Pram~aa~ Vol. 1, No. 2, 1973. Printed in India.~) 1973 The law of rectilinear diameter for the liquid-gas phase transition A C BISWAS Tata Institute of Fundamental Research, Bombay 400005 MS received27 April 1973 Abstract. A rigorous semi-microscopicderivation of the law of rectilinear diameter for the liquid-gas phase transition has been provided. Keywords. Liquid,phase transition. 1. Introduction The law of rectilinear diameter for the liquid-vapour equilibrium phase was discovered by Cailletet and Mathias-in 1886. Since then it has been used by many experimentalists to measure the critical density. Apart from its intrinsic significance the law is of tremendous practical importance since the determination of the critical density by direct measurement is extremely difficult due to the fact that small changes in pressure produce large changes in density along the critical isothermal. A theoretical derivation of the law of rectilinear diameter, however, is missing except for a very specific lattice model (Lee and Yang 1952). Widom and Rowlinson (1970) suspected the validity of the law for continuum models. Recently, Cornfeld and Cart (1972) have provided strong experimental evidence supporting the validity of the law for continuum fluids. 2. Analysis In what follows we propose to derive the law of rectilinear diameter from a recently developed theory (Biswas 1973) of the liquid-gas phase transition on the basis of Mayer's cluster expansion (Munster 1969). In fact it has been shown (Ruelle 1969) that for any stable potential ,~u(I ri--rj I) where the summation is carried out for i>j with the particles having hard repulsive cores the following theorem holds for a manyparticle system. Theorem: If there exists a temperature T = T c at and below which all Mayer cluster 1 bts (except perhaps a finite number of them) are positive then a first order phase transition does occur and the isotherms for T ~ Tc are given as ~ '-X kT 1=1 00 tZ ; - = O XV' for ~<1 defining the gas state. (1) /=1 109 however. T ) ~ . T)/V as V~oo. It. the radius of convergence of the series 2~ bl2l for T~< To. D i s c u s s i o n In this note.N B ( T ) ) N (7) 0o(7") . Since. 1 ~/g/(l)l .2po log ~" -I- ~r O0 (1/. IT. l=1 CO gl P -. the relation (5) has been derived from a rigorous microscopic theory. V. T) = 0 for a given volume V and temperature T (see Biswas 1973). T) is defined as the least positive root of Z(At.= 2po-v 1~1 (3) /=1 l for ~ > 1 defining the liquid-state where gl=blz ° and -z-=z/zo being the fugacity and CO z0. p and v have the usual l=l meaning of pressure and specific volume. T) ='g2~u exp (V Z b / ) / x = 0 (6) l=l vanishes and then evaluate the limit N(V.( I . The existence of such a value of N is ensured by the hard-core nature of the interaction at short distances. T) Oo=p°(T) = V-~oo (4) V where At(V.~ lIB(T) (8) so that where ~o u(_r)0 B(T) =2zr J" dr r~[1--e kT] 0 (9) .2-.110 A C Biswas 00 o0 o0 /=1 /=l for ~ = 1 defining the saturated vapour region. In this approximation it is easy to see (Mandle 1971) that Z(N. From equation (2) it follows that ½[PL+oa] = po( T) (5) In principle one can evaluate po(T) by finding the least value of At=At for which dN Z(At. We shall be interested in determining P0 in the neighbourhood of T-~ Tc (the critical temperature) to the lowest order in To-. we are more interested in finding the qualitative nature of the relation (5) from the microscopic point of view and as such we shall calculate p0(T) from a van der Waal's type approximation of the partition function.. 3. Equivalently one can say that At is the least number of particles contained in the volume V at temperature T at which the free energy becomes infinite. by a van der Waal's type calculation (which is exact upto the second virial coefficient) of p0(T) we do not sacrifice any essential qualitative feature of the physical system. Lt At(V. again. Paris 104 1563 Cornfeld A B and Carr C Y (1972) Phys. we find that a=4"55 fits reasonably well the experimental data obtained by Cornfeld and Carr (1972) for xenon as well as the experimental data for hydrogen obtained by Mathias et al (1921) (see Hirschfelder et al 1954). . 87 410 Mandle F (1971) Statistical physics (London: John Wiley & Sons) pp 203-205 Munster A (1969) Statistical thermodynamics Vol. Phys. For kT~." po(Tc) =Pc] we have k T¢=aa/b (12) We now expand po(T) in the neighbourhood of T=-T c and retain only the lowest power of T-. uo where u0 is the minimum value of the potential u(~) (I0) B( T) "~ b--a]k T b and a are as defined by Mandle (1971). Note that in both these experimental results temperature is plotted in 'centigrade '. Rigorous results (New York: W A Benjamin Inc. 29 28 Hirschfelder J O. R. Now. 5 549 Cailletet and Mathias E (1887) C. Acad. Rev.2-3 Lee T D and Yang C N (1952) Phys. 52 1670 *Note that our critical parameters To. Lett. Curtiss C F and Bird R B (1954) Molecular theory of gases and liquids (New York: John Wiley & Sons) see p 362 fig 5. the assumption (I 1) on the dependence ofv c on b has been intuitively guided by van der Waal's theory. have got no connection with the van der Waal's theory of condensation. Paris 102 1202 Cailletet and ]¢f_athi~ E (1886b) o7 Phys. 1 (New York: Academic Press) Ruelle D (1969) Statistical mechanics. rigorously speaking T c should be determined from oo We shall rather start with the assumption* that the critical specific volume vc = l!pc = b(a--1)la (11) Then where a is a number greater than unity and will be treated as a parameter.The law of reailinear diameterfor the liquid-gasphase transition I 11 is the second virial coefficient. Rev.l----b-a k T c _bk a Tc--T [ a--ll Tc--T]~ ~ =Pc 1 + -+ (I -ba_kTc)9 ] (13) so that (5) becomes [ ½L%+pc]:pc 1 + 1 rc-r] Tc ] (14) giving the law of rectilinear diameter. of course. Phys. The relation (14) has the unknown parameter a. Sd. Sd.7 Chem. highly over~timatcs vc.) Widom B and RowIinson J S (1970) . R. Acad. 7 131 Oailletet and Mathias E (1886a) 6'. vc etc. the latter. References Biswas A O (1973) o7 Statist. One should convert the temperature to ' absolute' scales before comparing with the formula (14). from equation (8) ['. However.T c getting _bkrc a P°(T)-.
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