Obtaining mercury from mercury(II) nitrate

Contrary to Andselisk's opinion, mercury nitrate $\ce{Hg(NO3)2}$ is not decomposed into $\ce{HgO}$ in water. Mercury nitrate is hydrolyzed into a basic salt, whose formula is $\ce{Hg(OH)NO3}$, and the equation is:

$$\ce{Hg(NO3)2 + H2O -> HNO3 + Hg(OH)NO3}$$

To get pure mercury from mercuric nitrate, Andselisk recommends heating their substance to 400 °C. This technique will produce huge clouds of nitrogen dioxide which is brown, toxic and corrosive. An easy way of avoiding this drawback is to mix thoroughly mercury nitrate $\ce{Hg(NO3)2}$ with an equal mass of sodium carbonate $\ce{Na2CO3}$, and heat this mixture. It will produce carbon dioxide $\ce{CO2}$ instead of nitrogen dioxide $\ce{NO2}$. The equation of the reaction is:

$$\ce{2 Hg(NO3)2 + 2 Na2CO3 -> 4 NaNO3 + 2 Hg + 2 CO2 + O2}$$

It should be mentioned that, above 380 °C, $\ce{NaNO3}$ will be slowly decomposed. So it is not recommended to heat to such a high temperature. The reaction starts at 150 °C. This operation should be done in a retort. The mercury will condense in the spout of the retort, and may be recovered. But mind! Mercury is toxic. And all operations with mercuric compounds are forbidden by the law.

The process you described would be more appropriately called "reduction of mercury(II) to elemental mercury". Unfortunately, the trick with iron likely won't work (something more inert like copper would be a better choice though).

Mercury(II) oxide is weakly basic, so mercury salts in general would easily undergo hydrolysis and form basic oxosalts in aqueous solution unless acidified. Mercury(II) nitrate quickly hydrolyses to poorly soluble yellow mercury(II) oxide upon dilution or addition of alkali:

$$\ce{Hg(NO3)2(s) + H2O(l) -> HgO(s) + 2 HNO3(aq)}\label{rxn:R1}\tag{R1}$$

which can be converted back to nitrate by adding excessive amount of nitric acid, which in turn won't leave any chance for iron not to be oxidized, so you end up with both metals in solution.

I didn't want to dive deep into the discussion of how mercury(II) salts hydrolyze and leave it simple, but after receiving critique from Maurice claiming $\ce{Hg(OH)NO3}$ to be the “real” product, I think I allow myself to add a paragraph or two. Raman spectroscopy and x-ray scattering studies in late 1960 demonstrated that hydrolysis of mercury(II) salts produces series of polynuclear oxo-bridged species of types $\ce{Hg2OH(H2O)2^3+},$ $\ce{Hg3O(H2O)3^4+}$ or $\ce{Hg4O(OH)(H2O)3^5+}$ [1, 2]. Formation of mercury(II) hydroxide nitrate $\ce{Hg(OH)NO3}$ as a hydrolysis product was taught in 1940–1950s era, and it stuck in several textbooks published later probably because it's listed in all editions of Pauling's General chemistry up to 1988. However, not only it's oversimplified (granted, reaction \eqref{rxn:R1} is also an oversimplification in a sense it's a boundary case), but is also an incorrect concept.

The most up-to-date summary of what is really happening when $\ce{Hg(NO3)2}$ is dissolved in water can be found in Mercury handbook [3, p. 115]:

$\ce{Hg(NO3)2}$ solutions are only stable in the presence of a certain amount of nitric acid, which prevents hydrolysis. $\ce{Hg(NO3)2}$ quickly hydrolyzes in excess water and produces a precipitate of $\ce{Hg3O2(NO3)2 · H2O}$ or, when boiled in dilute solutions, forms mercury(II) oxide $(\ce{HgO}).$

Regarding the reduction of mercury(II), there are two pathways: $\ce{Hg^0}$ may be obtained from mercury(II) nitrate using a dry or a wet method.

Speaking of a dry method, the most straightforward way to obtain metallic mercury from nitrate is to heat $\ce{Hg(NO3)}$ in a distillation apparatus (b.p. of mercury is 357 °C). Above $\pu{400 °C}$ it readily decomposes:

$$\ce{Hg(NO3)2(l) ->[\pu{400 °C}] Hg(g) + 2 NO2(g) + O2(g)}\label{rxn:R2}\tag{R2}$$

\eqref{rxn:R2} is a brutto-reaction; the nitrate first decomposes to the red mercury(II) oxide at lower temperatures (which, in turn, decomposes to the elements):

$$\ce{2 Hg(NO3)2(l) ->[\pu{350 °C}] 2 HgO(s) + 4 NO2(g) + O2(g)}\tag{R3}$$

Wet method suggest mild conditions and a reaction in solution. For example, formic acid (also used in silver refining) being a strong reducing agent would cause a precipitate from ammoniacal mercury(II) nitrate solution.

Since nitrogen dioxide, mercury vapors as well as mercury salts and oxides are highly toxic, the reactions must be carried in a fume hood which makes it a poor fit for a talent show. Considering the possible risks and your level or preparation (no offence), I'd strongly advise to take extreme caution doing mercury chemistry and avoid public demonstrations until you become more experienced.

Note: chemical reactions are adopted from [4, p. 310–312]

References

1. Cooney, R.; Hall, J. Raman Spectra of Mercury(II) Nitrate in Aqueous Solution and as the Crystalline Hydrate. Aust. J. Chem. 1969, 22 (2), 337. https://doi.org/10/b6t3h2.
2. Johansson, G.; Haugsten, K.; Rasmussen, S. E.; Svensson, S.; Koskikallio, J.; Kachi, S. An X-Ray Investigation of the Hydrolysis Products of Mercury(II) in Solution. Acta Chem. Scand. 1971, 25, 2787–2798. https://doi.org/10/bn5j2g. (PDF)
3. Kozin, L. F.; Hansen, S.; Kit, M. Mercury Handbook: Chemistry, Applications and Environmental Impact; RSC Publ: Cambridge, 2013. ISBN 978-1-84973-409-7.
4. R. A. Lidin, V. A. Molochko, and L. L. Andreeva, Reactivity of Inorganic Substances, 3rd ed.; Khimia: Moscow, 2000. (in Russian)

Here is a possibly interesting safe path to try for a Zinc (or Aluminum) Hg amalgam, which can be used in organic synthesis (e.g., for the Clemmensen reduction) courtesy of the field of Hydrometallurgy. It starts by first placing a sheet of, say, pure Zinc in HCl to create some surface absorbed H atoms (or Al in NaOH, where purity is important to avoid toxic hydrogenated gases) in a fume hood. Remove and rinse with distilled water. Add the treated sheet of metal to a dilute solution of $\ce{Hg(NO3)2}$. This is expected to result in a product of Mercury(I) nitrate in dilute nitric acid. Reaction mechanics:

$\ce{H•(surface)⇌e-(aq) + H+(aq)}$

$\ce{Hg(2+)(aq) +e-(aq)->Hg(+)}$

$\ce{H+ + NO3- = HNO3}$

Apparently, the H• radical functional behaves (per its seemingly reversible formation reaction: e- + H+ = H• ) as apparently an (e-, H+) operating pair. See this source, 'Hydrometallurgy 2008: Proceedings of the Sixth International Symposium', p. 818, which cites a commercial leaching process, to quote Equation (5):

" PbS + 2 H• = Pb + H2S (5) "

which one could view functionally as:

$$ Pb(+2)S(2-) + 2 (e-, H+) = Pb + H2S (g) $$ Further, per Wikipedia on Mercury(I) nitrate the action of light (or less safe, boiling) results in the liberation of mercury, to quote:

"If the solution is boiled or exposed to light, mercury(I) nitrate undergoes a disproportionation reaction yielding elemental mercury and mercury(II) nitrate:[2]

$$ Hg2(NO3)2 ⇌ Hg + Hg(NO3)2 " $$ where the Hg(NO3)2 can be further attacked by hydrogen atoms.

Note, if an amalgam is not formed, one just has successfully removed Mercury from the nitrate.