One method is to balance equations by the oxidation number method .
Step 1. Calculate the oxidation numbers of every atom:
#stackrelcolor(blue)(+1)("K")stackrelcolor(blue)(+7)("Mn")stackrelcolor(blue)("-2")("O")_4 + stackrelcolor(blue)(+1)("Na")_2stackrelcolor(blue)(+3)("C")_2stackrelcolor(blue)("-2")("O")_4 + stackrelcolor(blue)(+1)("H")_2stackrelcolor(blue)(+6)("S")stackrelcolor(blue)("-2")("O")_4 → stackrelcolor(blue)(+1)("K")_2stackrelcolor(blue)(+6)("S")stackrelcolor(blue)("-2")("O")_4 + stackrelcolor(blue)(+1)("Na")_2stackrelcolor(blue)(+6)("S")stackrelcolor(blue)("-2")("O")_4 + stackrelcolor(blue)(+2)("Mn")stackrelcolor(blue)(+6)("S")stackrelcolor(blue)("-2")("O")_4 + stackrelcolor(blue)(+4)("C")stackrelcolor(blue)("-2")("O")_2 + stackrelcolor(blue)(+1)("H")_2stackrelcolor(blue)("-2")("O")#
Step 2. Identify the atoms that change oxidation number.
#"Mn: +7 → +2; Change = -5"# #"C: +3 → +4; Change = +1"#
Step 3. Equalize the changes in oxidation number.
You need #"1 atom of Mn"# for every #"5 atoms of C"# or #"2 atoms of Mn"# for every #"10 atoms of C"# . This gives us total changes of #+10# and #"-10"# .
Step 4. Insert coefficients to get these numbers.
#color(red)(2)"KMnO"_4 + color(red)(5)"Na"_2"C"_2"O"_4 + "H"_2"SO"_4 → "K"_2"SO"_4 + "Na"_2"SO"_4 + color(red)(2)"MnSO"_4 + color(red)(10)"CO"_2 + "H"_2"O"#
Step 5 Balance #"K"# .
#color(red)(2)"KMnO"_4 + color(red)(5)"Na"_2"C"_2"O"_4 + "H"_2"SO"_4 → color(blue)(1)"K"_2"SO"_4 + "Na"_2"SO"_4 + color(red)(2)"MnSO"_4 + color(red)(10)"CO"_2 + "H"_2"O"#
Step 6 Balance #"Na"# .
#color(red)(2)"KMnO"_4 + color(red)(5)"Na"_2"C"_2"O"_4 + "H"_2"SO"_4 → color(blue)(1)"K"_2"SO"_4 + color(orange)(5)"Na"_2"SO"_4 + color(red)(2)"MnSO"_4 + color(red)(10)"CO"_2 + "H"_2"O"#
Step 7. Balance #"S"# .
#color(red)(2)"KMnO"_4 + color(red)(5)"Na"_2"C"_2"O"_4 + color(olive)(8)"H"_2"SO"_4 → color(blue)(1)"K"_2"SO"_4 + color(orange)(5)"Na"_2"SO"_4 + color(red)(2)"MnSO"_4 + color(red)(10)"CO"_2 + "H"_2"O"#
Step 8. Balance #"O"# .
#color(red)(2)"KMnO"_4 + color(red)(5)"Na"_2"C"_2"O"_4 + color(olive)(8)"H"_2"SO"_4 → color(blue)(1)"K"_2"SO"_4 + color(orange)(5)"Na"_2"SO"_4 + color(red)(2)"MnSO"_4 + color(red)(10)"CO"_2 + color(brown)(8)"H"_2"O"#
Every substance now has a coefficient, so the equation should be balanced.
Step 9. Check that all atoms balance.
#bb"Atom"color(white)(m) bb"Left Hand Side"color(white)(m) bb"Right Hand Side"# #color(white)(m)"K"color(white)(mmmmm) 2 color(white)(mmmmmmmmmm)2# #color(white)(m)"Mn" color(white)(mmmml)2 color(white)(mmmmmmmmmm)2# #color(white)(m)"O"color(white)(mmmmll)60 color(white)(mmmmmmmmml)60# #color(white)(m)"Na" color(white)(mmmm)10 color(white)(mmmmmmmmml)10# #color(white)(m)"C"color(white)(mmmmll)10 color(white)(mmmmmmmmml)10# #color(white)(m)"H"color(white)(mmmmll)16color(white)(mmmmmmmmml)16# #color(white)(m)"S"color(white)(mmmmml)8color(white)(mmmmmmmmmm)8#
The balanced equation is
#"2KMnO"_4 + "5Na"_2"C"_2"O"_4 + "8H"_2"SO"_4 → "K"_2"SO"_4 + "5Na"_2"SO"_4 + "MnSO"_4 + "10CO"_2 + "8H"_2"O"#
Here's a video on balancing equations by the oxidation number method.
VIDEO
