The main target of this study is to develop a theoretical method for determining small contents of dodecanol in samples of sodium dodecyl sulfate (SDS) by a detailed analysis of surface-tension isotherms. As a tool for our analysis, we employ the van der Waals model. Its application to data for alkanols and anionic surfactants gives an excluded area per adsorbed molecule equal to the geometrical area of the molecular cross section and adsorption energies consonant with Traube's rule. Because the dodecanol and SDS have different excluded areas, we extended the van der Waals model for the case of a two-component adsorption layer, with account for the counterion binding in the Stem layer. General-expressions for the surface free energy, two-dimensional equation of state, surface chemical potentials, adsorption isotherms, and surface dilatational elasticity are derived. The experimental surface-tension isotherms are fitted by varying only one adjustable parameter. The model was successfully tested against data for solutions of SDS with a known content of dodecanol. Knowing the parameters of the model, we computed various properties of the surfactant adsorption layer. The results show that the presence of a small amount of dodecanol leads to a considerable increase of the total adsorption and surface elasticity. Even a relatively small (0.2 mol %) fraction of dodecanol in SDS may lead to a predominant content (up to 86 mol %) of dodecanol in the mixed adsorption layer. We applied the model for determining unknown contents of dodecanol in SDS samples at different stages of purification. The addition of NaCl may lead to a significant reduction in the mole fraction of dodecanol in the adsorption layer. The developed theoretical model and computational procedure are also appropriate for a quantitative analysis and computer modeling of the adsorption from other mixed ionic-nonionic surfactant solutions, at both air-water and oil-water interfaces.