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A New Perspective for Kinetic Theory and Heat Capacity Volume 13 (2017) PROGRESS IN PHYSICS Issue 3 (July) A New Perspective for Kinetic Theory and Heat Capacity Kent W. Mayhew 68 Pineglen Cres., Ottawa, Ontario, K2G 0G8, Canada. E-mail: [email protected] The currently accepted kinetic theory considers that a gas’ kinetic energy is purely trans- lational and then applies equipartition/degrees of freedom. In order for accepted theory to match known empirical finding, numerous exceptions have been proposed. By re- defining the gas’ kinetic energy as translational plus rotational, an alternative explana- tion for kinetic theory is obtained, resulting in a theory that is a better fit with empirical findings. Moreover, exceptions are no longer required to explain known heat capacities. Other plausible implications are discussed. 1 Introduction Various explanations for equipartition’s failure in describ- The conceptualization of a gaseous system’s kinematics orig- ing heat capacities have been proposed. Boltzmann suggested inated in the writings of the 19th century greats. In 1875, that the gases might not be in thermal equilibrium [8]. Planck Maxwell [1] expressed surprise at the ratio of energies (trans- [9] followed by Einstein and Stern [10] argued the possibility lational, rotational and/or vibrational) all being equal. Boltz- of zero-point harmonic oscillator. More recently Dahl [11] mann’s work on statistical ensembles reinforced the current has shown that a zero point oscillator to be illusionary. Lord acceptance of law of equipartition with a gas’s energy being Kelvin [12–13] realized that equipartition maybe wrongly de- equally distributed among all of its degrees of freedom [2–3]. rived. The debate was somewhat ended by Einstein claiming The net result being that the accepted mean energy for each that equipartition’s failure demonstrated the need for quantum independent quadratic term being kT=2. theory [14–15]. Heat capacities of gases have been studied The accepted empirically verified value for the energy of throughout the 20th century [16–19] with significantly more a /textitN molecule monatomic gas is kT=2 with its isomet- complex models being developed [20–21]. It becomes a goal of this paper to clearly show that an ric molar heat capacity (Cv) being (3R=2). An implication is that a monatomic gas only possesses translational energy alternative kinetic theory/model exists. A simple theory that [4–5]. The reasoning for this exception is that the radius of a correlates better with empirical findings without relying on monatomic gas is so small that its rotational energy remains exceptions while correlating with quantum theory. negligible, hence its energy contribution is simply ignored. 2 Kinetic theory and heat capacity simplified Mathematically speaking equipartition based kinetic the- ory states that a molecule with n00 atoms has 3n00 degrees of Consider wall molecules 1 through 8, in Fig. 1. The total freedom (f ) [5–6] i.e.: mean energy along the x-axis of a vibrating wall molecule is 00 f = 3n : (1) Ex = kT: (3) This leads to the isometric molar heat capacity (Cv) for large Half of a wall molecule’s mean energy would be kinetic en- polyatomic molecules: ergy, and half would be potential energy. Thus, the mean 3 kinetic energy along the x-axis, remains C = n00R: (2) v 2 kT Interestingly, the theoretical expected heat capacity for N di- E = : (4) x 2 atomic molecules is 7NkT=2. This is the summation of the following three energies a) three translational degrees, i.e. In equilibrium, the mean kinetic energy of a wall molecule, as 3NkT=2. b) three rotational degrees of freedom, however defined by equation (4) equals the mean kinetic energy of the since the moment of inertia about the internuclear axis is van- gas molecule along the same x-axis. Herein, the wall in the ishing small w.r.t. other moments, then it is excluded, i.e. y-z plane acts as a massive pump, pumping its mean kinetic NkT. c) Vibrational energy, i.e. NkT. This implies a molar energy along the x-axis onto the much smaller gas molecules. heat capacity Cv =7NkT=2 = 29.3 J/(mol*K). However, em- In equilibrium each gas molecule will have received a pirical findings indicate that the isometric molar heat capacity component of kinetic energy along each orthogonal axis. Al- for a diatomic gas is actually 20.8 J/(mol*K), which equates though there are six possible directions, at any given instant, to 5RT=2 [6]. This discrepancy for diatomic gases certainly a gas molecule can only have components of motion along allows one to question the precise validity of accepted kinetic three directions, i.e. it cannot be moving along both the pos- theory! In 1875 Maxwell noted that since atoms have internal itive and negative x-axis at the same time. Therefore, the parts then this discrepancy maybe worse than we believe [7]. total kinetic energy of the N molecule gas is defined by 166 Kent Mayhew. A New Perspective for Kinetic Theory and Heat Capacity Issue 3 (July) PROGRESS IN PHYSICS Volume 13 (2017) Fig. 1: Ideal monoatomic gas at pressure Pg and temperature Tg Fig. 2: Ideal diatomic gas at pressure Pg and temperature Tg sour- sourrounded by walls at temperature Tw = Tg. Gas molecules have rounded by walls at temperature Tw = Tg. Gas molecules have vi- no vibrational energy. brational energy. equation (4) i.e. 3NkT=2. Up to this point we remain in it cannot rotate hence both energies can only result in agreement with accepted theory. vibrational energy of the wall molecules along its three Consider that you hit a tennis ball with a suitable racquet. orthogonal axis. If the ball impacts the racquet’s face at a 90 degree angle, then After numerous wall impacts, our model predicts that an the ball will have significant translational energy in compar- N molecule monatomic gas will have a total kinetic energy ison to any rotational energy. Conversely, if the ball impacts (translational plus rotational) defined by the racquet at an acute angle, although the same force is im- 3 parted onto that ball, the ball’s rotational energy can be sig- EkT(t;r) = NkT: (5) nificant in comparison to its translational energy. The point 2 being, in real life both the translational and rotational energy, Fig. 2 illustrates a system of diatomic gas molecules in a con- are due to the same impact. tainer. The wall molecules still pass the same mean kinetic Now reconsider kinetic theory. Understandably, momen- energy onto the diatomic gas molecule’s center of mass with tum transfer between both the wall’s and gas’ molecules re- each collision. Therefore the diatomic gas’ kinetic energy is sult in energy exchanges between the massive wall and small defined by equation (5). The diatomic gas molecule’s vibra- gas molecules. Moreover, the exact nature of the impact will tional energy would be related to the absorption and emis- vary, even though the exchanged mean energy is constant. sion of its surrounding blackbody/thermal radiation. There- fore, the mean x-axis vibrational energy within a diatomic gas Case 1: Imagine that a monatomic gas molecule collides molecule remains defined by equation (3) and the total mean head on with a wall molecule, e.g. the gas molecule energy for a diatomic gas molecule becomes defined by hitting wall molecule no. 3 in Fig. 1. Herein, the gas molecule might only exchange translational energy 3 5 E = E + E = kT + kT = kT: (6) with the wall, resulting in the gas molecule’s mean ki- tot kT(t;r) v 2 2 netic energy being purely translational, and defined by Therefore the total energy for an N molecule diatomic gas equation (4). becomes Case 2: Imagine that a monatomic gas molecule strikes wall 3 5 molecule no. 1 at an acute angle. The gas molecule E = E + E = NkT + NkT = NkT: (7) tot kT(t;r) v 2 2 would obtain both rotational and translational energy from the impact such that the total resultant mean en- For an N molecule triatomic gas: ergy of the gas molecule would be the same as it was 3 7 E = E + E = NkT + 2NkT = NkT; (8) in Case 1, i.e. defined by equation (4). tot kT(t;r) v 2 2 Case 3: Imagine a rotating and translating monatomic gas n00 signifies the polyatomic number. Therefore for N molecule striking the wall. Both the rotational and tran- molecules of n00-polyatomic gas, the vibrational energy is slational energies will be passed onto the wall molecu- 00 le. Since the wall molecule is bound to its neighbors, Ev = (n − 1)NkT: (9) Kent Mayhew. A New Perspective for Kinetic Theory and Heat Capacity 167 Volume 13 (2017) PROGRESS IN PHYSICS Issue 3 (July) 00 Therefore, the total energy for a polyatomic gas molecule is: acetylene (C2H2, n = 4, Cv= 35.7) are linear bent molecules 00 and good fit, while pyramidal ammonia (NH3, n = 4, Cv = 3 E = E + E = NkT + (n00 − 1) NkT 27.34) is not. Could the gas molecule’s shape influence how tot kT(t;r) v 2 ! it absorbs surrounding thermal radiation, hence its vibrational 1 (10) = n00 + NkT: energy? 2 Table 2 shows the accepted adiabatic index versus our theoretical adiabatic index for most of the same substances Dividing both sides by temperature and rewriting in terms of shown in Table 1. Our theoretical adiabatic index compares per mole (N=6:02 × 1023) then equation (10) becomes: rather well with the accepted empirical based values, espe- ! ! cially for low n00 < 4 and high n00 > 11, as is clearly seen Etot 1 1 = nk n00 + = R n00 + : (11) in Fig.
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