\(\text{HNO}_{3}(\text{ℓ}) + \text{PbS}(\text{s})\) \(\to\) \(\text{PbSO}_{4}(\text{s}) + \text{NO}_{2}(\text{g}) + \text{H}_{2}\text{O}(\text{ℓ})\)
\(\text{Pb}^{2+}\) is a spectator ion.
The unbalanced reduction half-reaction is:
\(\text{NO}_{3}^{-}(\text{aq})\) \(\to\) \(\text{NO}_{2}(\text{g})\)
Add water molecules to the right and \(\text{H}^{+}\) ions to the left (acid medium) to balance the oxygen and hydrogen atoms:
\(\text{NO}_{3}^{-}(\text{aq}) + 2\text{H}^{+}(\text{aq})\) \(\to\) \(\text{NO}_{2}(\text{g}) + \text{H}_{2}\text{O}(\text{ℓ})\)
Balance the charge by adding an electron to the left (this makes sense as this is the reduction half-reaction, and \(\text{N}^{5+}\) \(\to\) \(\text{N}^{4+}\)):
\(\text{NO}_{3}^{-}(\text{aq}) + 2\text{H}^{+}(\text{aq}) + \text{e}^{-}\) \(\to\) \(\text{NO}_{2}(\text{g}) + \text{H}_{2}\text{O}(\text{ℓ})\)
The unbalanced oxidation half-reaction is:
\(\text{S}^{2-}(\text{aq})\) \(\to\) \(\text{SO}_{4}^{2-}(\text{aq})\)
Add water molecules to the left and \(\text{H}^{+}\) ions to the right to balance the oxygen and hydrogen atoms:
\(\text{S}^{2-}(\text{aq}) + 4\text{H}_{2}\text{O}(\text{ℓ})\) \(\to\) \(\text{SO}_{4}^{2-}(\text{aq}) + 8\text{H}^{+}(\text{aq})\)
Balance the charge by adding eight electrons to the right (this makes sense as this is the oxidation half-reaction, and \(\text{S}^{2-}\) \(\to\) \(\text{S}^{6+}\)):
\(\text{S}^{2-}(\text{aq}) + 4\text{H}_{2}\text{O}(\text{ℓ})\) \(\to\) \(\text{SO}_{4}^{2-}(\text{aq}) + 8\text{H}^{+}(\text{aq}) + 8\text{e}^{-}\)
We multiply the reduction half-reaction reaction by \(\text{8}\) to balance the number of electrons in both equations:
\(8\text{NO}_{3}^{-}(\text{aq}) + 16\text{H}^{+}(\text{aq}) + 8\text{e}^{-}\) \(\to\) \(8\text{NO}_{2}(\text{g}) + 8\text{H}_{2}\text{O}(\text{ℓ})\)
Adding the two equations together gives the balanced equation (electrons are equal on both sides and can be removed):
\(8\text{NO}_{3}^{-}(\text{aq}) + 16\text{H}^{+}(\text{aq}) + \text{S}^{2-}(\text{aq}) + 4\text{H}_{2}\text{O}(\text{ℓ})\) \(\to\) \(\text{SO}_{4}^{2-}(\text{aq}) + 8\text{H}^{+}(\text{aq}) + 8\text{NO}_{2}(\text{g}) + 8\text{H}_{2}\text{O}(\text{ℓ})\)
Putting the spectator ions back into the equation and removing any extra \(\text{H}^{+}\) ions and water molecules we get:
\(8\text{HNO}_{3}(\text{ℓ}) + \text{PbS}(\text{s})\) \(\to\) \(\text{PbSO}_{4}(\text{s}) + 8\text{NO}_{2}(\text{g}) + 4\text{H}_{2}\text{O}(\text{ℓ})\)