Calculation of the internal energy of neon, argon, krypton and xenon at their critical points using a thermomechanical model

Cryogenics 38 (1998) 1267–1268  1999 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0011-2275/99/$ - see front matter

PII: S0011-2275(98)00089-7

Calculation of the internal energy of neon, argon, krypton and xenon at their critical points using a thermomechanical model S.A. Ulybin†, V.I. Sukhov*‡ and V.D. Chernetsky§ †The Moscow State Academy of Chemical Engineering, 107066, Moscow, Russia ‡Joint Stock Company of Cryogenic Engineering (JSC ‘CRYOGEN-MASH’), 143900, Balashikha, Moscow Region, Russia §Institute of Theoretical and Experimental Physics, 117259, Moscow, Russia

Received 25 February 1998; revised 17 August 1998 The energy balance equation of phenomenological thermomechanics has been used to calculate thermal, repulsive, attractive, vibrational and spin energies of Ne, Ar, Kr, and Xe at their critical points. The obtained results show a good agreement with analogous calculated data for the thermomechanical substance. The discrepancies between them never exceed 4.9%.  1999 Elsevier Science Ltd. All rights reserved

Keywords: phenomenological thermomechanics; energy balance equation; atomic substances; critical state; individual energies In previous works1–3 the analytical ratios of phenomenological thermomechanics have been used for theoretical strict subdivision of the internal energy of the model thermomechanical substance into its six separate components: thermal, repulsive, attractive, vibrational, spin and surface. It was also pointed out that the thermophysical properties of such atomic cryogenic products as neon, argon, krypton and xenon are close mostly to thermophysical properties of

*To whom correspondence should be addressed.

thermomechanical substance. In present article the calculating results of subdivision internal energies of the thermomechanical substance and enumerated above real atomic substances at their critical points are presented. The calculations have proved out the high model qualities of thermomechanical substance4. So, the discrepancies between experimental and calculated data never exceed 4.9%. We suppose, that it is very satisfactory for such simple model6. The energy balance equation served as the initial relationship for the calculations:

冋 冉冊

3 ∂␳ PV PV ⫹ 3PV ⫺ 3 1/3 ⫺ 3 ⫺ V2 2 Z ∂V ⫹

冉冊

⫺ RT

T

册

3 3 ␳ ⫹ 2RT(1 ⫺ Z1/3 ) ⫺ RTB ⫽ 0 ZRTB 2 ␳0 2

(1)

where P ⫽ pressure; V ⫽ specific volume; T ⫽ absolute temperature; R ⫽ gas constant; Z ⫽ PV/RT ⫽ compressibility factor; ␳ ⫽ 1/V ⫽ density; ␳0 ⫽ density on line of ideal gas under P ⫽ 0 and T ⫽ 0; TB ⫽ Boyle’s temperature. As at critical point the partial derivative (∂P/∂V)Tc ⫽ 0, Equation (1) is simplified to form:

冉 冊 3 PV 2 c c

therm

⫹ (3RTc )vibr ⫹ ⫹

冉

⫺

冉 冉冊

⫹ (3PcVc )rep ⫹

3 RT 2 B

冊

冋

冊

⫺ 3PcVc Z1/3

attr

册

3 ␳c Z RT ⫹ 2RTc(1 ⫺ Z1/3 c ) 2 c B ␳0 ⫽0

sp

(2)

sur

Expression (1) was obtained under assumption that in equilibrium thermomechanic macrosystem the volumetric and surface energies are numerically equal each other and opposite in sign. Since value of the later is equal to ⫺ 3/2RTB, the specific volumetric energy is equal to 3/2RTB. In relation to it, all internal energy components of the thermomechanical substance are computed (bottom line in Table 1). For it Zc ⫽ 8/27, ␳c ⫽ 9/32 ␳0, Tc ⫽ 3/8 TB, and Equation (2) can be transformed to:

冉 冊 4 RT 9 c

therm

⫹

冉 冊 冉 8 RT 9 c

⫹

rep

⫹ (3RTc )vibr ⫹ (RTc )sp ⫽

⫺

冊

4 RT 3 c

3 RT ⫽ 4RTc 2 B

attr

(3)

Numerical significances of the individual energies can not be determined from Equation (3) because temperatures Tc, TB and gas constant R are not known separately for thermomechanical substance. But their relative contributions can be found from (3) rather easily. The thermal

Cryogenics 1998 Volume 38, Number 12 1267

Research note: calculation of the internal energy of neon, argon, krypton and xenon: S.A. Ulybin et al. Table 1 Fractions of the individual components of the internal energy of thermomechanical substance and cryogenic atomic substances, calculated relative to UB (1), Uc (2), Utot (3) Energy/substance

Ne flow rate, % Ar flow rate, % Kr flow rate, % Xe flow rate, % Thermomechanical substance

Thermal

Repulsive

Attractive

1

2

3

1

2

3

1

0.109 ⫺1.8 0.108 ⫺2.7 0.107 ⫺3.6 0.106 ⫺4.5 0.111

0.113 1.8 0.109 ⫺1.8 0.109 ⫺1.8 0.108 ⫺2.7

0.112 0.9 0.109 ⫺1.8 0.109 ⫺1.8 0.108 ⫺2. 7

0.218 ⫺1.8 0.215 ⫺3.2 0.214 ⫺3.6 0.211 ⫺4.9 0.222

0.225 1.4 0.218 ⫺1.8 0.218 ⫺1.8 0.216 ⫺2.7

0.224 0.9 0.218 ⫺1.8 0.218 ⫺1.8 0.216 ⫺2.7

⫺0.326 ⫺0.336 ⫺2.1 0.9 ⫺0.325 ⫺0.330 ⫺2.4 ⫺0.9 ⫺0.324 ⫺0.329 ⫺2.7 ⫺1.2 ⫺0.32 ⫺0.327 ⫺3.9 ⫺1.8 ⫺ 0.333

Table 2 Absolute values of the individual energies [J/kg]

2

Vibrational

Spin

3

1

2

3

1

2

3

⫺0.334 0.3 ⫺0.329 ⫺1.2 ⫺0.329 ⫺1.2 ⫺0.327 ⫺1.8

0.727 ⫺3.1 0.740 ⫺1.3 0.738 ⫺1.6 0.733 ⫺2.3 0.750

0.750 0.0 0.750 0.0 0.750 0.0 0.750 0.0

0.745 ⫺0.7 0.749 -0.1 0.750 0.0 0.750 0.0

0.247 ⫺1.2 0.250 0.0 0.249 ⫺0.4 0.247 ⫺1.2 0.250

0.255 2.0 0.253 1.2 0.253 1.2 0.253 1.2

0.253 1.2 0.253 1.2 0.252 0.8 0.253 1.2

Table 3 Initial values

5

Substance/ energy

Ne

Ar

Kr

Xe

Substance/Value

Ne

Ar

Kr

Xe

Utherm Urep Uattr Uvibr Usp Utot UB Uc

8238 16475 ⫺24602 54869 18625 73605 75445 73159

13714 27429 ⫺41382 94191 31792 125744 127295 125588

9052 18104 ⫺27335 62321 20991 83133 84453 83094

7938 15876 ⫺24029 55039 18577 73401 75129 73385

Atomic weight R,J/kg*K ␳0,g/cm3 Pc,bar Vc,cm3/g ␳c,g/cm3 Tc,K Zc TB,K

20.1 411.93 1.673 26.53 2.070 0.4830 44.40 0.3003 122.10

39.9 208.12 1.870 48.98 1.867 0.5357 150.86 0.2912 407.76

83.80 99.21 3.210 54.96 1.098 0.9107 209.39 0.2905 567.50

131.30 63.32 3.890 58.21 0.9092 1.0990 287.74 0.2884 791.00

energy contribution is equal 1/9; repulsive ⫺ 2/3, attractive ⫺ 1/3 (with the minus sign), vibrational ⫺ 3/4, spin ⫺ 1/4 of internal energy. The maximum contribution falls on the vibrational energy, the minimum-on the thermal; they differ almost by a factor of seven. For cryogenic products the calculations were made (Table 1): 1. 2. 3.

relative to energy UB ⫽ 3/2RTB at Boyle’s point; relative to energy Uc ⫽ 4RTc at critical state; relative to total energy Utot in volume of macrosystem.

The summarized experimental data5, given in Table 3, served as initial values for calculations. Substituting them in Equation (2), all individual energies, given in Table 2, have been found for Ne, Ar, Kr and Xe. Their isotope components are not shown in 5. One peculiarity follows from comparison of Table 2 and Table 3. All parameters, given in Table 3, are varying monotonically as the substance atomic mass increases. Quite a different situation is observed for individual energies in Table 2. It is obvious that here exists extremum for argon.

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Cryogenics 1998 Volume 38, Number 12

As soon as possible, we are planning also to calculate the individual energies of neon, argon, krypton and xenon at another states. And, in particular, to discuss a probable influence of quantum effects in neon on precision of thermomechanical calculations for it.

References 1. Ulybin, S.A. and Sukhov, V.I., Subdivision of internal energies of neon, argon, krypton and xenon into separate. Cryogenics, 1997, 37, 61–62. 2. Ulybin, S.A. and Sukhov, V.I., Some possibilities of using phenomenological thermomechanics in cryogenics. Chem. Petrol. Eng., 1994, 30, 175–179. 3. Ulybin, S.A. and Sukhov, V.I., Thermomechanics interpretation of Clausius Entropy. Chem. Petrol. Eng., 1995, 31, 554–559. 4. Ulybin, S.A., Thermomechanical components of internal energies of atomic substances at critical states. Chem. Petrol. Eng., 1997, 4, 45–47. 5. Thermophysical Properties of Neon, Argon and Xenon. 1976, pp. 636. (in Russian). 6. Ulybin, S.A., Phenomenological Thermomechanics leads to macrodeterminism. Herald of the Russian Academy of Sciences, 1995, 65(6), 497–501.