= Entropy of mixing
{title2=$\Delta_{\rm mix}S=-nR\sum_i x_i\ln x_i$}
{wiki}
The entropy of mixing is the <entropy> change between initially separate components and their final <mixture>, with initial and final <temperatures>, <pressures> and component <amount of substance> values specified. For an <ideal solution> made from pure <liquids> at the same <temperature> and <pressure>, let $n=\sum_i n_i$. Then
$$
\Delta_{\rm mix}S=-R\sum_i n_i\ln x_i
=-nR\sum_i x_i\ln x_i.
$$
The right-hand expression is the <ideal solution> result; the general definition does not require ideality. Each interior <mole fraction> has a negative <logarithm>, so mixing distinct components gives positive <entropy of mixing>. At a pure endpoint, use the <limit> $x\ln x\to0$ as $x\to0^+$.
For one <mole> of the chosen A/B <ideal solution> with $x_A=0.4$,
$$
-0.4\ln0.4-0.6\ln0.6\approx0.673011667,
\qquad
\Delta_{\rm mix}S\approx5.59573\,\mathrm{J\,K^{-1}}.
$$
For the same total <amount of substance>, equal <mole fractions> maximize this binary <entropy of mixing>: the composition function $s(x)=-x\ln x-(1-x)\ln(1-x)$ has
$$
s'(x)=\ln\frac{1-x}{x},\qquad
s''(x)=-\frac{1}{x}-\frac{1}{1-x}<0,
$$
so its unique maximum is $s(1/2)=\ln2$. This is a comparison between different prepared compositions, not a prediction that a closed sample changes its component inventory to become equimolar.
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