$9.$ The Gibbs$-$Duhem equation relates changes in the chemical potentials of components, ensuring that the $\_\_\_\_\_$ condition of a system is maintained.

$10.\_\_\_\_\_$ is the excess Gibbs energy per mole of mixture arising from non$-$ideal mixing.

$11.$ The Margules equation provides a relationship between activity coefficients and composition for $\_\_\_\_\_$ mixtures.

$12.$ The Van Laar equation is another model used to estimate activity coefficients, focusing on $\_\_\_\_\_$ binary systems.