= Binary activity-coefficient consistency
{title2=$x\,d\ln\gamma_A+(1-x)\,d\ln\gamma_B=0$}
For a nonideal <chemical solution>, write its <thermodynamic activities> on a <mole fraction> basis as $a_A=x\gamma_A$ and $a_B=(1-x)\gamma_B$. Use composition-independent pure-component reference <chemical potentials> at the same <temperature> and <pressure>:
$$
\mu_A=\mu_A^*+RT\ln(x\gamma_A),
\qquad
\mu_B=\mu_B^*+RT\ln[(1-x)\gamma_B].
$$
The <activity coefficients> describe departure from the <ideal solution> under that reference convention.<Thermodynamics and Chemistry by Howard DeVoe>[Thermodynamics and Chemistry, §9.5.1, printed p. 259, reference states; §9.5.3, printed p. 261, equation 9.5.14]{ref}
At fixed <temperature> and <pressure>, insert these expressions into the <Gibbs-Duhem equation>. The <ideal solution> terms cancel because $x\,d\ln x+(1-x)\,d\ln(1-x)=dx-dx=0$. What remains is
$$
x\,d\ln\gamma_A+(1-x)\,d\ln\gamma_B=0.
$$
Thus two independent fits to measured <activity coefficients> can each look smooth while failing to describe one possible <Gibbs free energy>. This is a necessary consistency condition; satisfying it alone does not establish experimental accuracy or <thermodynamic stability>.
For an original algebraic example, suppose at one fixed <temperature> and <pressure> that $\ln\gamma_A=\alpha(1-x)^2$, with dimensionless constant $\alpha$. The constraint requires $d\ln\gamma_B/dx=2\alpha x$. With the pure-B normalization $\gamma_B(0)=1$, <integration> gives
$$
\ln\gamma_B=\alpha x^2.
$$
For $\alpha=1/2$ and $x=1/4$, the <derivatives> of the two <logarithms> are $-3/4$ and $1/4$; their weighted sum is $-3/16+3/16=0$. Choosing $\ln\gamma_B=\alpha x$ instead would give a weighted sum $3/16$ at that composition, contradicting the <Gibbs-Duhem equation>. These are invented model functions, with no claim to fit a particular <mixture>.
The fixed-<temperature>, fixed-<pressure> restriction matters. Along a path with changing <temperature> or <pressure>, the general equation contains $S\,dT-V\,dp$, and the reference <chemical potentials> and $RT$ also change. Across <phase coexistence>, apply the relation separately within each smooth <thermodynamic phase>; do not differentiate a discontinuous switch between phases as if it were a single composition curve.
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