How do we balance redox reactions by ion electron method?

To balance redox reactions you have to look at each of the
elements in the reaction and see which ones have changed their oxidation number. To do
this, assign an oxidation number to each element and compare the oxidation number of the
element on the reactant and product sides of the
equation.

If the element has gained one or more electrons
it is said to have been reduced.  If the element has lost one or more electrons it is
said to have been oxidized.  The rule is that the "oxidizing agent is reduced and gains
electrons" while the "reducing agent is oxidized and gains
electrons.

Once you have identified the elements whose
oxidation numbers have changed you can begin to balance the oveall
reaction.

The other thing you will need to determine is if
the reaction is taking place under acidic or basic conditions. This will allow you to
properly completely balance any redox reaction.  If the reaction is acidic, use H+ on
the left and HOH on the right. If the reaction is basic, use OH (-1) on the left and HOH
on the right to balance atoms.

The first step is to take
each element separately and balance the "half-reaction" for that element.  Depending on
the bare bones, unbalanced half-reaction you may be able to just balance with electrons
or you may have to first balance atoms before you add the
electrons.

Lets take an example so you can see what is
happening:

FeSO4 + KMnO4 --> Fe2(SO4)3 +
HOH

The important part of this reaction involves the ions
actually involved in the redox reaction.  The other ions are considered "spectator
ions".  The net ionic reaction is:

Fe (2+)  +  MnO4 (-1) 
+  H (+1) -->  Fe (+3) + Mn (+2) + HOH

The iron has
been oxidized and the permanganate reduced:

Fe (+2)
--> Fe (+3)

MnO4 (-1) --> Mn
(+2)

Now balance the iron reaction by adding one electron
to the product side.

Fe (+2) --> Fe (+3) + 1
electron

The second half reaction requires balancing the
atoms first, then the electrons.  Because it is under acidic conditions, H+ can be added
to the left side as the source of hydrogen ions.

MnO4 (-1)
+ H+ --> Mn(+2) + HOH

B alancing the atoms gives
us:

MnO4(-1) + 8 H+ --> Mn(+2) + 4
HOH

Now balance the electrons in  this half
reaction:

MnO4 (-1) + 8 H+ + 5 electrons --> Mn(+2)
+ 4 HOH

This is important:
Check both half reactions so each has the same number of electrons. If
not, use coefficients on one or both half reactions so the total number of electrons is
equal in each of the half reactions.

In this case 5
electrons were added to the reactant side for the permanganate so you need 5 electrons
on the product side for the iron.  Multiply through by 5, and you
get:

5 Fe (+2) --> 5 Fe(+3) + 5
electrons

You are almost
done!

The final step is to add the two half
reactions giving you:

MnO4(-1) + 8 H+  + 5 Fe(+2) + 5
electrons --> Mn(+2) + 4 HOH + 5 Fe(+3) + 5
electrons.

The electrons cancel out and you have your
balanced equation.

In general,