Chemical formulas with non-integer coefficients [duplicate]








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Let's consider the composition $\ce{Hg\_{0.7}Cd\_{0.3}Te}$ which appears in my data.


As I remember from my chemical knowledge it's not possible having non-integers numbers as coefficients of elements. Moreover, this is clearly not the percentage representation, since we have $1$ atom of tellurium. So is this some sort of reduced chemical formula that should be multiplied by a smallest suitable integer to get an appropriate result? If not, what does this exactly represent?




**The rule that coefficients have to be integers applies to compounds containing discrete molecules, but some classes of compounds don't**


There is an important rule in chemistry that the formulae of *molecules* must contain integer ratios of elements. But not every substance in the (very broad) field of chemistry is made of discrete molecules.


There are at least two major exceptions where the units don't contain discrete molecules: many minerals and metal alloys.


Metals don't consist of molecules. They are more like a sea of ions in a soup of free electrons. And many metals can freely mix in any ratio (though in many cases it is even more complicated and some alloys consist of a fine-grained mix of multiple alloys of different composition). But for the simpler cases virtually any ratio of several metals is possible (think of this as being like a solution of both salt and sugar dissolved in water: a wide variety of ratios of sugar and salt are possible).


This appears to be the case that prompted the question. Many metal alloys are, in effect, solid solutions of one or more metals in another metal and any ratio of the components is possible.


Another class where non-integer ratios are common is minerals. Many consist of a variety of continuous chains of silicate (Silicon-oxygen) structures making up sheets, chains or other complex structures. These have a net negative charge which is balanced in the mineral by a variety of different metal ions. But, because any metal with the right charge and size can do that job, they often consist of non-integer ratios of metals.


So, for some classes of substance where the building blocks are not discrete molecules, non-integer ratios of components are perfectly possible.


 18

Formulas with non-stoichiometric rational number coefficients do not represent molecular structures, but just empirical molar ratio of involved elements. For solid matter, unless it is composed from molecules, even formulas with small integer coefficients do not represent molecular structure, but are expressions of element molar ratios.


E.g. $\ce{CaF2}$ does not mean the respective mineral is composed from $\ce{CaF2}$ molecules, but it means the ratio of $\ce{Ca^2+}$ and $\ce{F-}$ ions is in the ratio 1 : 2.


Some natural or man-made compounds or materials do not have element ratios in small integer numbers, so their formulas use rational numbers. Formally, any multiple of given ratio can be used, but practically at least one coefficient is normalized to integer number. For cases 2 elements are interchangeable in composition, their coefficients often represent their mutual molar fraction, or its integer multiple. In other cases, there is taken an ideal formula as a template and non-integer coefficients express deviation from it.


For your case of $\ce{Hg\_{0.7}Cd\_{0.3}Te}$, the mixed cadmium/mercury telluride has both metals mutually interchangable with molar fraction coefficients. Note that they may not be quite nice and round 0.7 or 0.3, but generally any rational number between 0 and 1.


Take it as a particular formula for the general empirical formula $\ce{Hg\_{x}Cd\_{1-x}Te}$, $x \le 1$, $x \ge 0$.


For $x=0$, it is $\ce{CdTe}$, for $x=1$, it is $\ce{HgTe}$.


It is quite common form for formula notation of mixed chalkogenides where the coefficients are not stoichiometric.


Typical cases are metal oxides with more than 1 oxidation states, where coefficients are empirical. $\ce{FeO}$ is rather $\ce{Fe\_{0.9x}O}$ as there is, aside of $\ce{Fe^{II}}$, also $\ce{Fe^{III}}$. The general formula could be $$\ce{Fe^{II}\_{1-x}Fe^{III}\_{x}O\_{1+0.5x}}$$ or $$\ce{Fe^{II}\_{\frac{1-x}{1+0.5x}}Fe^{III}\_{\frac{x}{1+0.5x}}O}$$.


It could be formally written as $\ce{a FeO . b Fe2O3}$, but such a notation does not bring any advantage, as it does not represent any real molecular structure.


$\ce{Fe2O3}$ does not represent any real molecular structure either. It just happens that the molar ratio of elements is in this (idealized) case expressed in small integer numbers.


 18

Maurice's answer is correct, but let me try to explain my own way.


The fractional coefficients aren't just picked out of a hat: they add up to 1. That's not an accident! The point is that this is a solid solution with Te2- ions and X2+ ions, where X is Hg2+ *or* Cd2+, potentially occupying the same position in the crystal lattice. You could have a 69% and 31%, or 71% and 29%, and it would work just as well. Converting the percentages to integers would be misleading, because it would give the impression that they are, or should be, *exact* proportions, which have to be maintained to avoid some fundamental change in structure. Also, using the fractional coefficients typically makes it quite clear who has to match up with who to fill up a specific position in the lattice together.


 2

The formula $\ce{Hg\_{0.7}Cd\_{0.3}Te}$ means that $\ce{Hg}$ and $\ce{Cd}$ have the same dimension, nearly same radius, and same valence. They can be interchanged, in any proportion between $0$ to $100$%. Here there are $70$% mercury and $30$% cadmium. But it could also have been possible to get any other coefficient $x$ for mercury ($x<1$}, provided the coefficient of cadmium is $(1 - x)$.


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