinetic which contains three quadratic terms. Thus equipartition states that the mean translational kinetic energy [Ekinetic] of an N molecule ideal gase is2,3:

 

           Ekinetic =3NkT/2                               (6)

For N monatomic gas molecules the accepted vae for its total energy is defined by equation (6). Hence the isometric molar heat capacity is 3R/2. Which is to illogically accept no consideration to rotational energy!!

The total mean vibrational energy [Evibrational] of a one-dimensional harmonic oscillator is the summation of the kinetic (kT/2) and potential (kT/2) energies, that being:

 

           Evibrational=kT/2+k”T/2=kT                      (10)

For a diatomic gas the accepted degrees of freedom are: three degrees of freedom associated with translational energy, two degrees of freedom with rotation and one degree accepted for vibration. Based upon this one would expect that isometric molar heat capacity for a diatomic gas is 6R/2. However, the empirically proven value is 5R/2, which has led to the illogical act of the vibrational degree of freedom being disregarded, thus enabling the theoretical value to equate to the empirical values. The accepted reasoning is that vibrational energy at room temperatures makes negligible contributions to a gas’ heat capacity, which may strike some as odd since vibrational and potential energy are the total contribution to the heat capacity of condensed matter, at room temperature. It is interesting that no one has even queried this before.

In our new perspective (next page) we shall present a theory that has no such illogical requirements at the above traditionally beheld theory obviously requires. Returning to the traditional accepted theory.   

The accepted theoretical isometric specific heats for monatomic through triatomic gases are given in Table 1:

 

Table 1: Accepted theoretical values

 

Type of gas

J/(mol*K)

Monatomic

3R/2

12.5

Diatomic

5R/2

20.8

Triatomic

6R/2

24.9

For larger polyatomic gases consisting of n">2 atoms/elements all three rotational degrees of freedom are now considered resulting in the degrees of freedom (3n”) being defined by:

            3n”=3+3-(3n”-6)                                           (11)

Equation (11) means that for larger polyatomic molecules that there are three degrees of freedom for translational and three degrees of freedom for rotation plus (3n”-6) vibrational degrees of freedom.

Illogically, large linear molecules are treated slightly differently with there being  (3n”-5) vibrational degrees of freedom but only two degrees rotational of freedom, hence giving the same overall value. Based upon degrees of freedom arguments the isometric molar heat capacity (Cv) for large polyatomic molecules would be:

           Cv=3n”R/2                                                     (12)

Although heat capacities of gases have been extensively studied throughout the 20th century4,5,6, no one has seen fit to challenge the accepted traditional theory even though it is obviously filled with exceptions, all  designed to make theory fit empirical data. It is amazing how the sciences have gotten away with such continuous illogical fudge factoring all in the name of making their pet theories fit empirical data.

 

  In part 3 we will explain the grandiose mistake that is beheld in traditional kinetic theory and provide a theory that better fits all emperical data without relying upon an of the illogical fudge factoring that traditional theory requires.

 

Mistake of interest.

 

You will find in many texts concerning traditional kinetic theory that they derive their theory by starting with the kinetic energy and momentum of a gas molecule moving along the various orthogonal axis (x,y,z). You will then find the flux in any one of the six possible directions (+x,-x,+y,-y,+z,-z) is taken to be 1/6 of the system's total flux. In reality it is 1/4 of the flux (see "PDF file showing flux calculation Reif"  to the right). Of course this traditional falacy in flux is deemed acceptable because it allows a for a derivation that seemingly matches accepted kinetic theory. A case of two or more wrongs making a right. Perhaps but then again "Not".   

 

 

Copyright Kent W. Mayhew

 

 

Some referenced used herein

 

2. “Fundamentals of Statistical and Thermal Physics”, F. Reif, McGraw-Hill, New York, 1965

3."Statistical Thermodynamics and Microscale Thermophysics", V. Carey, Cambridge U 1999

 

                    Link to Kinetic Theory Part 3

 

                     Return to Part 1

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