I collected a small stash of 14k jewelry scrap plus several gold crowns and I wanted to convert it to 24k pure gold. Here's the process, with pictures.
The total scrap weight was about 29 grams. Assuming it was all 14k, the estimated yield of 24k gold should have been around (29 x 14)/24 = 16.9 grams. I think the dental gold is a bit lower karat but this weight should be a reasonable estimate.

In the past I've used aqua regia to dissolve gold but concentrated nitric and hydrochloric acids aren't readily available since I no longer work in a lab. So, I tried the crude chemist's method of combining concentrated bleach (8.25% sodium hypochlorite) plus swimming pool muriatic (hydrochloric) acid. The mixture generates free chlorine in acidic solution which attacks gold and other metals. The process was very slow. Here's my scrap after a couple hours:

The liquid assumed a green color presumably from copper and/or nickel in the 14k jewelry alloy. I changed the solution many times as it lost potency. The solution gradually became more yellow as the copper/nickel components were depleted. Here's what it looked like after 3 days:

Periodically I combined extract solutions and concentrated the mixture by boiling off water and excess HCl. I then diluted the concentrate (which at this point was mostly sodium chloride, with the dissolved gold present as gold chloride) with distilled water, ran it through a paper coffee filter to remove particulates, and precipitated the gold with sodium nitrite solution. Nitrogen dioxide gas is generated which boils off as a brown vapor. The gold precipitates as a granular solid, called gold sponge. Base metal oxides and hydroxides may also precipitate, mixing with the gold. Here's a photo of the reaction as the sodium nitrite is added.

The last pieces to dissolve were the gold crowns. I guess that makes sense; you want your dental work to withstand lots of wear so the alloy is likely harder and more corrosion-resistant than jewelry alloy. It took a whole month before the last pieces of crown were dissolved.
Eventually all the extracts were precipitated and the sponge combined. The total weight was 17.46 grams and the product was visibly contaminated with dark specks and white debris.

To purify the gold I re-dissolved it in bleach and acid, filtered it, and re-concentrated it, again, to drive off excess HCl. Too much residual acid in the solution will consume sodium nitrite, making the gold precipitation less effective. The nearly-dry residue shows beautiful crimson layers of gold chloride mixed with a thick crust of sodium chloride at the bottom of my beaker:

After a final filtration and dilution the gold concentrate was ready for precipitation. I really like the color!

Here's the beaker after the final addition of sodium nitrite. The gold sponge has separated nicely, forming a coherent layer on the bottom.

I drained off the liquid and boiled the sponge repeatedly with distilled water, then performed one boiling cycle with dilute nitric acid (which will dissolve any stray silver or base metal oxides), then boiled twice more with distilled water. Finally, I drained off the water and dried the sponge in my kitchen oven. Total sponge weight: 16.84 grams.

The last step was to use an ordinary Bernz-o-Matic propane torch to carefully fuse the sponge into metallic gold. My three 24k nuggets have a total weight of 16.78 grams; pretty close to my original estimated yield!
This project was very time-consuming but fun. Now I'll have to collect more scrap so I can do it all over again!

