Iron metal reacts with oxygen problem

[skepticwulf]

Homework Statement

When the supply of oxygen is limited, iron metal reacts with
oxygen to produce a mixture of FeO and Fe2O3. In a certain
experiment, 20.00 g iron metal was reacted with 11.20 g oxygen
gas. After the experiment, the iron was totally consumed,
and 3.24 g oxygen gas remained. Calculate the amounts of
FeO and Fe2O3 formed in this experiment.

Homework Equations

The Attempt at a Solution

3 Fe + 2 O2 ------> FeO + Fe2O3
20g 0,12mol 0,12mol
0,36mol

As each 3 mol Fe produces 1 mol FeO and Fe2O3 , just by multplying moles to their molar masses it gives:
8,64g FeO and 19,2g Fe2O3. In my solution no need for O2 in calculations.
But, solution manual disagrees :(
Firts of all it states each reaction separately (WHY?)
Then it calculates O2's mole by subtracting 3.24 from 11.2 and by dividing it to its molar mass and then:
"
Let’s assume x moles of Fe reacts to form x moles of FeO. Then 0.3581 – x, the remaining
moles of Fe, reacts to form Fe2O3. Balancing the two equations in terms of x:... " etc etc
But why?? What's wrong with my calculation? I assumed when it says "3.24 g oxygen gas remained" as O2 is surplus, excess.
Did I assume wrong?

 

[skepticwulf]

Besides, if 3 mole Fe react with 2 mol O2 then 0,36 mol react with 0,24 mol and that's 7,68g, so (11,2-7,68), that 3,24g is indeed excess amount. Right?

 

[Borek]

3 Fe + 2 O2 ------> FeO + Fe2O3

Why do you assume FeO and Fe₂O₃ are produced in equimolar quantities? They don't have to and they are not guaranteed to. Just because there are several reaction occurring doesn't mean they can be described by a single reaction equation. Quite the opposite - each process is separate.

Please note your reaction - being actually a sum of two reactions - is not unique. It can be balanced in infinitely many ways, for example

13Fe + 9O₂ → 3FeO + 5Fe₂O₃

or

35Fe + 23O₂ -> 13FeO + 11Fe₂O₃

and so on (check these are indeed correctly balanced). With many "correct" reaction equations, which one are you going to select to solve the question? You won't have this problem if you treat each reaction separately.