The overall equation for a lead-acid battery should be in your book. Ultimately the lead-acid battery features lead in oxidation states 0 and +4, then a redox reaction happening in between them to achieve a compound of lead with an oxidation state of +2.
is the one in oxidation state +4. To get to +2, it requires 2 electrons.
is the one in oxidation state 0. To get to +2, it needs to lose 2 electrons.
You should see that there is clearly a 1:1 molar relationship between lead and lead (IV) oxide because they both need to gain/lose 2 electrons each.
} = \mbox{n(Pb)} = \frac{50}{207.2} = 0.241\mbox{ mol})
} = 0.241 \times (207.2 + 16.0\times 2) = 57.7\mbox{ g})
Now, to find the mass of lead sulphate:
It should be clear that the reaction will look something like:
This is unbalanced, and clearly missing some reactants and products, but it will do for our purposes. What I have done is basically an elemental balance: I know that my reaction only has two sources of lead - the lead (IV) oxide and the pure lead. On the products, the only place where lead is produced is in the lead sulphate.
In non-nuclear chemistry, the elements at the beginning balance with the elements at the end, so therefore there must be 2 times as much lead sulphate than there was lead (IV) oxide or pure lead (because they both combined in a
ratio to form
moles of lead sulphate).
} = 2\cdot \mbox{n(Pb)} = 2 \times 0.241 = 0.483\mbox{ mol})
A clearer way, and more explicit way of thinking about the elemental balance is this:
} = \mbox{n(Pb)} + \mbox{n(PbO}_2\mbox{)})
This will give you the same result as above, since
} = \mbox{n(PbO}_2\mbox{)})
} = 0.483 \times (207.2 + 32.1 + 16.0\times 4) = 146\mbox{ g})
Learning how to utilise the elemental balance to save yourself from writing a full equation is also very helpful sometimes. I used it in my actual exam (VCAA Chemistry 2006) for Unit 3.
By the way, we could have looked up this equation from our textbook, but it's more fun to deduce things with minimal information. It's also useful if you're actually supposed to know this equation (I'm not sure), you can just remember the bare minimum, and build the rest up as you go along, like I did here.
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