Scaling Laws for the Critical Properties

By applying the conditions for the critical point of pure fluids, we obtain a set of equations for the critical temperature and density  

Making a Taylor expansion in powers of the density, the first and second virial coefficients predicted by TPTl are found to be  

in order to solve Eq. (106) for the critical temperature we will need to linearize the virial coefficients with respect to the temperature. To do so, let us assume that there is a finite asymptotic critical temperature in the limit of infinite chain length, which we call 0, in analogy to the polymer + solvent case. We now make a series expansion of B2 and Bs in powers of A r = 6 — T up to first order, and consider the limit of this expression for large N, leading to 

This equation shows that Arcrit(N) must reach an asymptotic finite value, since the right-hand-side term should ultimately vanish for large N. The requirement for Tciit(N) to attain a finite asymptotic critical temperature equal to 0 is then obeyed provided that C2 vanishes. If we now notice that 

The case of the critical density is much simpler. Substitution of the expression for B3 in the condition for the critical density, Eq. (107), shows that  

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