Mercury is below hydrogen in the activity series, so #"Hg"# does not displace hydrogen from nitric acid.
However, the nitrate ion is a strong oxidizing agent.
In acid solution it oxidizes mercury to the mercury(II) ion and becomes reduced to nitrogen monoxide.
The balanced equation is
#"3Hg + 8HNO"_3 → "3Hg"("NO"_3)_2 + "2NO" + 4"H"_2"O"#
You can find the general technique for balancing redox equations in acid solution here.
Step 1: Write the two half-reactions.
#"Hg" → "Hg"^"2+"#
#"NO"_3^"-" → "NO"#
Step 2: Balance all atoms other than #"H"# and #"O"#.
Done.
Step 3: Balance #"O"#.
#"Hg" → "Hg"^"2+"#
#"NO"_3^"-" → "NO" + 2"H"_2"O"#
Step 4: Balance #"H"#.
#"Hg" → "Hg"^"2+"#
#"NO"_3^"-" + "4H"^"+" → "NO" + 2"H"_2"O"#
Step 5: Balance charge.
#"Hg" → "Hg"^"2+" + "2e"^"-"#
#"NO"_3^"-" + "4H"^"+" + "3e"^"-"→ "NO" + 2"H"_2"O"#
Step 6: Equalize electrons transferred.
#3×["Hg" → "Hg"^"2+" + "2e"^"-"]#
#2×["NO"_3^"-" + "4H"^"+" + "3e"^"-"→ "NO" + 2"H"_2"O"]#
Step 7: Add the two half-reactions.
#"3Hg" → "3Hg"^"2+" + "6e"^"-"#
#"2NO"_3^"-" + "8H"^"+" + "6e"^"-"→ "2NO" + "4H"_2"O"#
#stackrel(————————————————————)("3Hg + 2NO"_3^"-" + "8H"^"+" → "3Hg"^"2+" + "2NO" + 4"H"_2"O")#
Step 8. Add the spectator ions
#"3Hg + 2NO"_3^"-" + "8H"^"+" → "3Hg"^"2+" + "2NO" + 4"H"_2"O"#
#color(white)(mm)+ "6NO"_3^"-" color(white)(mmmm)+"6NO"_3^"-"#
#stackrel(————————————————————)("3Hg + 8HNO"_3 → "3Hg"("NO"_3)_2 + "2NO" + 4"H"_2"O")#
∴ The balanced equation is
#"3Hg + 8HNO"_3 → "3Hg"("NO"_3)_2 + "2NO" + 4"H"_2"O"#