  Phase behavior of liquid crystals is of long-standing interest due to
numerous applications, with one of the key issues being how the presence of
impurities affects the liquid crystalline order. Here we study the
orientational order of 4-cyano-4$^{'}$-pentylbiphenyl (5CB) and
4-cyano-4$^{'}$-hexylbiphenyl (6CB) nematic liquid crystals in the presence of
varying concentrations of water and n-hexane molecules serving as impurities,
by carrying out both fully atomistic simulations and experiments. Our results
reveal that mixing of the impurities (in case of hexane) with the host liquid
crystals causes a nematic-to-isotropic phase transition with hexane
concentration as the control parameter while demixing (in case of water)
results in only weak impurity-induced perturbations to the nematic
liquid-crystalline order. We develop a coarse-grained model illustrating the
general nature and entropic origin of the mixing-induced phase transition.
