# Fractional index in a chemical formula of metal oxide

I have just read an article in which they used $$\ce{WO3}$$ and $$\ce{WO_{2.9}}$$ as precursors, whose objective was to form $$\ce{WS2}$$, in an atmosphere of argon by CVD. In one boat there is the powder of $$\ce{S}$$ and in another boat there is $$\ce{WO3}$$ or $$\ce{WO_{2.9}}$$, in an oven with a double temperature zone. In the article they conclude that $$\ce{WO_{2.9}}$$ forms larger layers of $$\ce{WS2}$$ in the substrate giving the following justification:

They found that $$\ce{W^6+}$$ cannot be directly sulfated by $$\ce{S},$$ unless some intermediates are formed due to the high energy of the $$\ce{W-O}$$ bond. The reduction from $$\ce{W^6+}$$ to $$\ce{W^5+}$$ is mandatory for the incorporation of sulfur in the $$\ce{WO3}$$ network. For $$\ce{WO_{2.9}}$$ in our case, its partial $$\ce{W^6+}$$ ions have been reduced to $$\ce{W^5+}$$ or $$\ce{W^4+}$$ ions. Therefore, we find that replacing $$\ce{W^5+}$$ or $$\ce{W^4+}$$ with $$\ce{W^6+}$$ in the initial stage facilitates the growth of the single crystal $$\ce{WS2}$$ film.

I didn't realize what the difference is between $$\ce{WO3}$$ and $$\ce{WO_{2.9}}$$ and where these ions come from that they talk about in this part of the article. I also don't understand what a fractional index means in a chemical formula. Could you help me to interpret that part of the article and explain these doubts that I have please?

There is nothing special about fractional indices in chemistry, it is just a pointer signifying we are dealing with a non-stoichiometric compound. Group 6 metal are known not only for their regular oxides existing in numerous polymorphic modifications, but also for forming homologous series of nonstoichiometric oxides $$\ce{\ce{M^{VI}_nO_{3n-1}}}$$ and $$\ce{\ce{M^{VI}_nO_{3n-2}}}$$ with the general formula $$\ce{M^{VI}O_{3-x}}$$ $$(\ce{M} = \ce{Mo}, \ce{W};$$ $$x \in (0; 1)).$$

Now, to make things clear, notation "$$\ce{WO_{2.9}}$$" does not refer to a molecule; $$\ce{WO_{2.9}}$$ is a formula unit showing composition of a network solid. To further illustrate the origin of the fractional coefficients here, let's refer to the crystal structures of the stoichiometric and non-stoichiometric tungsten(VI) oxides $$(\ce{WO3}$$ and $$\ce{WO_{2.9}},$$ respectively).

Depending on the temperature, $$\ce{WO3}$$ adopts several polymorphs with various crystal structures; at the room temperature the most common structure it adopts is the one of distorted $$\ce{ReO3}$$ type, being made of tilted $$\ce{[WO6]}$$ octahedra cross-linked via the corners (for the illustration, see e.g. Wikipedia).

On the other hand, $$\ce{WO_{2.9}}$$ is an oxygen-deficient compound in comparison. To account for that and eliminate the presence of point defects, crystal structure undergoes a transformation resulting in increased number of edge-sharing groups of $$[\ce{WO6}]$$ polyhedra and reduced number of the corner-sharing ones, e.g. the structure gets more "packed", and, more importantly, a crystallographic shear is introduced (which is not an impurity). Let's have a look at the asymmetric unit of $$\ce{WO_{2.9}}$$ of the crystal structure determined by Magnéli [1]:

Figure 1. Crystal structure of $$\ce{WO_{2.9}}$$ [1, ICSD-24736] showing an asymmetric unit $$\ce{W10O29}.$$ Color code: $$\color{#FF0D0D}{\Large\bullet}~\ce{O}$$; $$\color{#2194D6}{\Large\bullet}~\ce{W}$$.

Due to the asymmetric unit $$\ce{W10O29},$$ the chemical formula (formula unit) is reduced to $$\ce{WO_{2.9}}.$$ One can also fill the entire unit cell knowing the number of formula units $$(Z)$$ per unit cell; here, $$Z = 2$$ and this results in equal formula $$\ce{W20O58},$$ often seen in the literature:

Figure 2. Crystal structure of $$\ce{WO_{2.9}}$$ [1, ICSD-24736] showing packed unit cell. Color code: $$\color{#FF0D0D}{\Large\bullet}~\ce{O}$$; $$\color{#2194D6}{\Large\bullet}~\ce{W}$$.

Finally, to illustrate the location of the aforementioned shear plane, let's complete coordination shells on all tungsten atoms and view the structure along the $$b$$ axis:

Figure 3. Crystal structure of $$\ce{WO_{2.9}}$$ [1, ICSD-24736] showing polyhedral representation ($$[\ce{WO6}]$$ octahedra) with the shear plane along $$a$$ axis denoted with red arrow. Color code: $$\color{#FF0D0D}{\Large\bullet}~\ce{O}$$; $$\color{#2194D6}{\Large\bullet}~\ce{W}$$.

As for the assignment of the formal oxidation numbers for the non-stoichiometric compounds like this, this is typically considered a non-trivial task of a questionable value. Also, see the answer of mine to the related question Oxidation state of tungsten in $$\ce{W3O8}$$.

### References

1. Magnéli, A. Structure of β-Tungsten Oxide. Nature 1950, 165 (4192), 356–357. DOI: 10.1038/165356b0.
• There are also references to a $\ce{W_{18}O_{49}}$ such as this one. Would this be a combination of $\ce{W_nO_{3n-1}}$ and $\ce{W_nO_{3n-2}}$ with $n=6$? – Oscar Lanzi Feb 15 at 10:47
• @OscarLanzi Judging from the structure (ICSD 202488 by Lamire et al. 1987), $\ce{W18O49}$ is a different story: there is a tungsten with C.N. 7 (pentagonal bipyramid, like in many uranium(VI) componds) forming pentagonal assemblies with other $[\ce{WO6}]$ octahedra, which is even more efficient packing (than shear plane) that can afford to account for further O-deficiency. – andselisk Feb 15 at 11:04
• Thanks. One minor point: "Further O deficiency" seems to be a judgment call. $\ce{W_{18}O_{49}}$ is less deficient than $\ce{W3O8}$; the latter, on an equal tungsten basis, would be $\ce{W_{18}O_{48}}$. – Oscar Lanzi Feb 15 at 11:17
• @OscarLanzi True, let's just agree on oxygen deficiency. – andselisk Feb 15 at 11:20