We study the chemical ordering in Bi$_2$Te$_{3-x}$Se$_x$ grown by molecular beam epitaxy on Si substrates. We produce films in the full composition range from x = 0 to 3, and determine their material properties using energy dispersive X-ray spectroscopy, X-ray diffraction and Raman spectroscopy. By fitting the parameters of a kinetic growth model to these results, we obtain a consistent description of growth at a microscopic level. Our main finding is that despite the incorporation of Se in the central layer being much more probable than that of Te, the formation of a fully ordered Te-Bi-Se-Bi-Te layer is prevented by kinetic of the growth process. Indeed, the Se concentration in the central layer of Bi$_2$Te$_2$Se$_1$ reaches a maximum of only $approx$ 75% even under ideal growth conditions. A second finding of our work is that the intensity ratio of the 0 0 12 and 0 0 6 X-ray reflections serves as an experimentally accessible quantitative measure of the degree of ordering in these films.
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