  In this paper we show that the virial expansion up to third order for the
osmotic pressure of a dilute polymer solution, including first-order
perturbative corrections to the virial coefficients, allows for a scaling
description of phase-separation data for polymer solutions in reduced
variables. This scaling description provides a method to estimate the
Theta-temperature, where demixing occurs in the limit of vanishing polymer
volume fraction $\phi$ and infinite chain-length $N$, without explicit
assumptions concerning the chain-length dependence of the critical parameters
$T_c$ and $\phi_c$. The scaling incorporates three limiting regimes: the Ising
limit asymptotically close to the critical point of phase separation, the pure
solvent limit and, the tricritical limit for the polymer-rich phase
asymptotically close to the Theta point. We incorporate the effects of critical
and tricritical fluctuations on the coexistence curve scaling by using
renormalization-group methods. We present a detailed comparison with
experimental and simulation data for coexistence curves and compare our
estimates for the Theta-temperatures of several systems with those obtained
from different extrapolation schemes.
