Balance the following equation stepwise:
`Cr_(2)O_(7)^(2-) + Fe^(2+)++H^(o+)rarrCr^(3+) + Fe^(3+) + H_(2)O`

Explanation

Separation of the equation in two half reactions
(i) Write the O.N of all the atoms involved in the skeleton equation
`(overset(+6)Cr_(2) overset(-2)O_(7))^(2-)+(overset(+2)Fe)^(2+)+overset(+1)(H)^(+)rarroverset(+3)(Cr)^(3+)+overset(+3)(Fe)^(3+)+overset(+1)H_(2)overset(-2)O`
(ii) Identify the atoms which undergo change in O.N.
`(overset(+6)Cr_(2)O_(7)^(2-))+(overset(+2)Fe)^(2+)+(H))^(+)rarroverset(+3)(Cr)^(3+)+(overset(+3)Fe)^(3+)+H_(2)O`
(iii) Find out the species involved in the oxidation and reduction half reactions.

Thus, the two half reactions are:
Oxidation half reaction : `Fe^(2)+rarrFe^(3+)`
Reduction half reaction :`(Cr_(2)O_(7))^(2)rarrCr^(3+))`
Step II Balancing of oxidation half reaction :
The oxidation half reaction is :`Fe^(2+)rarrFe^(3+)`
(i) As the increase in O.N as a result of oxidation is 1, add one `e^(-)` on the the product side to balance change in O.N.
`Fe^(2+)rarrFe^(3+)+e^(-)`
(ii) The charge is already balanced and thus the equation is also balacned
`Fe^(2+) rarrFe^(3+)+e^(-)`
Step III Balancing of reduction half reaction
The reduction half reaction is : `(overset(+6)Cr_(2)O_(7))^(2-)+6e^(-)rarr(overset(+3)Cr)^(3+)`
(i) The decrease in O.N per Cr atom is 3 and the total decreasae in O.N for two Cr atoms is 6 .Therfore, add `6e^(-)` on the reactant side
`(Cr_(2)O_(7))^(2-)+6e^(-)rarrCr^(3+)`
(ii) Balance Cr atoms on both sides of the equation
`(Cr_(2)O_(7))^(2-)+6e^(-)rarrCr^(3+)`
(iii) In order to balance O atoms add seven `H_(2)O` molecules on the product side and then to balance H atoms add 14 `H^(+)` on the reactant side.
`(Cr_(2)O_(7))^(2-)+6e^(-)+14H^(+)rarr2Cr^(3+)+7H_(2)O`
Step IV.Adding the two half reactions:
In order to equate the electrons, multiply the equation (i) by 6 and then add to equation (ii) order to get the final equation .
`[Fe^(2+)rarrFe^(3+)+e^(-)]xx6`
`({:(Fe^(2+)rarrFe^(3+)+e^(-)"]"xx6),(Cr_(2)O_(7)^(2-)+14H6(+)+6e^(-)rarr2Cr^(3)+7H_(2)O):})/(6Fe^(2+)+Cr_(2)O_(7)^(2-)+14H^(+)rarr6Fe^(3+)+2Cr^(3+)+2Cr^(3+)+7H_(2)O)`