In this paper we investigate the standard one-dimensional model for adsorption of a contaminant in a packed column. Initially we focus on two mass sink models: a full nonlinear Langmuir equation, including adsorption and desorption and the linearised version which neglects the local concentration. Travelling wave solutions are developed for both cases, providing exact analytical formulae for the concentration and adsorbed fraction throughout the column. Comparison is made with experimental data for the adsorption of toluene on activated carbon (dominated by physical adsorption) and Cr(III) on zeolite NaX (dominated by ion exchange). For toluene the nonlinear model provides good agreement but it is less accurate for Cr(III). The linear model apparently works well for Cr(III) but only at the expense of violating the model assumptions. This motivates extending the travelling wave solution to a Sips sink model, which has a power law dependence on local concentration. Comparison with data provides excellent agreement in all cases and also indicates why the previous models fail. Following from the investigation of the linear sink model, we prove that previously accepted results frequently fail due to a violation of assumptions made in their derivation. Specifically we challenge the belief that widely used models such as Bohart–Adams, Yoon–Nelson, Bed Depth Service Time, etc can accurately predict breakthrough data. Their success instead may be attributed to: at times, having a similar shape to the breakthrough curve rather than being the correct solution; treating measurable quantities as fitting parameters and a willingness to accept that constant parameters vary. In contrast to these previous models, the travelling wave solution of the present paper, based on a Sips sink, provides analytical solutions consistent with the model assumptions for a wide range of experimental data.