We define a reduced pressure p ∗ = βpR 3 1 and reduced chemical potentials µ ∗ i = βµ$_{i}$ . Eqs. (31) and (32) are three equations for four unknowns (two state points each characterised by two densities). Therefore, regions of two-phase coexistence depend parametrically on one free parameter. Eqs. (33) and (34) are solved numerically with a Newton-Raphson procedure [60]. The resulting set of solutions yields the binodal. I - N - N triple points are located where the I - N and N - N coexistence curves cross. Therefore, regions of two-phase coexistence depend parametrically on one free parameter (which can be chosen arbitrarily, e.g. as the value of concentration x in one of the phases). Analytic expressions for the free energy, pressure and chemical potentials for the isotropic phase may be found on insertion of Ψ( θ ) = 1 / (4 π ) into the ideal (5) and excess (22) parts of free energy functional. For this purpose, we use where the integrals I 1 and I 2 are calculated in Appendix A. The expressions for the FMT isotropic free energy, βF$_{iso}$/V , pressure, p ∗ iso and chemical potentials µ ∗ i, iso are and µ ∗ 1 , iso = ln ( c (1 − x ) 4 π ) + cπ 2 2 [ 2(1 − x ) + x ( λ 2 + λ ) ] + c $^{2}$π$^{2}$[(1 − x ) 2 + 2 x (1 − x ) λ 2 + x $^{2}$λ$^{4}$] , (39) p ∗ iso = c + c $^{2}$π2 2 [ (1 − x ) 2 + x $^{2}$λ3 + ( λ 2 + λ )( x − x $^{2}$) ] + 2 π $^{2}$c3 3 [ x $^{3}$λ6 + 3( x 2 − x $^{3}$) λ 4 + 3( x − 2 x 2 + x $^{3}$) λ 2 + (1 − x ) 3 ] , (38) βF$_{iso}$ V R 3 1 = c (1 − x ) ln ( c (1 − x ) 4 π ) + cx ln ( cx 4 π ) − c + π 2 2 c 2 [ (1 − x ) 2 + x $^{2}$λ3 + x (1 − x )( λ 2 + λ ) ] + π 2 3 c 3 [ (1 − x ) 3 + x $^{3}$λ6 + 3 x (1 − x ) $^{2}$λ2 + 3 x $^{2}$(1 − x ) λ $^{4}$] , (37) ∫ π 2 0 dθ sin θ ∫ π 2 0 dθ $^{′}$sin θ $^{′}$K ( θ, θ $^{′}$) = I 1 8 π = π 2 2 (35) ∫ π 2 0 dθ sin θ ∫ π 2 0 dθ $^{′}$sin θ ′ ∫ π 2 0 dθ $^{′′}$sin θ $^{′′}$L ( θ, θ $^{′}$, θ $^{′′}$) = I 2 16 π = π 3 2 (36) G. Equation of State for the Isotropic Phase 13