# Molecular formula and decimal number

Today I did some stoichiometry exercise about finding the ratio between the elements. What confused me was that while the ratio was right, but it was decimal.

Ex.

Reducing $\pu{0.800 g}$ of lead oxide with an excess amount of hydrogen, $\pu{0.725 g}$ of elementary Pb formed. What formula of the oxide?

In the chapters I've learnt so far, no oxidation number or stuff was mentioned. So I did as the examples states. I did this:

$$m(\ce{PbO})=\pu{0.800 g},$$ $$m(\ce{Pb})= \pu{0.725 g},$$ $$m(\ce O)= 0.800-0.725=0.072(\pu g).$$

The ratio is therefore $x/y$ where $x =0.725/207.2$ and $y = 0.072/15.9994$

which is equal to $11.59/14.91 =0.7$, so the formula should be $\ce{Pb_{0.7}O}$ and if I multiply it by 2 I get $\ce{Pb_{3\text/2}O2}$; Multiply it by 2 again I get $\ce{Pb3O4}$, the answer.

Why should I have to do this? Is there anyway to go from decimal to fraction with the correct answer?

Now, I didn't find the answer until later in another problem, where the same thing happens. I was stuck on the decimal ratio. It was:

$\pu{10.0 g}$ of elementary phosphorus reacts completely with $\pu{7.75 g}$ of sulfur. What is the molecular formula?

I did the same thing: number of moles of $\ce{P}$ is 0.32, number of moles of $\ce{S}$ is 0.24. I calculated the ratio and it was 1.33:1, so the formula should be $\ce{P_{1.33}S1}$, but if I multiply by 3, I get the answer neatly. Why is that? Am I doing this wrong?

• What is supposed to be the question here? Why proportions of elements aren't written as fractions? – Mithoron Feb 2 '18 at 0:51
• well 0.800-0.725 = 0.075 not 0.72. The ratio also shouldn't be rounded to 0.7. You got lucky that you got the right answer. – MaxW Feb 2 '18 at 18:53

You must find integers as for your are requested to find a molecular formula, or, in case, a minimal or a formal formula for ionic compounds.

Obviously each atom must enter in that formula as an integer, because a molecule is composed of atoms and not by parts of them.

If you were requested to find the ratio between elements in a sample, then does not really matter if the answer is given as 1.5:2 or as 3:4.

Cases exists in which the ratio is not that of two integers, these compounds are called non stoichiometric and results from vacancies in ionic lattices.

Doping is another case in which that happens.

But you will never be asked to find the molecular formula of something that has none.

For an unknown sample, you would find what case applies. A ratio of integers would indicate a possible molecular nature of one compound.

By convention, chemists have decided to use whole numbers for molecular compounds and most simple solid-state chemistry. (Exceptions are doped compounds, where the exact composition is not fully fixed and the fractions involved are tiny.)

So, no, I do not think that the work shown in the question is wrong. By convention, you multiply by the lowest number that will give you integer numbers until you get an indication (usually by a higher molecular mass, think mass spectrometry) that the molecule is even larger than that. Consider the example of ethane ($\ce{C2H6}$), wherein a ratio calculation like the one carried out in the question would yield $\ce{CH3}$ -- clearly a wrong formula for that compound.

• To TAR86 and most importantly to @Federico. " by convention" is an unhappy choice or words. It is at the basis of modern atomism. There is 1 atom of H. Half of it doesn't exist. – Alchimista Feb 2 '18 at 10:02
• @Alchimista I see your point for molecules, for FeO/Fe2O3 systems, not so much. – TAR86 Feb 2 '18 at 10:10
• Doping is different story as well as alloys and ionic crystals. Obviously I refer to molecular formula. This is what I want to highlight especially to the OP as dramatically important for a beginnert. I had no doubt you know what you mean, but in general the incipit of your answer is potentially misleading. Better to point out the existence of non stoichiometric compounds, for which no molecular formula can be attained. Else the question make little sense, as does not relly matter if not just integeres can be obtained. – Alchimista Feb 2 '18 at 10:27
• For the iron system you mentioned, indeed the only way to use whole numbers is in FeO/Fe2O3 as you did. – Alchimista Feb 2 '18 at 10:44