Thermit is just Iron oxide also known as ferrate, rust or (Fe3O4) mixed with Aluminum (Al). Thermit was used for welding before we had arc welders or electricity. It was most widely used to weld train tracks and train parts, steam ships and automobiles. Now days it is used as rocket fuel for space ships. One reason you might want to make some would be to turn a pile of rust into relatively pure Iron or steal if you add some carbon to it. Carbon can be added by putting a small amount of green wood in your thermit. Thermit can also be made out of rust and calcium from lime or rust and silica but it is extremely harder to light. The formula for a thermit reaction is 8Al+3Fe3O4=9Fe+4Al2O3.
Cation: Thermit burns at 3000 deg. Celsius or 5432 deg. Fahrenheit well over the temperature of molten steal 1350 deg. Celsius or 2462 deg. Fahrenheit so where welding gloves goggles, long sleeves and keep at least five gallons of water near by even if you do this on cement. Also unless you are a rocket scientist do not try to make a rocket out of this. It will explode and it could kill you and one or all of your friends.
Rust can be made easily by putting some steal wool in water for a few days but steal wool contains carbon. The better way to make rust is to with electrolysis. Put an old nail with a lot of Iron in it into a jar of water and find a 4-6 volt DC power source with 6-10 amps like a transformer from an old train set. Attach the positive wire to the nail and put the negative in the water. Be sure to hang the nail over the side so as to keep the end of the positive wire out of the water that way the current has to travel through the nail. This process causes the water molecules to separate into hydrogen and oxygen and attracts the oxygen atoms to the nail. This will also make hydrogen so do it in a well-ventilated area away from flames. It could look like this.
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Once the nail has rusted take it out, scrape off the rust and put the nail back into the water then put the rust in some other water just incase you scraped off some un rusted iron. You will need about 10 grams of rust.Next make some Aluminum filings with a fine file you can use pop can aluminum but 01Aluminum is pure and it works best. You will need about 3 grams of aluminum. If you are lazy you can buy alumaseal it is a fine powder used to seal radiators on cars.
Use a scale to weigh out 3 grams of aluminum and about 8-10 grams of rust. The thermit is easer to light if you use less rust but the iron wont be as pure when you are done. To make it easer to light you can add magnesium or barium peroxide to your thermit. Set the magnesium or barium peroxide on top of the Thermit and run a 1 foot fuse to the thermit. If you do not have any fuse you can use an electric model rocket igniter or a strip of magnesium ribbon. Before you light the magnesium ribbon use some sand paper to scrape of the oxidation. Light the magnesium ribbon with a torch or set it on a rock, heat it up as much as possible with a lighter then smack it with a hammer and it will light. As soon as you light the fuse you should run at least 15 feet away because once the thermit goes there will be molten metal everywhere.
Note: Never light thermit in a closed container it will explode and probably kill some one. I know rockets are cool but you are probably no rocket scientist. I know from experience that I am not.
If you are using themit to weld or to purify rust then you should read (Thermit welding process by Richard N. Hart).