The nematic to neat-soap (N-NS) and nematic to isotropic (N-I) phase transitions in binary solutions of cesium per-fluoro-octanoate (CsPFO) and water have been studied by high-precision calorimetry at 39.3, 43.6, 48.3, 49.0, 50.7, 52.0, and 55.3 wt. % CsPFO concentrations. The ratio of the heat-capacity amplitudes, above and below the N-NS transition, and the critical exponent-alpha remain near 1 and 0, respectively, suggesting three-dimensional XY-type behavior. The coexistence range and the latent heat at the first-order N-I transition decrease with decreasing concentration. Extrapolation of the N-I coexistence width suggests existence of a Landau point at approximately 28 wt. % concentration. The coefficients of expansion in the Landau-de Gennes free energy at the N-I transition and their concentration dependence have been determined. Coefficients of the cubic and the biquadratic terms become vanishingly small at low concentrations underscoring the need to use higher-order terms in the free energy.