JOURNAL OF RESEARCH of the Notional Bu reau of Standa rds - A. Ph ysics and Chemistry Vol. 72A, No. 2, March- April 1968 Second Virial Coefficient of He 4 in the Temperature Range from 2 to 20 oK Marjorie E, Boyd, Sigurd y, larsen, and Harmon Plumb Institute for Basic Standards, National Bureau of Standards, Washington, D. C. 20234 ( December 7, 1967) We p reliminary values for Ihe seco nd viria l coeffic ient of He' in the te mp eratu re range from 2 to 20 K. 1 h ey wele de rI ved from rece nt so und ve lOC It y meas ur e m e nt s in Ih e gas made by Plumb a nd Cata la nd uSIn g a n ultra sonIc Int erfe rometer. Key Word s: He', low te mp era tur e, seco nd viri a l. In thi s c ommuni ca ti on we pr ese nt valu es for the seco nd vi rial coe ffi cient of He 4 in the tempe ratur e ran ge from 2 to 20 oK. Th ey were derived from rece nt so und velocit y measurements in the gas mad e by Plumb a nd Ca taland using an ultr aso ni c interferom- eter [1] .1 Our valu es for the virials must be consid- ered as prelimin ary s in ce the me thod for determining them from so und velocity data pl aces such stringent de mand s on th e numerical accuracy of the l atter that it.will be n ecessa ry to ac c umul ate a much larger body of dat a than is now ava il ab le to ac hi eve more exact virials. H owever, th e pr oblem of discre pan ci es existing betw ee n various provisional te mp e ratur e scales below 20 oK is c urr e ntly of strong int er es t [2] and contrib- utes a note of ur gency to the probl em of obtai nin g new meas ured values of th e second virial coe ffi cient in thi s t e mp e ratur e ran ge. C urr e ntl y, th e o nl y va lu es ava il a bl e are obtained e ith er by extrapolating upw ard from th e PVT data of Keller [3] in the 2 to 4 oK region, by relying on a very few point s obtained by Keeso m and Walstra [4] mor e than twe nt y years ago, or by us in g th eo re ti c al calculations ba sed on ass umed inter- mol ecu l ar potential function s. We thus be li eve that o ur values, although th ey mu st be r ega rd ed as te nt a- tiv e, are probably the b es t now available and may therefore prove useful as guid es to ward resulvin <f th e inte rn a ti onal temperature · scale difficulties. meas ur ements of sou nd velocity isotherms wh ic h should enable more accurate and re li ab le valu es to be determined are now in progr ess in this laboratory. Isothe rm s of the spee d of so und as a function of pr ess ur e were obtained [1 1 at pressures low e noua h that the isoth er ms could be re pr ese nt ed by a rel at ionship, W' 1=C( l +a p) , where W is th e so und 1 Figures in . br ac kets indicate the lit eratu re referen ces at the end of this I}aper. 155 veloc it y. C RT/M , a nd Ca is the s lor e of th e iso- th e rm o Th e coefficient a is related to the seco nd virial coefficient B by [51 the e quation a(T) = (RT) - I [2B +± TB ' +± TlB"] 3 15 where th e pnmes denote differe ntiation with respect to T, the te mp e ratur e. To o btain the seco nd virial from the a 's it is thu s necessa ry to ass ume a fu nctional form for B(T) with coefficients to be det erm ined, differe ntiat e it twice, and det erm ine the cons tant s by leas t squares fittin g of the data for a(T). This approxi- mation of B by an arb itr ar il y chose n ex press ion intro- du ces a stro ng un ce rtainty in th e r es ults in ad dition to the un certainti es involved in dete rminin g the a's from the isoth er ms. Many exper im e ntali sts hav e u se d th e form B = a + bi T , and recent experience of two of th e authors (MEB and SYL) [6J in fittin g results of an accurate phase-shift ca lculation of B fo r He 4 , using a Lennard-Jon es potential with deBoer param eters, indi- ca t es th at this is a r easo nabl e form in the t empera tur'e range of this d ata. H owever, c urvatur e as a fun ction of liT does b eco me in cr eas in gly a pp are nt for the L-J vi rials at about 20 OK, as well as at the low T end. Using, then, B = a. + bi T we find th e res ults shown in t ab le 1. An id ea of the accuracy ma y be o bt a in e d by both observing the c hanges in th e res ults when fittin g to different te mp e rature ranges se parat ely and by noting th e effec t of addin g additional terms in th e functional form for B. Of th ese two t ypes of changes the first is th e l east se rious in its e ff ec ts on th e re sults. Co mparin g the virials obtained when fittin g th e mi ddl e te mp e ratur e range to those obtain ed fittin g the high and low ranges separately we find them to differ by