# Are the units of mole of oxygen molecules the same with the units of mole of nitrogen molecules?

Suppose we have in a box $$3 \,{\rm mol}$$ $$\ce{O}$$ and $$3 \,{\rm mol}$$ $$\ce{N}$$. Is the addition of these two quantities meaningfull? We do it all the time when we want to find the molar fraction. I can't understand if it dimensionally correct to add these two things. If a mole is like a dozen then a mole of eggs and a mole buildings can't add up.

I think a crucial step that is omitted when we want to find the molar fraction is that we should convert these moles of arbitrary entities (eggs, buildings, molecules) to same entities. I can't find anything else to justify such a process (addition of different kind of moles). Would be the following way a proper way to find the mole fraction? $$n_{_{T}}= \frac{\,{\rm mol} \, \text{of molecules}}{\,{\rm mol} \, \text{of O molecules}}\cdot n_{_{O}} + \frac{\,{\rm mol} \, \text{of molecules}}{\,{\rm mol} \, \text{of N molecules}}\cdot n_{_{N}}$$

So $$n_{_{T}}= 6\,{\rm mol}$$. What about the molar fraction of oxygen? $$\chi_{_{O}}=\frac{n_{_{O}}}{n_{_T}}= \frac{\text{3 mol of O molecules}}{\text{6 mol of molecules}}=0.5 \text{ of what?}$$

We know the ratio must be dimensionless but why? We deviding moles of $$\ce{O}$$ molecules by moles of molecules. Are these units different or not?

• Are integers expressing number of apples the same as integers expressing number of pears ? Are metres expressing length of roadd the same as metres expressing height of trees ? – Poutnik Nov 26 '20 at 13:37
• It's a deep question, and perhaps one could say that the "of molecules" and "of O" are merely labels, and not algebraic objects like the actual unit "mol". But in any case, it works. In science, sometimes we must settle for something inexplicably working ("shut up an calculate"). – electronpusher Nov 26 '20 at 13:43
• Adding on to what @Poutnik said, when you’re finding mole fraction of oxygen, you’re finding the fraction of apples in a box of apples and pears. For this, you divide the number of apples by the number of apples and pears. – Aniruddha Deb Nov 26 '20 at 14:10
• If the box contains 3 mol of O2 and 3 mol of N2, there is 6 mol of diatomic molecules. The oxygen molar fraction is then x_O2 = n_O2 / ( n_O2 + n_N2 ). BTW mol is the symbol of mole, as kg is the symbol of kilogram. The former is often confused. 0.5 of nothing. It is unitless. It is the same like ( 3 kg of apples ) / ( 3 kg of apples + 3 kg pear pears ) gives 0.5 as the mass fraction of apples. – Poutnik Nov 26 '20 at 14:56
• You can add 1 mole eggs and 1 mole buildings and this gives you 2 moles objects. Back to Chemistry, think about perfect gases. As far P V T are concerned, it is perfectly sensible. But even with no ideality at all, you can always count total number of entities and basically define the relative composition. It is just that I don't see anything useful on adding eggs to buildings. But if you find a property of the world that depends on the molar ratio of eggs in city centre, why not? – Alchimista Nov 27 '20 at 9:37

## 1 Answer

It is a good question which has already been somewhat addressed in the 1880s. This "field" was called quantity calculus. Calculus here is not the integration / differentiation, but rather the Latin calculus implying a method of calculation.

There is a very nice article "Quantity Calculus: Unambiguous Designation of Units in Graphs and Tables" by Mary Anne White in the Journal of Chemical Education. Please read this if you are seriously interested. Search on Google Scholar and it is free to download from there.

In quantity calculus Each physical quantity as the product of a numerical value and a unit:

physical quantity = numerical value × unit

This approach was introduced by British scientists and many leading physicists used it. Now there is there is nothing less or nothing more. Therefore your ambiguity arises from introducing another factor such as "oxygen" or "nitrogen". The unit mol does not know whether it belongs to oxygen or nitrogen.

As explained in the comments, suppose we write L symbolizing the height of a tree, then I can only write, L = 10 m. For mathematical purposes, I will not introduce "tree" anywhere in this equation. The tree is already incorporated in L (in your mind) but not in the mathematical equation. One can also write L/m =10. Now you have a pure number on both sides.

For fun: Think about taking the derivatives or log of quantities with units. What is the unit then?

• Another fun is with $\ln{\frac{L_1}{L_2}}=\ln{L_1} - \ln{L_2}$ – Poutnik Nov 26 '20 at 19:13
• Yes, indeed. That proves that L's must be dimensionless numbers as L/m. – M. Farooq Nov 26 '20 at 19:58
• @M.Farooq What troubles me is that we can use mole to measure anything. Trees, eggs, buildings etc. But we use mass to measure a specific property. It doesn't matter if it is kilograms from eggs or buildings is still mass. That is what confuses me. – Anton Nov 29 '20 at 12:40
• No you shouldn't use moles to measure or count anything. It is good for atomic and sub-atomic scaled objects. We cannot have a mole of trees or eggs on this Earth (big big number)!! Now mol as a unit is fine just like kg. Can we not buy 1 kg of carrots? 1 kg of iron, 1 kg of paper? – M. Farooq Nov 29 '20 at 16:50