Chemistry - Turn lead into gold via radioactive decay
Solution 1:
Interesting idea, but it has already been done, and not cheaply - read on.
How could we get a great quantity of $\ce{_{82}^{197}Pb}$ ?
There would be two problems with getting a large amount of $\ce{_{82}^{197}Pb}$. First, the parent nuclide of $\ce{_{82}^{197}Pb}$ is $\ce{_{83}^{197}Bi}$ which is unstable and has a half-life of only 9.33 minutes - so you can't get a large quantity of $\ce{_{82}^{197}Pb}$'s precursor to begin with. Second, once $\ce{_{82}^{197}Pb}$ is formed, it has a half-life of 8.1 minutes, so it transmutes quickly to $\ce{_{81}^{197}Tl}$.
How long would it take for it to turn into gold?
$\ce{_{82}^{197}Pb}$ half-life = 8.1 minutes
$\ce{_{81}^{197}Tl}$ half-life = 2.84 hours
$\ce{_{80}^{197}Hg}$ half-life = 64.14 hours
After 10 half-lives, ca. 0.1% of the starting material will be left $$\ce{(1/2)^10 = $0.0009766$}$$ The last step is the slowest by far, so after about 641.4 hours (26.73 days), you should have something around 99.9% pure gold.
Can that be accelerated?
Unlike chemical reactions that can heated, catalyzed, etc., this type of nuclear transformation keeps a set schedule.
Once most of it has been turned into gold, how can we extract the gold ? (there are still remaining Pb,Tl,Hg molecules)
As noted above, you can get whatever purity you desire, just wait.
What would be the yield of the process?
It would be high for the 3 nuclear transformations you listed. Each of the elements you listed decays directly and only to the daughter isotope you've shown. However, as noted above, you can't start with $\ce{_{82}^{197}Pb}$, you generate it from $\ce{_{83}^{197}Bi}$, the decay of which adds some impurity along with lead. And then since the bismuth isotope is not long-lived you'd probably start with its precursor, and so on until you find something that has a long enough life that you could assemble a reasonable quantity.
Back around 1980 Glenn Seaborg actually transmuted bismuth to gold, but only a few thousand atoms (see this reference also).
How much money would it approximately cost compared to how much we win in gold?
The Wikipedia article I referenced directly above notes, "the expense far exceeds any gain." There are other ways (fission and fusion) to produce gold, but at least with the methods available today, the cost would be astronomical.
Solution 2:
One thing Ron didn't cover is the side effects of all that decay. All that short-half-life stuff will be rather radioactive, to the point where you might not survive to see the results. It's also possible that if you get enough together to be profitable that it will simply vaporize itself. The stable gold should eventually condense so as long as you contain the vapor it might work. Or you'll have a very pretty gold-lined flask.
The stable isotopes of lead are 204, 206, 207 and 208, you'd have to knock a lot of protons and neutrons out to get down to 197.
Shame! Knock a proton off of lead and it turns into thallium. But as that's the point of the exercise just bump off two more and we're done.
Solution 3:
I can’t guess at the cost, but consider this: if it were profitable, many people would do it, and then the value of gold would quickly reduce to a level consistent with the cost of producing it.