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I have been told that applying POAC in any equation means conserving the number of moles of an element in both reactant and product . For example : $$\ce{H2 + O2 -> H2O}$$ Here when we apply POAC on oxygen we write :

Equation 1 :

No. Of Atoms(or moles) of "O" in reactant = No. Of Atoms(or moles ) of "O" in product

Equation 2: (obtained from equation 1)

2(moles of $\ce{O2}$) = 1(moles of $\ce{H2O}$)

Where equation 2 gives us the necessary data required in solving questions of stoichiometric calculations . I understand how equation 2 comes from equation 1 but I don't understand why equation 1 is correct even when the chemical equation is unbalanced ?

As I think equation 1 can be applied only when the chemical equation is balanced . As here :

No. Of Atoms of "O" in reactant =2

No of Atoms of "O" in product = 1

Thus by equation 1 we can say that $1=2$ .

And my book says POAC can be applied even when equation is not balanced . So here equation is not balanced and when I apply POAC I get $1=2$ . So what's is wrong with my reasoning?

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    $\begingroup$ Atoms cannot be created or destroyed in a chemical reaction, hence we must balance equations for all the types of atoms involved, i.e. there have to be the same number of each type on either side. In your example $\ce{2H2 + O2 = 2H2O}$ $\endgroup$ – porphyrin Jul 10 '17 at 12:58
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    $\begingroup$ If I am interpreting your question correctly, equation 1 is basically a demand that you balance your equation. So: $$ H_2 + O_2 = H_2O $$ is not balanced, therefore equation 1 doesn't really apply. I would say that "equation 1" is not even really an equation, more of a demand, or statement that must be fulfilled. $\endgroup$ – iammax Jul 10 '17 at 12:59
  • $\begingroup$ @porphyrin I know my chemical equation is not correct and must be balanced but the point I want to make is while applying POAC we don't need to balance the equation(so says my book ) !! So that is why while writing equation 1,2(applying POAC ) I have not balanced the chemical equation $\endgroup$ – Freelancer Jul 10 '17 at 13:11
  • $\begingroup$ @iammax If equation 1 is incorrect ( which I too think it is ) then equation 2 is also incorrect but that is what my book , teacher has been applying as POAC in questions were balancing the chemical equation is difficult (to use mole-mole analysis ) by trial and error $\endgroup$ – Freelancer Jul 10 '17 at 13:14
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    $\begingroup$ @Freelancer In your linked question, their reaction is unbalanced, but they essentially balance it in the calculations. Note the line: $$ Moles of O_2 = \frac{3}{2} x Moles KClO_3 $$ This step is to account for the fact that the reaction is unbalanced, but in your example in the first post where you "prove" 2 = 1, you don't include a step like this, which is why you are getting a contradiction. $\endgroup$ – iammax Jul 10 '17 at 13:27
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As said by many others , The equation 1 given above is simply put very wrong . As it can be applied only when a balanced chemical equation is given to us , otherwise we will reach at all sorts of contradictions as in the question above .

However many non- rigorous teachers , textbooks often write equation 1 in POAC (which is always true in a balanced chemical equation ) and then say now equation 2 is obtained from equation 1(which is partially incorrect) . It is true equation 2 comes naturally from equation 1 , but only for a balanced chemical equation.

So then how come we apply equation 2 in POAC in all sorts of question ? And still come up with the right answer ?

The answer is that equation 2 is true in itself always whether the chemical equation given ( in POAC ) is balanced or not . So it is not related to equation 1 for its validity .

How is equation 2 always valid ?

As pointed out by @iammax in the comments actually we (sort of ) essentially balanced the chemical equation given to us in the equation 2 itself .

For example in the $KClO_3$ question linked above , the moles of $O_2$ , moles of $KClO_3$ given to us are actually according to the stoichiometric coefficients , which enable us to use POAC without balancing the reaction (even though we have done that indirectly by using No. of moles of $O_2$ , $KClO_3$).

To conclude we can say after using No. Of moles of reactant , product we have indirectly balanced the chemical equation as the Ratio of their number of moles will be Equal to Ratio of their respective stoichiometric coefficients . After which we just use Conservation of Atoms( as the name says it) and multiply the number of moles of the molecule with the number of Atoms present on which POAC is being applied ( For example we multiply no. Of moles of $O_2$ by 2 when we apply POAC on oxygen atom ).

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