Entropy change in non-ideal solutions
The detailed calculation for $\Delta G$ is here,  Derive expression for internal energy of mixing and entropy of mixing using statistical thermodynamics  which gives $$\Delta G = RT[n_1\ln(x_1)+n_2\ln(x_2)] +(n_1+n_2)x_1x_2w$$

and as $\Delta S= -d\Delta G/dT$ then 

$$\Delta S = -R[n_1\ln(x_1)+n_2\ln(x_2)] - (n_1+n_2)x_1x_2\frac{dw}{dT}$$

where $w \approx (2E_{12}- E_{11} - E_{22})$ and $E_{12}$ is the interaction energy between molecules of type $1$ and $2$ when they are adjacent and similarly for $E_{11}$ and $E_{22}$ all of which are negative. From a consideration of the energies involved $w$ is expected to be positive.

The first term is the  Raoult's law expression for an ideal solution and the second term the deviation, which is sometimes also called the excess  entropy of mixing. For benzene cyclohexane when $n_1=n_2=x_1=x_2=1/2$  the excess entropy  $\displaystyle -\frac{1}{4}\frac{dw}{dT}\approx 1.7$ J/mol/K at $298$ K and $1.2$ at $343$ K.