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An ideal solution is a <chemical solution> whose component <chemical potentials> depend on composition through
$$
\mu_i(T,P,\{x_j\})=\mu_i^*(T,P)+RT\ln x_i,
\qquad x_i=\frac{n_i}{\sum_j n_j}.
$$
Here $n_i$ is an <amount of substance>, $x_i$ its <mole fraction>, $R$ the <molar gas constant>, and $\mu_i^*$ the pure-component <liquid> reference at the same <temperature> $T$ and <pressure> $P$. This model applies to every component across the composition range; it is stronger than an approximation valid only near a pure <solvent>.<Thermodynamics and Chemistry by Howard DeVoe>[Second edition, version 10, §9.4.2, printed pp. 249–250, equations 9.4.5 and 9.4.8: pure-liquid reference and full composition range]{ref}

With this pure-<liquid> <standard state> convention, the <thermodynamic activity> satisfies $\mu_i=\mu_i^*+RT\ln a_i$, so an <ideal solution> has $a_i=x_i$, or <activity coefficient> $\gamma_i=a_i/x_i=1$.<IUPAC Compendium of Chemical Terminology, fifth edition (Gold Book) by International Union of Pure and Applied Chemistry/ideal mixture>[IUPAC Compendium of Chemical Terminology, fifth edition (Gold Book) by International Union of Pure and Applied Chemistry; Ideal mixture, definition: activities equal mole fractions, equivalently unit activity coefficients; I02938, online version 5.0.0]{ref} This statement concerns the <liquid>. Assuming that a coexisting <vapor> is an <ideal gas> <mixture> is an additional approximation. Its <mole fractions> $y_i$ generally differ from the <liquid> <mole fractions> $x_i$.

For an original two-component example, stipulate $T=300\,\mathrm K$ and pure-component <vapor pressures> $p_A^*=80\,\mathrm{kPa}$ and $p_B^*=20\,\mathrm{kPa}$. A and B are invented distinguishable components; these numbers are chosen model inputs, not material data. Combine an <ideal solution> with an <ideal gas> <vapor>, neglect the pure-<liquid> reference's <pressure> correction, and exclude other <gases>. At <liquid> <mole fractions> $x_A=0.4$ and $x_B=0.6$, <Raoult's law> and addition of <partial pressures> give
$$
p_A=0.4(80)=32\,\mathrm{kPa},\qquad
p_B=0.6(20)=12\,\mathrm{kPa},\qquad
P=p_A+p_B=44\,\mathrm{kPa},
$$
$$
y_A=\frac{p_A}{P}=\frac{32}{44}=\frac{8}{11},
\qquad y_B=\frac{3}{11}.
$$
Thus the coexisting <vapor> contains about $72.73\%$ A by <amount of substance>, although the <liquid> contains only $40\%$. The total <pressure> is neither pure-component <vapor pressure>, and $x_A$ cannot be substituted for $y_A$. This enrichment is the basis of <distillation>.

\Image[/ideal-solution-pressure-and-vapor-composition.png]
{title=Chosen ideal binary model: partial pressures and vapor enrichment}
{description=At the stipulated <temperature> $300\,\mathrm K$, the invented pure-component <vapor pressures> are $80$ and $20\,\mathrm{kPa}$. Left: <partial pressures> and their total versus the <liquid> <mole fraction> $x_A$. Right: the coexisting <vapor> <mole fraction> $y_A$; the diagonal marks equal compositions. Both panels highlight $x_A=0.4$, giving $P=44\,\mathrm{kPa}$ and $y_A=8/11$. These are model curves, with no measured data.}

The usefulness of the <ideal solution> is that one composition law connects <chemical potentials>, <vapor-liquid equilibrium>, <entropy of mixing> and <Gibbs free energy of mixing>. Its <liquid> components can interact strongly: the defining condition is their composition dependence, rather than the absence of all interactions as in an <ideal gas> model.


 Ideal solution (source code)