by the entropy term only: For an ideal solution, the Gibbs free energy of mixing is always negative, meaning that mixing of ideal solutions is always spontaneous. The lowest value is when the mole fraction is 0.5 for a mixture of two components, or 1/n for a mixture of n components. Solutions and temperature dependence of miscibility Ideal and regular solutions The above equation for the entropy of mixing of ideal gases is valid also for certain liquid (or solid) solutions—those formed by completely random mixing so that the components move independently in the total volume. Such random mixing of solutions occurs if the interaction energies between unlike molecules are similar to the average interaction energies between like molecules. The value of the entropy corresponds exactly to random mixing for ideal solutions and for regular solutions, and approximately so for many real solutions. For binary mixtures the entropy of random mixing can be considered as a function of the mole fraction of one component. For all possible mixtures, , so that and are both negative and the entropy of mixing is positive and favors mixing of the pure components. Also the curvature of as a function of is given by the second derivative This curvature is negative for all possible mixtures , so that mixing two solutions to form a solution of intermediate composition also increases the entropy of the system. Random mixing therefore always favors miscibility and opposes phase separation. For ideal solutions, the enthalpy of mixing is zero so that the components are miscible in all proportions. For regular solutions a positive enthalpy of mixing may cause incomplete miscibility (phase separation for some compositions) at temperatures below the upper critical solution temperature (UCST). This is the minimum temperature at which the term in the Gibbs energy of mixing is sufficient to produce miscibility in all proportions. Systems with a lower critical solution temperature Nonrandom mixing with a lower entropy of mixing can occur when the attractive interactions between unlike molecules are significantly stronger (or weaker) than the mean interactions between like molecules. For some systems this can lead to a lower critical solution temperature (LCST) or lower limiting temperature for phase separation. For example, triethylamine and water are miscible in all proportions below 19 °C, but above this critical temperature, solutions of certain compositions separate into two phases at equilibrium with each other. This means that is negative