# Regular solution

In chemistry, a regular solution is a solution that diverges from the behavior of an ideal solution only moderately.[1] Its entropy of mixing is equal to that of an ideal solution with the same composition, due to random mixing without strong specific interactions.[2][3] For two components

$\Delta S_m = -nR(x_1\ln x_1 + x_2\ln x_2)\,$

where $R\,$ is the gas constant, $n\,$ the total number of moles and $x_i\,$ the mole fraction of each component. The enthalpy of mixing is non-zero, unlike for an ideal solution.

A regular solution can also be described by Raoult's law modified with a Margules function with only one parameter $\alpha$:

$\ P_1= x_1P^*_1f_{1,M}\,$
$\ P_2= x_2P^*_2f_{2,M}\,$

where the Margules function is

$\ f_{1,M}= {\rm exp}(\alpha x_2^2)\,$
$\ f_{2,M}= {\rm exp}(\alpha x_1^2)\,$

Notice that the Margules function for each component contains the mole fraction of the other component. It can also be shown using the Gibbs-Duhem relation that if the first Margules expression holds, then the other one must have the same shape.

The value of $\alpha$ can be interpreted as W/RT, where W = 2U12 - U11 - U22 represents the difference in interaction energy between like and unlike neighbors.

In contrast to the case of ideal solutions, regular solutions do possess a non-zero enthalpy of mixing, due to the W term. If the unlike interactions are more unfavorable than the like ones, we get competition between an entropy of mixing term that produces a minimum in the Gibbs free energy at x1= 0.5 and the enthalpy term that has a maximum there. At high temperatures the entropy wins and the system is fully miscible, but at lower temperatures the G curve will have two minima and a maximum in between. This results in phase separation. In general there will be a temperature where the three extremes coalesce and the system becomes fully miscible. This point is known as the upper critical solution temperature or the upper consolute temperature.

In contrast to ideal solutions, the volumes in the case of regular solutions are no longer strictly additive but must be calculated from partial molar volumes that are a function of x1.

The term was introduced in 1927 by the American physical chemist Joel Henry Hildebrand.[4]