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“Prediction of enthalpy for nitrogen gas” revised I.H. Umirzakov Institute of Thermophysics, Lavrentev prospect, 1, 630090 Novosibirsk, Russia e-mail: [email protected]
Abstract It is shown that the energy spectrum of the pure vibrational levels of the molecule consisting of two atoms interacting with each other via the modified Rosen-Morse potential, the analytical expressions for the vibrational partition function and enthalpy of the diatomic molecule, obtained in the paper “Prediction of enthalpy for nitrogen gas, M. Deng, C.S. Jia, Evr. Phys. J. Plus 133, 258 (2018)”, are incorrect. Keywords: modified Rosen-Morse potential; oscillator, vibrational energy spectrum, partition function; enthalpy; diatomic molecule, dimer; nitrogen N 2 . Introduction The modified Rosen-Morse potential is given by  2
ere q , V (r ) De 1 r e q
where De is the dissociation energy, De 0 , r is the interatomic separation, r 0 , re is the equilibrium bond length, re 0 , q and are the adjustable parameters, 0 and q 0 . Here the energy is measured from the minimum of the potential well, so V (re ) 0 and
V (r ) De . The expression for the energy spectrum of the pure vibrational levels of the molecule consisting of two atoms interacting with each other via the modified Rosen-Morse potential was used in  to obtain analytical expressions for the vibrational partition function. The expression for the enthalpy of the ideal gas of the diatomic molecules was obtained using the vibrational partition function and rigid rotator model in order to take into account the rotations of the molecule. The calculated enthalpy for the nitrogen molecule N 2 was compared with the experimental data. We show in the present paper that the energy spectrum of the pure vibrational levels of the diatomic molecule and the analytical expressions for the vibrational partition function and enthalpy of the diatomic molecule obtained in  are incorrect. Main consideration 1. According to  the energy E (v) of the pure vibrational level with the vibrational quantum number v is defined from
where vmax is the most vibration quantum number, 8De / 2 2 , 0 and a ere / q . Using the condition q 0 we obtain from a ere / q that a 0 . Here X integer is equal to X for an integer X , and X integer is equal to the greatest integer less than X for a non-integer X . According to  the value of vmax given by Eq. 6 can be obtained using Eq. 2 from the condition dE (v) / dv v v
The partition function of the vibrational states ( Q v ) of the diatomic molecule is equal to v max
Q e E ( v ) / kT , v
where k the is Boltzmann’s constant and T is the temperature . Eq. 2 gives E (vmax ) 0 in the case when
1 a 1 1 (a 2 1) 2n , 2
where n 0, 1, 2... . The value E (vmax ) 0 corresponds to the un-bonded state of the diatomic molecule. Therefore Eq. 6 cannot be used in order to calculate the partition function of bonded vibrational states of the diatomic molecule, and it is necessary replace Eq. 6 by the correct one. For example, Eq. 6 can be replaced by 2 . vmax 1 a 1 1 (a 2 1) / 2 1/ 2 integer
in order to avoid the inclusion of the un-bonded state to the partition function. 2. It is easy to see that Eq. 6 is incorrect in the case: a) 0 a 1 . We have from Eq. 6 that vmax 0 in the case: b) a (1 ) /(1 ) and 1 . Therefore in the cases a) and b) the solution of the Schrodinger equation, energy spectrum, given by Eq. 2, analytical expressions for the partition function and enthalpy of the diatomic molecule, obtained in  from Eq. 2, loss their physical sense. 3. The vibrational partition function was calculated in  using the approximate relation
1 v Qappr [e E ( 0) / kT e E ( vmax 1) / kT ] 2
v max 1 E ( x ) / kT
The enthalpy of the bonded vibrational states is equal to the mean energy of the states , so v v v Eappr we have H v E v and H appr for the exact ( H v ) and approximate ( H appr ) enthalpies of the bonded vibrational states of diatomic molecule. Using the exact relation E kT 2 ln Q / T we obtain from Eqs. 8 and 11 v max
E H E (v)e E ( v ) / kT , v
1 v v Eappr H appr [ E (0)e E ( 0) / kT E (vmax 1)e E ( vmax 1) / kT ] 2
v max 1
E ( x )e
E ( x ) / kT
for the exact E and approximate Eappr mean energies
of the bonded vibrational states,
respectively. The enthalpy H of the molecule is equal to the sum of the rotational H rot , translational H tr and vibrational H v enthalpies : (14) H H v H r H tr , tr (15) H 5N AkT / 2 , H rot N AkT
where N A is the Avogadro’s number and Trot 2 / 2re2k . The detailed analysis of the derivation of Eqs. 8 and 15  from Eq. 7  shows that Eqs. 8 v v and 15  for the partition function Qappr and enthalpy H appr are incorrect. For example: -
it is necessary to replace the incorrect 21 1 and 21 2 by the correct 21 12 and
21 22 , respectively, in the right hand side of Eqs. 8 and 15 ; 2 / kT erfi ( 2 1 2 ) / kT the incorrect term 21e 1 must be replaced by the correct one 21e 2 1 / kT erfi
(2 ) / kT in the numerator at the right hand side of Eq. 15 ; 2 2
the dimensionalities of the arguments of the exponents e1 / kT and e 2 / kT in Eq. 15  are equal to that of an inverse of an energy while the dimensionalities of the arguments of the exponents must be dimensionless, and etc. Therefore the data for the enthalpy presented on Fig. 1  are incorrect if they were obtained using Eq. 15 . The correct Eqs. 8 and 15  are: -
v Qappr e De / kT G / 2 ,
v H appr De ( 22e 2 / kT 12e 1 / kT ) / G kT / 2 [1 (e 1 / kT e 2 / kT ) / G ] 2
4. According to  E (0) 0 and dE(v) / dv 0 , where 0 v vmax , for the nitrogen molecule
N 2 . Therefore we obtain from Eqs. 12-13 Hv 0,
v H appr 0
for T 0 , and v H v H appr E (0) 0
in the limit T 0 (see also Fig. 3). According to Eqs. 15-16 the rotational and translational enthalpies are positive at T 0 , and these enthalpies vanish in the limit T 0 . Therefore we have from Eqs. 12-14 using Eqs. 15-16 and 20-22 H 0,
H appr 0
for T 0 , and H H appr E (0) 0
in the limit T 0 . But according to Fig. 1 
H appr 0 at low temperatures. Therefore the data presented on
Fig. 1  for H appr could be incorrect, and the experimental data presented on Fig. 1  for the enthalpy of the nitrogen molecule could be incorrect too. v / Qv of the approximate vibrational partition 5. The temperature dependence of the ratio Qappr v function Qappr obtained from Eq. 11 to the exact one Q v obtained from Eq. 8 for the nitrogen
molecule N 2 is presented on Fig. 1. One can see that the ratio increases from 1/ 2 to 1 with increasing temperature, and Eq. 11 gives incorrect values of the vibrational partition function, especially at low temperatures. v / E v 1) 100% between the 6. The temperature dependence of the relative difference ( Eappr v approximate mean vibrational energy Eappr obtained from Eq. 13 to the exact one E v obtained
from Eq. 12 for the nitrogen molecule N 2 is presented on Fig. 2. We established that increases from 50% to 2% with increasing temperature from zero to 3168.35 K , and it decreases from 2% to 0% with increasing temperature from 3168.35 K to infinity, Eq. 13 underestimates the mean vibrational energy at temperatures lower than 1863.86 K , and Eq. 13 slightly overestimates the mean vibrational energy at temperatures higher than 1863.86 K .
Fig. 1. Temperature dependence of ratio v Qappr / Qv of approximate vibrational v partition function Qappr obtained from Eq.
11 to exact one Q v obtained from Eq. 8.
Fig. 2. Temperature dependence of relative v ( Eappr / E v 1) 100% difference between approximate mean vibrational v energy Eappr obtained from Eq. 13 to exact one E v obtained from Eq. 12 for the nitrogen molecule N 2 .
7. The temperature dependencies of the approximate H appr defined from Eqs. 13-16 and exact enthalpies H defined from Eqs. 12 and 14-16 for the nitrogen molecule N 2 are presented on Fig. 3.
Fig. 3. Temperature dependencies of approximate H appr defined from Eqs. 13-16 (solid red line) and exact enthalpies H defined from Eqs. 12 and 14-16 (dashed blue line) for the nitrogen molecule N 2 .
As evident from Fig. 4 the values of the approximate and exact enthalpies for the nitrogen molecule are very close to each other despite the considerable difference between the approximate and exact vibrational enthalpies given by Eqs. 13 and 12, respectively, at temperatures below 1863.86 K . As one can see from Fig. 4 the ratios of the approximate and exact vibrational enthalpies to the sum of the rotational and translational enthalpies are small. The smallness of these ratios at temperatures below 1863.86 K is the reason of the good agreement between H appr and H .
Fig. 4. Temperature dependencies of: ratio H appr / H of approximate enthalpy H appr defined from Eqs. 13-16 to exact enthalpy H defined from Eqs. 12 and 14-16 for the nitrogen molecule N 2 (dashed black line); ratio
v H appr /( H rot H tr )
v vibrational enthalpy H appr defined from Eq.
13 to sum H rot H tr of H tr and H rot defined from Eqs. 15-16 (solid red line); and ratio H v /( H rot H tr ) of exact vibrational v enthalpy H appr defined from Eq. 13 to sum
H rot H tr (dotted blue line).
Conclusion It is shown that that the energy spectrum of the pure vibrational levels of the molecule consisting of two atoms interacting with each other via the modified Rosen-Morse potential, the analytical expressions for the vibrational partition function and enthalpy of the diatomic molecule obtained in  are incorrect. References  M. Deng, C.S. Jia, Evr. Phys. J. Plus 133, 258 (2018).  L. D. Landau, E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory. Vol. 3, 3rd ed. (Pergamon Press, New York, 1977).