Question

In this problem we examine the second virial coefficient of square-well potentials. Calculate the second virial coefficient B2(T) of a square-well (SW) fluid with hard-core diameter a, well depth -E<0, and well width Ao with A > 1). Compare/check the limiting cases e 0 and A -+1 against the hard-sphere result.

Answer 1

Solutien: 1 Virial Eguatien Stete con write the paessure as an iinite sesis we We ass ume hod an density PPRT KT B,CT) n2 fomction Maclaurin expansien abeut osi the virial cofsicient , wich is enly where, 6n (T) is emperature his just ceffi cientsgien b Bm+1 CT ude The vinial coefficuts be igoreusly cal culatd from tatis can eal me chanics. The Pho tessis muh teo leng for this clas buct can phove that Bn depends ly molecules. The Second visial coftient 8 the esult is that we Bm the intera ctions between two molecules, etc depends enly en inteaetiens between given y The Second visial cofpicient ,(T) 2 V 2V 5Sd de sine f 2 whehe, i efj So, B, depends nly molecules The vial Cofficients potential and potential and Hard sphere: and is called the May-funeiem. en the potential between an ipo lated pair of -1 be expressed the temperatue Bi the tmperature we junctien the Bi (uir),T). we know the can as a can calaulate Bi =21-exp(pal)] * dr with for

for eutr) for be split into two pauts exp (-pu))dr -exp(put))vid The can 1- 2n we e Phuging in the potential, 2-3 +0 3 haid sphene second vinial cofficient is independos hard phue fluid Note hat the femperatue. This is to be erpe cted be cause the iAatheumal ie, its pAoperties do not depend en tempevatune Square well potential :- B 2 -e xp(-putr))d using for tor -E for e-putr) epE for r> + C-efe)d to В, Т % 2 T 2T 2n[+ C-efe) C- 3 fanchon tential is a po B fom the sapare well ND that Hemperature