This article gives a mathematical treatment of the batch rectification of a binary mixture in the range of low concentrations of the lighter component in the distillate, both for the case in which the amount of holdup in the column may be neglected (Part I) and for the case in which this holdup has to be taken into consideration (Part II). It is assumed that at phase equilibrium the concentration of this component in the vapour is directly proportional to the concentration in the liquid. The relations derived in Part I show the importance of choosing a high reflux ratio in the process of removing the lighter component. A graph is given which facilitates the finding of suitable values of the reflux ratio and of a suitable number of “transfer units” or theoretical plates for the column. A comparison is made between removal of the lighter component by batch and by continuous rectification. It is found that, economically, batch rectification is inferior to continuous rectification if a high degree of purity of the heavier component is to be reached. The amount of holdup is taken into consideration by assuming that the column is continuously in a state characterized by constant concentration ratios at a constant ratio between the amount of holdup and the amount of material contained in the still. This state is termed the stationary state of the second order. The influence of the holdup is illustrated in a number of graphs, which bring to light several particularities. Finally, a discussion is given of some effects connected with the holdup, which are termed the accumulation, accommodation, pinch and dead space effects (the first two correspond respectively to the depletion and flywheel effects of Pigford. An example is given of a numerical computation.