# Non-integer amounts of atoms

## Main Question or Discussion Point

Hello.

I have a non-stoichiometric reaction that balances out to:

C12H23.4 + 12.265(O2) ---> 11.7(H2O) + 6.415(CO2) + 5.585(C)

The problem started with the compound C12H23.4 which is the published formula for a high-grade kerosene of the form CnH1.95n. So, in the beginning, there exists a non-integer value of hydrogen atoms (which seems crazy), and this propagates through the entire equation to create even MORE "nonsensical" fractional particles.

What is going on here? How is one to interpret this result?

It seems like the only "solution" is to multiply all by 1000 and divide by 5 to get:
200(C12H23.4) + 2453(O2) ---> 2340(H2O) + 1283(CO2) + 1117(C)

But that still leaves the question of the 23.4(H)!? How can you have a .4(H) atom clinging to a carbon atom? It can't really exist to do ANY combining.

It's an average composition. You're dealing with a mixture of several different substances, and some have more hydrogen than others.

That makes sense. If so, when dealing with an actual molecule and a real reaction, do you simply round to the nearest integer, always round up...or always round down, once you know what (n) is, and therefore, 1.95(n)?

The interpretation of

H2 + 1/2O2 => H2O

isn't "one molecule of hydrogen reacts with half molecule of oxygen to form one molecule of water"

The correct way to read is : " one mole of hydrogen molecules react with half a mole of oxygen molecules to form one mole of water molecules". The same reasoning goes to your equation. So, there are no fractional atoms involved.

It's a common mistake, though.

Borek
Mentor
The interpretation of

H2 + 1/2O2 => H2O

isn't "one molecule of hydrogen reacts with half molecule of oxygen to form one molecule of water"

The correct way to read is : " one mole of hydrogen molecules react with half a mole of oxygen molecules to form one mole of water molecules".
Yes and no. When there are fractional ceofficients used situation gets out of control as there are no fractional atoms/molecules (no doubt about it), but what you wrote suggests that reaction equations are written only in terms of moles - but I don't think there is anything wrong in interpreting them in terms of atoms/molecules, when all coefficients are integer.

(BTW: use sub/sup to format formulas).

The same reasoning goes to your equation. So, there are no fractional atoms involved.
No fractional atoms, but this is a different problem. Reaction you wrote can be easily converted to integer coefficients, reaction posted by treddie doesn't have and will never have integer coefficients, as it is only an approximation of the observed case.

Borek
Mentor
That makes sense. If so, when dealing with an actual molecule and a real reaction, do you simply round to the nearest integer, always round up...or always round down, once you know what (n) is, and therefore, 1.95(n)?
You don't do neither.

Imagine you have mixture of ethane C2H6 and ethene C2H4. You burn them in oxygen:

2C2H6 + 7O2 -> 4CO2 + 6H2O
C2H4 + 3O2 -> 2CO2 + 2H2O

Now, if you know your mixture of them is 1/3 of ethane (by mass) and 2/3 of ethene, and you know you have a 1 kg of mixture, you can either split the mass into two substances and calculate results of each reaction separately, then add them, or you can use a trick - add these reactions (in correct proportions!) getting something "reaction equation like" - and use this "reaction equation" for further calculations.

In this particular case masses are 1:2, so moles are 1:2.14. You multiply the first reaction by 1, second by 2.14 and add them. To make things more complicated, first reaction has to be divided by 2 first, so that we start with coefficient of 1 for ethane:

C2H6 + 3.5O2 -> 2CO2 + 3H2O

and second reaction becomes

2.14C2H4 + 6.42O2 -> 4.28CO2 + 4.28H2O

Sum is:

2.14C2H4 + C2H6 + 9.92O2 -> 6.28CO2 + 7.28H2O

Now, we can go even further and combine both ethane and ethene:

2.14C2H4 + C2H6 = C4.28H8.56 + C2H6 = C6.28H14.56

and finally we have "reaction equation"

C6.28H14.56 + 9.92O2 -> 6.28CO2 + 7.28H2O

that describes observed stoichiometry of reaction of 1:2 mass by mass mixture of ethane and ethene burnt completely in oxygen. This is just a mathematical trick that simplifies further calculations, and all coefficients and formulas should be not treated the same way they are treated in normal circumstances (there is no compound with formula C6.28H14.56 - this formula nicely describes composition of the mix used) - but they can be used in normal stoichiometric calculations.

"Reaction equation" you have listed is even more complicated, as it was most likely not calculated from the basic stoichiometry as I did above, but it was determined experimentally - amount of soot is not something that can be easily predicted.

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@Borek: I agree with you when you say there's no problem interpreting a equation in term of molecules instead of moles when all coefficients are integers. However, all my chemistry teachers have told me that the correct way to read it is using moles.

My intention with my post was to clarify that a fractional integer doesn't mean fractional molecules. I find using moles a extremely useful way to get around this problem, but each one should follow its tastes for rigor.

Then, in the final analysis, what would you do in the real world with
C6.28H14.56 + 9.92O2 -> 6.28CO2 + 7.28H2O?

It seems that at some point, this has to be "reduced" to reflect reality by making some changes to the equation. I understand the facility of working with a "perfect" mathematical formula, but it ultimately fails to explain reality unless some changes are made to it. Is the answer to split it all back up somehow, to get back to the original two equations, or would that be impossible by ending up with an infinite amount of possible solutions? Much like 3 + 4 always = 7, but 7 can be broken up into an infinite amount of two values that sum to 7.

Then again, in most real-world situations, I would suppose the amount of particles actually involved in a reaction would probably never be known to such accuracy that it even matters...it would all reduce to a statistical average. Unless you were dealing with some study of quantum amounts, then it would definitely matter.

Then, in the final analysis, what would you do in the real world with
C6.28H14.56 + 9.92O2 -> 6.28CO2 + 7.28H2O?
That C6.28H14.56 is non-standard and I suspect represents an averge which I suppose stoichmetrically would agree with empirical results. However, I've never seen a molecular formula written with non-integer numbers. And the non-integer coefficients in a balanced equation represents moles like $6.28 CO_2$ represents 6.28 X 44 grams of $CO_2$. It's an empirical expression. The 6.28 does not represent molecules.

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alxm
Then, in the final analysis, what would you do in the real world with
C6.28H14.56 + 9.92O2 -> 6.28CO2 + 7.28H2O?
You could probably get an okay approximation of the heat of combustion from that, if you interpolated the hydrocarbon value between, say, C6H12 and C7H16.

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Borek
Mentor
That C6.28H14.56 is non-standard and I suspect represents an averge
Ah, the joy of answering without reading the thread. Two posts earlier I have shown how and when you can get this equation.

--
methods

Borek
Mentor
Then, in the final analysis, what would you do in the real world with
C6.28H14.56 + 9.92O2 -> 6.28CO2 + 7.28H2O?
It can be easily used to calculate mass ratio of hydrocarbon mix to oxygen, so you can use it to predict amount of oxygen necessary to burn given mass of of the mixture. It will also let you calculate mass of products. That's what the stoichiometry is about, isn't it?

It seems that at some point, this has to be "reduced" to reflect reality by making some changes to the equation.
Actually it is quite the opposite - this is the "reduced" equation that reflects the reality. Ideal case was two separate equations, but they were difficult to use.

Then again, in most real-world situations, I would suppose the amount of particles actually involved in a reaction would probably never be known to such accuracy that it even matters...
No. Many mixtures you will deal with have relatively well known composition. Equation you have started this thread with is just an example of such situation:

compound C12H23.4 which is the published formula for a high-grade kerosene of the form CnH1.95n.
Kerosene is produced to meet some parameters, very likely composition is one of them (perhaps not directly, but for sure heat of combustion is one of the parameters, that sets limits to possible composition variation). So you can be sure if you are working with kerosene that meets the standard, it has composition that follows the formula, and it burns according to the reaction given.

So then, going back over this thread, would you say that the fraction in C12H23.4 (and similar compounds) results because not all of the Kerosene molecules in the mixture would have the same exact amount of H atoms attached. But that the average comes out to 23.4?

Borek
Mentor
Exactly, kerosene is a mixture.

That was the first answer to your question that you got in this thread - and it was spot on.

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Well, it took me awhile, but it finally sunk in! Thanks everybody for all of your generous help. Much appreciated!