Indian Journal of Chemistry Vol. 43A, July 2004, pp. 1385-1 392

Diatomic potential energy function for helium

Elaheh K Goharshadi* & Mohsen Abbaspour

Dept. of Chemist ry , Ferdowsi University, Mashhad 91779, Iran

E- mail: gohari @ferdowsi .um.ac.ir

Received 4 November 2003; revised 12 March 2004

The diatomic potential energy function for helium has been determined via the in version of reduced viscosity collision integrals at corresponding states and zero pressure. The resulling potential has been fitted to obtain an analytical form. A comparison of newly determined potential with the previously determined potentials is also included. The transport properties of he lium such as viscosi ty , self-diffusion, and thermal conductivity coefficients at different temperatures and pressures have been calculated and compared with experimental data and they are found to be in good agreement with each other. The second virial coeffic ient of helium has been also calculated using the potential at different temperatures.

IPC Code: Int.C l. 7 COlB 23/00

The intermolecular forces are of interest in a wide range of disciplines as the interaction between molecules controls the progress of molecular collisions and determines the bulk properties of matter l

.

Specific information about a potential energy function can be extracted by the inversion of experimental data. In particular, direct inversion techniques for the determination of potential from data on molecular transport properties and second virial coefficients have been developed that do not require any explicit assumption to be made about the fundamental form of the potentiaI2

-4. The main aim of the present paper is to determine the helium potential energy function by inversion of the reduced viscosity collision integrals at zero pressure.

The second objective of the work is to present a table of specific collision integrals at zero pressure in evaluating the transport coefficients of helium at any pressure and temperature.

Finally, we have calculated the transport and equilibrium properties of helium at different temperatures. The results of these calculations are in an excellent accord with experiment.

Theory

Transport properties The macroscopic properties of gases can be

explained in terms of the motion and interaction of molecules. Enskog and Chapman5

,6 developed the theory for transport properties of monoatomic dilute gases. The most significant feature of their theory is

that each transport coefficient of the gas or gas mixture can be expressed in terms of a series of collision integrals, namely, Q(I,S)(T), characterized by the values of I and s. For example, viscosity of a pure monoatomic gas is related to Q (2.2) and the binary diffusion depends on Q(I,I ).

The kinetic theory expression for the coefficient of viscosity (1]) of a pure dilute monatomic gas of

molecular mass, m at a temperature, Tis

... (1)

where kB is the Boltzmann constant and If( is a constant which weakly depends on temperature6

.

The temperature-dependent collision integral Q (2,2)

for monoatomic gases is explicitly related to the potential, U(r) through the cl ass ical mechani cal expressions5

:

... (2)

.. . (3)

and

.. . (4)

1386 INDIAN J CHEM, SEC A, JULY 2004

where E is the relative kineti c energy of a pair of colliding molecules, Q(2)(E) is a transport cross section, b is the impact parameter, X is the scattering angle, and ro is the classica l di stance of closest approach in a collision.

It is convenient to define reduced collision integrals, Q(2.2 )*(T*) by

. .. (5)

where Qrs(2.2\n is the colli sion integral for rigid spheres of diameter d,

. .. (6)

and T*is the reduced temperature and is defined T* = kTI£, where £ is the potential well depth.

The initial density dependence of transport properties The determinati on of the density dependence of

transport coefficients such as viscos ity and thermal cond uctivity of gases is of both scientifi c and practical importance. In analogy into the dens ity expansion of equilibrium properties , viscos ity can be written as an expans ion in powers of density, p,

11 = 110 ( I + BIlP + ... ) .. . (7)

where 110 represents the zero-density viscosity coefficient. The parameter BIl is the second viscosity virial coefficient.

The viscosity of moderately dense gases(up to 2 mol dm·3

) at a speci fied temperature and densi ty can be calculated via the eq uation:7

... (8)

where N A is the Avogadro number, a is the collision diameter at which the intermolecular potential is zero, BIl* is the reduced second viscosi ty virial coefficient, BIl '= BIl /a

3, the values of which have been given in

ref. 7. According to the theory of Rainwater and Friends.9, the reduced second viscosity virial coefficient is assumed to consist of three contri butions:

B• - B (2)* B (3)* B (M -D)' 11- 1'] + 1'] + 1'] ... (9)

where B1'] (2)* is the contribution of two free monomers, B1'] (3)' represents three monomer contribution, and BIl (M-D)* is the contribution from monomer-dimer collisions.

Najafi and hi s colleagues7 have calculated the reduced second viscosity virial coefficient for noble gases based on the Rainwater-Friend theory using the most accurate diatomic potentials for these systems. Their results implied that there could be a universal function for the calculation of B' 1'] for all noble gases. The con'esponding states behavior of B' 1'] over the entire range of T* for the noble gases can be attributed to the superimposability of their reduced potenti als. According to this correlation, BIl' is the universal function of (T' r i in the form of a sixth order pol ynomial:

6 .

B1'] ' = 'fbi (T* f' . .. (10) ;=1

where b i is a constant and its values for the noble gases were g iven in ref. 7 .

Najafi et a /. lo have ex tended Eq .(8) to a very high density range (up to 40 mol dm-3 and 900 MPa) by the following equation:

11=11o(1+NAaB1']p)+(1+an 1 ... ( 11 ) 3 • 2 ( bl P + b2 P 2 )

1+ clP + c2 P-

where the values of parameters a, b" b2, C" and C2

have been given in ref. 10 . The thermal conductivity (It) of moderately dense

gases (up to 2 mol dm-3) at a specified temperatu re and density can be calculated via the equation II:

.. . ( 12)

where BA' is the reduced second thermal conductivity virial coefficient, assumed to be made up of three contributions:

... ( 13)

The superscripts represent the contribution from the nonlocality of monomer-monomer collisions, the effect of the presence of a third particle during a monomer-monomer collision, and monomer-dimer collisions, respectively.

Najafi et ai. 11 have shown that there exists a corresponding states behaviour for B\ for all noble gases:

.. . ( 14)

The values of parameters ao and al have been given in ref. II .

GOHARSHADI el al.: DIATOMIC POTENTIAL ENERGY FUNCTION FOR HELIUM 1387

The thermal conductivity of gases at high density up to 40 mol dm-3 at a specified temperature and pressure can be calculated II via Eq. ( 15):

.. . (15)

where DJ,. is a function of the fluid density l l and is represented by

DJ,. I)...*= L d .(plp*)j j=2.4 .5.8 J

.. . ( 16)

where the ex pansio n coe ffici ents, dj and the

coeffici ents )"'* and p *have been g iven in re f. II .

Determination of interatomic potentials via the inversion of reduced viscosity collision integrals at zero pressure

In the previ ous in vers ion a lgorithms fo r direct determ ination of interactio n potentials from the extended principle o f correspo nding states it has been assumed that the influence of the initi al density dependence of transport properties is small and negligibl e in compari son with the uncertainties assoc iated with the experimental methods2.3. In o the r words, the framework of the law of corresponding states l2.13 is based on experimental va lues at re lati vely low densities, since values at zero dens ity are not access ible to direct measure ments. In princi ple, such an assumptio n cannot be accepted because the di fference between the zero density and atmospheric viscosity data is not a lways negli g ib le, especiall y at low temperatures 7.

The reduced viscosity colli sio n integra l at zero pressure, Q y .2)* , can be obta ined by inserting Eq .(I) into Eq .(8):

(2, 2)* Q

cs

... ( 17)

where Q cs(2 .2)*has been taken fro m a correspond ing

states corre lation 12. The values of Q o(2.2)* can be used

to determine the inte ractio n potenti a l by an invers io n procedure as described be fore2-4. 14 . One of the peculi ar features of the inversio n procedu re is that it identi f ies a po int o n the experimenta l Q (2.2) versus temperatu re curve with a single po int on the U(r) fun cti o n. In view of the triple in tegra l that re lates the potentia l over its entire range to the colli sio n integral Q (2.2)(7), it is surpri sing that such an identificati on should be poss ible.

Results and Discussion It can be shown that the width of the potenti al

energy function may be defined in terms of the seco nd

viri al coefficient at temperatures for which T < £ / k B

(ref. 15) Therefore, knowing the inner branch of the potential well from the vi scosity , we can use thi s information in conjunction with the second viri a l coefficient data to determine the outer branch of the well uniquely. The equations used for this purpose are as follows 15:

V IE =T* -I ... ( 18)

r } - ,. 3 = -b (s * - I)N (T *) N L 0

... (19)

where r R and r L are the coordinates o f the ri ght (oute r) and left (inner) wall o f the potenti a l well , bo =

2 n NIP } 13, and N(T*) is g iven in ref 15 . The second

viri a l coeffi c ient o f a pa ir of noble gases is corre lated in the ex tended princ iple o f correspo nding states by the fo rmula l2:

OST* S I.I .. . (20)

* S * = -CT*) '12 E IIT [1 . 18623 + I .00824T* + 4.2557 I CT * )~

-1 8.6033(T * )} + 20.4732(T* )4 -8.7 1903(T*)'i

+ 1.1 4829(T *)6 ]

Thus, the potenti al fun cti on is not obtained directly . but if rL is known, say from viscosity data inversion, r R can be obta ined .

The di ato mi c potenti al energy function of he li um has been obta ined using the inversion of the reduced viscos ity colli sio n integra ls from the corresponding s tates corre latio n and zero pressure as desc ri bed above. At lo ng range, o nly the well -w idth of the potenti al obta ined fro m the second viria l coeffici ent data is ava ilable . T hi s has been used in conjuncti on with the inner coordinates of the we ll obtained fro m the viscos ity in vers ion to g ive the potential over a range of r. The results obta ined from th is way has been compared to the previously determined potenti als l6.17 (Fig. 1). The s li ght deviation between the potenti a l obtai ned by the invers ion of the corresponding s tates reduced viscosity coll isio n integrals and that o f zero pressure at long range where the attractive forces dominate is due to the fact that the density dependence of viscosity is very significant at low temperatures. Therefore, the potenti a l obtained

1388 INDIAN J CHEM, SEC A, JULY 2004

100 ,----------------------------------,

w :>

10

0.1 0.0

-0.3

-0.6

-0.9

-1.2

-1.5 -1------,--------,--------,------,-----,-------1

0.8 1.0 1.2 1.4 1.6 1.8 2.0

rIa

Fig. I- Reduced potential energy functi on o f he lium obtained by in versio n of corresponding Slates reduced viscosity colli sion integra ls (0 )1 7 and by inve rsion of reduced viscosity co lli sio n integral s at zero pressure (- ). The solid c urve is the Aziz et al. potential16

•

using the inversion of reduced viscosity colli sion integrals at zero pressure is superior representation of He-He interactio n potential than that provided by the inversion of the corresponding states red uced viscosity colli sion integrals l7

.

We have fitted our determined potential to an analytical form that is similar to the HFD-B (Hartree­Fock-dispersion) potenti al model:

• * * 6 * 8 * 10 V =A ex p(-OtX)-C6 Ix -Cs Ix -c /O Ix ... (2 1)

where x=rla, and V* = Vlt: . The va lues of the para meters of the potential are given in Table I .

The most commonly needed colli sion integra ls and their ratios for helium at zero pressure are given in Table 2 .

Figure 2 shows the viscosity of helium at zero pressure at different temperatures_ It is obvious that there exists a very good accordance be tween the experimenta l values l8 and o ur calcul ated va lues in the temperature range fro m 170 to 1050 K at zero

Table I-Parameters for the HFD- B like He- He Potential

Parameter Value

A' 1.0439x IOs

al 42.416

C6' 3.88778 Cs' 3.93271 C IO' -7. 83377

pressure . The experi mental and our calculated va lues o f viscosity obey the equation:

'1 = 1.49282 + 0.275109 T 0.737011 ... (22)

The vi scosity of helium at atmospheric pressure has been calculated from 50 to ] 073 K and compared with cOtTesponding experimental values l 9

-2 1 (Fig.3). The

curve follows the equatio n:

'1 = -0.068 1825 + 0.420847 T 0677855 ... (23)

Figure 4 shows the percentage dev iation of the ex perimental viscosity values from our calcu lated values aga inst pressure at different temperatures for helium.

The thermal conductivity of he lium at zero pressure from 170 to 1050 K obeys the equation :

A = -10.7407 + 4.04 195 T 0.653 184 ... (24)

The thermal conductivity of helium agai nst temperature at atmospheric pressure follows the equati on:

A = -1.67] 17 + 3.32407 T 0678055 (25)

Fig ure 5 shows the percentage deviatio n of the experimental thermal conducti vi ty values from our ca lcul ated values against pressure at different temperatures for helium. Figure 6 shows our calculated self-d iffusio n coefficient (D) values of helium at atmospheric pressure and experimental data in the temperature range from 50 to 523 K. It fo llows the equation :

D = -0. 1 13901 + .00281641 T + 1.04306x to-5 T 2

. .. (26)

Figure 7 shows the second virial coeffici ent (B2) of helium against temperature. T he resu lts are summarized in Table 3, including the pressure range,

GOHARSHADI et al.: DIATOM IC POTENTIAL ENERGY FUNCTION FOR HELIUM

Table 2-Dimensionless coll ision integrals n (l.s)* =n(l.s)/no2 and the related ratios for helium at zero pressure

10gT*

- 1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 .9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

n ( I .I)*

4.0869 3.4889 3.0269 2.66 19 2.3640 2.11 28 1.8956 1.7054 1.539 1 1.3950 1.27 17 1.1 675 1.0799 1.0065 0.9447 0.8922 0.8469 0.8070 0.77 10 0.7378 0.7067 0.6772 0.6492 0.6227 0.5980 0.5754 0.5556 0.5389 0.5256 0.5 156 0.5087

3. 10 15 2.7326 2.4370 2. 1885 1.9709 1.7763 1.602 1 1.4482 1.3 150 1.2020 1.1077 1.0293 0.9641 0.9093 0.8626 0.8220 0.7857 0.7524 0.72 12 0.69 14 0.6629 0.6356 0.6097 0.5856 0.5636 0.5446 0.529 1 0.5172 0.5088 0.5035 0.5005

nO.2)*

4.3500 3.72 13 3.2422 2.872 1 2.5768 2.3294 2. 111 6 1.9 135 1.7324 1.5697 1.4269 1.3046 1.20 18 1.l1 65 1.0460 0.9876 0.9389 0.8974 0.86 11 0.828 1 0.7970 0.7667 0.7368 0.7070 0.6770 0.6469 0.6 175 0.5900 0.5656 0.5453 0.5296

2.6486 2.3775 2.1454 1.9376 1.7479 1.5762 1.4239 1.2923 1.1 815 1.0899 1.0 144 0.95 19 0.8994 0.8546 0.8 155 0.7805 0.7482 0.7 175 0.688 1 0.6598 0.6326 0.6067 0.5825 0.5604 0.54 13 0.5258 0.5 142 0.5063 0.5015 0.499 1 0.4983

B*

1.2022 1.1 902 1. 1904 1.1 992 1.2 109 1.2 195 1.22 12 1.2 148 1.201 1 1.1 83 1 1.1642 1.1 469 1.1 324 1.1 208 1.11 24 1.1 072 1. 1050 1.1 053 1.1 069 1.1 085 1.1 095 1. 1092 1.1 07 1 1. 10 19 1.0920 1.0774 1.0595 1.0408 1.0240 1.0 I 06 1.00 10

c*

0.7589 0.7832 0.805 1 0.8222 0.8337 0.8407 0.8452 0.8492 0.8544 0.86 16 0.87 10 0.8817 0.8928 0.9035 0.9 13 1 0.92 13 0.9277 0.9323 0.9353 0.937 1 0.9380 0.9386 0.9392 0.9404 0.9426 0.9465 0.9523 0.9597 0.968 1 0.9764 0.9840

1389

50 ,---------------------------------, 60.----------------------------------.

45

40

35

<IJ

t':I 30 C. ¢ ~ 25

20

15

o 200 400 600

TIK 800

•

1000 1200

Fig. 2-Viscosity of helium against temperature at zero pressure for: experimental values (. )18, and our calcul ated values (solid line).

50 •

40

20

10

o~--~----._--~----~--_.----~--~

o 200 400 600 800 1000 1200 TIK

Fig. 3-Viscosity of helium against temperature at atmospheric pressure fo r: experimental va lues (. )19. ( _ )20. ( A )2 1. and our calcu lated values (solid line ).

1390 INDIAN J CHEM, SEC A, JULY 2004

3~----------------------------------~ ________ ~ T=77.36 K 19

• 77.45 22

2 0 113.80 22

0 140.1022

• 157.16 19

• a 171.40 22

~ • a 183.1523

0 • 194.66 19

0 • 200.30 22 ;;.. a Aa ~ A~ ~~ 0 .6. ~ .(/l t.~ 't 223.1523 Cl o ~~~ ~ 9 A V 0

Va A A a :A. A a &" ~ A A a T 234.1022

~A A g a 248.1524

V i a a!9' • a a v 273.00 22

• -1

~ • 273.16 19

~ 298.15 23 , ~ 299.10 22

-2 J • • 333.20 22

0 373.15 2'

0 373.90 22

• -3

0 20 40 60 80 100 120 140 160 180 200

P/atm

Fig. 4-Plot of the percent dev iation % Dev =( llc,,1c - llcxp / llcxp)X 100 of the experimental viscosity values with our calculated values against pressure at different te mperatures for he lium.

• T = 300.52 K25

0 300.5325

1.0 0 300.5525

• 300.61 25

0.8 a 300.62 25

a 300.6425

• 300.68 25

0.6 .6. 300.73 25

A 300.75 25

0.4 V 300.78 25

• V 300.83 25

0.2 • 300.9025

;;.. • ~ 300.94 25

~ • x ~ 301.01 25 ~ 0.0 <> •

~ • • 301 .08 25

~. 0 0 301 .17 25

-0.2 0 306.84 26

A 307.65 26 v V 0 307.97 26 -0.4

• • 309.0626

-0.6 JJ. T 309.3826

D a • 310.2226

• 0 0 310.31 26

-0.8 • • • 310.61 26 0

310.89 26

-1.0

40 60 80 100 120 140 160 180 200 220 240 260

Platm

Fig. 5- Plot of the per cent deviation %Dev =(Ac"lc - Acxp / Acxp)x 100 of the experimental va lues of thermal co nductivity with our ca lculated values against pressure at different temperatures for he lium.

GOHARSHADI el al.: DIATOMIC POTENTIAL ENERGY FUNCTION FOR HELIUM 1391

5 ,-----------------------------------~ Plllin- Plllax and the temperature range, T mi n - T lllax along with the minimum and maximum per cent deviation of the calculated values from their cOITesponding literature values.

•

o

o 100 200 300 400 500 600

TIK

Fig. 6--Self-diffusion coefficient of helium against temperature at atmospheric pressure for: experimental values (.)20, and our calculated values (solid line).

If we compare the values of viscosity at zero pressure with those at atmospheric pressure at the same temperature in Table 4 , it can be seen that there exits three different behaviors: 7 (i) At low temperatures, T* < 0.95, the viscosity at non-zero pressure is less than that at zero pressure since at thi s range both the two body and three-body contributions to viscosity (B1)*(2) and B1)*(3») are negative and increase rapidly with temperature, whereas the monomer­dimer contribution (B1)*(M.D») is positive and it

decreases at a slow rate. Hence, the full B/)* is negative but it increases rapidly in a positive manner with increasing temperature; (ii) at T*::::: 1, the viscosity is nearly independent of density since the three body and the monomer-dimer contributions and hence the magnitude of B1)* is very close to zero. Thi s temperature is similar to the Boyle temperature for a real gas, namely, the density dependence of the viscosity is negligible; and (iii) At an intermediate temperature range, 0.95~ T * ~ 10, the viscosity values at non-zero pressure are greater than the corresponding values at zero pressure, since B1)* is positive but decreasing smoothly with increasing temperature. At this range, the second reduced Boyle viscosity temperature occurs.

15

12

"0 E 9 ~

:E b .... 6 --I.tI

3

0

0 500 1000

TIK

--

1500 2000

Fig. 7-Second virial coefficient of helium against temperature for: experimental values (dashed line)27, and our calculated values ( solid line ).

]n summary, from the present results , the followin g conclusions can be made: (i)The remarkable feature of the inversion method employed in this work is that the specific information about potential energy function contained in experimental measurements is determined directly without assuming a function al form for the potential; (ii) the present paper provides the collision integrals and their ratios for helium at zero density which is needed to calculate the transport coefficients of helium in the moderate and high

Table 3--The pressure and temperature ranges, and minimum anrl maximum per cent deviations of calculated properti es

Propert y (x) Tmin-Tmax Pmin-Pmax I OO(Xc"lc- xex p,)! xex p' (K) (atm) min - max

Viscosity (" )1 8.24 170 - 1050 0 0.04 - 2.76

50 - 1073. 15 I 0.05 - 3.65

77.36 - 373.90 0.99 - 296.07 0.00 - 2.79 Thermal conductivity (,,-)1 8.20.25,26 170 - 1050 0 0.07 - 1.66

50 -1073 I 0.01 - 2.24

300.52 - 311.06 34.64 - 257.19 0.00 - 0.88 Self-diffusion coefficient.( D ) 20 50 - 523.15 1.41 - 5.04

Second virial coefficient ( B2 ) 27 87.65- 2000 0.13 - 14.39

1392 INDIAN J CHEM, SEC A, JULY 2004

Table 4-Calculated viscosity of he lium at zero and one pressure

0.90 0.95 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.50 3.00 3.50 4.00 4.50 5.00 5.50 6.00 6.50 7.00 7.50 8.00 8.50 9.00 9.50 10.00 10.50 11 .00 11.50 12.00 12.50 13.00 13.50 14.00

11 pO

(~lPa s)

1.60757 1.69093 1.7771 5 1.9280 2.0855 2.2395 2.3827 2.5301 2.6787 2.8 104 2.9444 3.08 11 3.2 196 3.8378 4.3977 4.9235 5.4318 5.8895 6.3492 6.7668 7.1826 7.5861 7.9670 8.3469 8.7236 9.0752 9.4255 9.7751 10.1244 10.4509 10.7763 11.101 0 11.4252 11.7494 12.057 1 12.3605 12.6632

11 pi

(IlPa s)

0.94611 1.706 10 2.30678 3. 1427 3.7006 4.0810 4.3357 4.5319 4.6883 4.7889 4.8779 4.9619 5.0436 5.3367 5.61 38 5.9192 6.2591 6.5821 6.9383 7.271 2 7.6197 7.9682 8.3034 8.6456 8.99 10 9.3158 9.6435 9.9740 10.3071 10.6195 10.9328 11.2471 11.5626 11 .8792 12. 1803 12.4780 12.7759

densities range at any temperature In a predictive mode within experimental errors; (iii) the work gives much confidence in the transport coefficients in the moderately and highly dense regions of helium than that of previous similar work. 17

; and (iv) the density dependence of the viscosity is much weaker at higher temperatures . This is due to the diminishing of the monomer-dimer contribution and the near cancellation of the two-body and three-body contributions at high temperatures7

.

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27 www.ni st.gov