Karki, Shankar S

Sinha, Suresh K

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InJianJ. Phys. 75A (1). 91-92 (2001)U P  Aan internutionat journalPhase equilibria of polar Lennard-Jones fluidsS h a n k a r  S K a rk i a n d  SurelBh K  S in h a *Department of Physics, L S. College,B. B A Bihar University, Mu/affarpup' 842 001, Bihar, IndiaAbstract • Liquid -  vapour equilibria of polar Lcnnard"Jones fluids is studied taking into account of dipole and quudrupole inoincnts We employ .in effective Lennard-Jones (ELJ) (12-6) potential to calculate cnticul point location for a range of values of both /i /(ecr^) and = Q' /(e.<7^)Our results are in good agreement with the Monte-Carlo simulation dataKeywords Polar Lennard-Jones fluids, critical point location, thermodynamic behaviourPACS Nos. . 05 60 -k. 05 70 Jk,47 10+g1. IntroductionT he p u rp o s e  o f  th e  p re s e n t  w o rk  is  to  s tu d y  th e  p h a se  e q u il ib r ia  of p o la r  i lu id  c o n s is t in g  o f  L e n n a r d - J o n e s  ( L J ) ( 1 2 - 6 )  s p h e re s  w ith e m b e d d e d  p o in t  d ip o le s  a n d  lin e a r  q u a d ru p o lc s . R e c e n tly , K arki et a\ [\\ h a v e  d e r iv e d  e f f e c t iv e  L J (1 2 -6 )  p o te n t ia l  fo r  e s tim a tin g  th e  th e r m o d y n a m ic  p ro p e r t ie s  o f  m o le c u la r  f lu id s .In th e  p re s e n t  w o rk , w e  im p ro v e  th e  e f f e c tiv e  L J (1 2 -6 )  p o ten tia l a n d  a p p ly  it to  s tu d y  th e  p h a se  e q u il ib r ia  o f  th e  p o la r  L cn n ard  -  J o n e s  f lu id s .2. Potential modelWc c o n s id e r  a  f lu id  w h o s e  m o le c u le s  in te ra c t  via p a ir  p o te n tia l o f the fo rmw(rfi), eo2) = Wo(/*) + M,,(rcu, ru2), (1)w here  r  =  I T j -  Tj  I is  th e  c e n tr e  -  to  -  c e n tr e  d is ta n c e  a n d  ft), re p re se n ts  th e  o r ie n ta t io n a l  c o o rd in a te s  (0, , 0 , )  o f  m o le c u le s  i. H ere, u^(r) is  th e  L e n n a rd  -  J o n e s  (L J )  (1 2 -6 )  p o te n tia lUo(r) =  4  € [ ( c r  I r)'~ - ( a  I r ) ^ ] , (2)(3)’ Corresponding AuthorFor correspondence . Ramani Mohan Garden, Kalambag Road Chowk. Muzaffarpur-842 002, Bihar, Indiaw h ere , an d  a rc  c o n tr ib u tio n s  d u e  to  d ip o le  -  d ipo le ,d ip o le  - q u a d r u p o lc  an d  q u a d ru p o lc  -  q u a d ru p o lc  in te ra c tio n s , re s p e c tiv e ly .3. Effective pair potentialK a rk i et al [\\ h a v e  e x p re s s e d  th e  o r ie n ta t io n  - in d e p e n d e n t 'p re a v e ra g e d ' p a ir  p o te n tia l in  th e  c o m p a c t fo rmV ^ ( r )  =  4 € | ( > 4 j 2 ( C T / / ) ’  ^ - [ / \ ^ ( c r / r ) ^ ’ 4  / V j ^ ( c T / r ) ^  4 -y 4 ,o ((T /r)’* ^ ] ) - l - ( / l , , ( ( T / r ) ’ ‘ -f y 4 ,3 (C T /r)’ ' - f / 4 ,5 ( ( T / r ) ' ‘^ ) | .(4)In  t e r m s  o f  th e  r e d u c e d  q u a n t i t i e s  7'* =  A T /€ , ju * "  = ) u ’ /(G <T^) a n d  l {ea^)  H ie e x p l i c i te x p re s s io n s  o f  th e  c o e f f ic ie n ts  a re  g iv e n  as :w h ere  e a n d  cr a re , re s p e c tiv e ly , th e  w e ll-d e p th  a n d  m o le c u la r  d iam e te r , a n d  is  th e  a n g le  -  d e p e n d e n t  p a r t , w h ic h  is ta k e n4^, =  l +  ( l / l 2 ^ * ) ( ; / » ’ ) ^ (5a)(5b)A , o = ( 7 / 2 0 7 • * ) ( G * ^ ) ^ (-5c)A,2 = l + ( 7 / l 8 ( K ) r * ^ ( A / * ^ ) \ (5d)4 , ,  = ( l / 5 r * ^ ) ( j U * ^ ) ^  ^ 2 * ^ (6a)©2001 lACS92 Shankar S Karki and Suresh K Sinha- 4 „ = ( 6 / 3 5 r * " ) / / * "  ( ^ ? * ^ ) ^/1,5 = (1 8 /2 4 5  r * " )  ( 0 * " )(6b)(6c)Karki et al[\ \ have neglected the terms appearing in the second bracket of the right hand side of eq. (4). In the presentEq. (15a) may be solved by the iterative process. Knowing 7^*, we obtain the reduced critical density * =  1 /  * forthe polar fluid.The values o f the critical temperature 7^* and critical density P (.*arc  com pared with the M onte C arlo (M C) data (31 instudy we consider these terms to obtain the effective pair Table I for various values o f and Q *".'fhe re.sulting 7^* isa bit higher than MC values except at p  = 2.0 and =2.0. where the situation is reversed. However, the resulting p f* is  higher than the MC values throughout the range of p  and Q*  ^considered here. On the whole, the agreement is good.potentialWe approxim ate a l r ^ ^ a l  r, - 1/6 [2] and using thec o e f f i c i e n t s w e  define the terms4 = (l/12)(p*-)^ +(l/4(2)''')(p*’ e*^ )+ (7 /20(2 )^ '■ ')((^* ‘ ) ^  (7)B = (2"^ /  5) (p  f  Q*^ + ( 6 /  35(2)'"^)p  (0 * ^  )^(8)+ ( 1 8 /2 4 5 (2 ) " - )C = ( 7 /8 0 0 ) ( p * - ) VThen eq. (4) can be expressed in the LJ (12-6) form V r(r) =  4 6 ^  [ (< T y ^ /r ) '^ - (C T , / r ) ^ ’] .wherea s a r l ( y =  F ~ '’\€ £ £ ;  / e  = ( l + ( » / 7 * - )  + ( C / 7 * ’ ) ) 7 -(9)( 10)( 1 1 )(12)andF  = (l + / 4 /7 * ) / ( l  + ( B /7 * ^ )  + ( C / 7 * ’ )), (13)Then the polar fluid can be treated as the LJ ( 12-6) fluid.4. Critical constantsIn order to test the theory we apply it to estimate the critical temperature 7^,, critical volume V^ .. For the LJ (12-6) fluid, they are given byT(*skT(;l€  =  1.26 Vc / Na  ' =3.1(14a)(14b)In order to estimate the critical constants for polar fluids, we replace €  -+  and <t ->  Cj  in cq. (14). Thus, we getTc*= 1.26 e , V^*= 3.lCT-\(15a)(15b)Table I. Values of the crilical icinperaiua- T,* and cntical density p* toi vanous dipole moment p and quadrupole moment Q*-.Tc* Pc*Q*-Presem MC Preseiu MC1,00 l.OO 1.82 1.79 ± 0 02 0.34 0.34 ± 0.01I 00 2.00 2 4.S 2 42 ± 0 02 0 ,16 0 3-t ±0012 00 1 00 2 06 2.00 ± 0 02 0 14 0 12 ± 0 012 00 2 00 2 62 2 65 ± 0 03 0 35 0 31 ± 0 01S. Concluding remarksThe effective pair potential for the polar LJ (12-6) fluids is expressed in the U (  12-6) form by simply replacing e  e ,  (7*)and a  - >  Of{T*).  The values o f €y  and (T ,, which decrease with increase o f T*, depend on p * ’ and Q*^. The effective LJ (12-6) potential is employed to estimate the critical temperature T* and critical density p(- * over a wide range o f p  ♦* and Q* '. The agreement with the M C |3 | data is good. This theory can be employed to calculate other thermodynamic properties, 'litis is not attempted here.AcknowledgmentsWe acknowledge the financial support and award of Emeritus Fellowship to SKS of the University Grants Commission, New Delhi.References[ 11 BP Karki. B K .Singh, T K l)ey and S K Sinha, Indian J Phys 615 (1999)|2] Om Singh and A W Joshi Pramana IS 487 (1980)13] G S Dubey and S F O'shca Phya. Rev. 49 2175 (1994)

Phase Equilibria of Polar Lennard-Jones Fluids

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Phase Equilibria of Polar Lennard-Jones Fluids

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