Consider a fluid near its critical point, with isotherms as sketched in Figure 12.3. Assume that the

Question:

Consider a fluid near its critical point, with isotherms as sketched in Figure 12.3. Assume that the singular part of the Gibbs free energy of the fluid is of the form
image text in transcribed \[
G^{(s)}(T, P) \sim|t|^{2-\alpha} g\left(\pi /|t|^{\Delta}\right)
\]
where \(\pi=\left(P-P_{c}\right) / P_{c}, t=\left(T-T_{c}\right) / T_{c}\) while \(g(x)\) is a universal function, with branches \(g_{+}\)for \(t>0\) and \(g_{-}\)for \(tarrow 0}=\) const.
(a) Using the above expression for \(G^{(s)}\), determine the manner in which the densities, \(ho_{l}\) and \(ho_{g}\), of the two phases approach one another as \(t \rightarrow 0\) from below.
(b) Also determine how \(\left(P-P_{c}\right)\) varies with \(\left(ho-ho_{c}\right)\) as the critical point is approached along the critical isotherm \((t=0)\).
(c) Examine as well the critical behavior of the isothermal compressibility \(\kappa_{T}\), the adiabatic compressibility \(\kappa_{S}\), the specific heats \(C_{P}\) and \(C_{V}\), the coefficient of volume expansion \(\alpha_{P}\), and the latent heat of vaporization \(l\).