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I want to be able to melt down typical glass (the kind you usually find at home) and mold it into other shapes. What temperatures would achieve this?
For comparison; if I were to take some sand and convert it to glass (as we know it), what temperatures would I have to generate?
Furthermore, would "heat-treating" glass in some way make it tougher? I mean "Tougher" along the lines of "bullet-proof".
The softening point of glass depends on its composition. For example,
Since you want to use glass found at home, it might be soft lead-glass (e.g. "crystal" tumbler), soda-lime glass (e.g. bottles), or borosilicate glass (e.g. coffee carafe). Melting together different types of glass is usually not successful because unless perfectly mixed, the difference in thermal coefficient of expansion will cause the object to shatter on cooling. One way to test the coefficient of expansion is to combine two different types of glass into a thread to see if it bends on cooling. This can be used to make a glass thermometer, similar to the bimetallic strip in oven thermometers. Calibration is left to the student.
If you start with fairly pure silica sand, then, you need at least 1650°C to get a glass. Although an inexpensive air-propane torch (not oxy-propane) can theoretically reach 1,995 °C, my own experience is that it takes a long time to melt incompletely just a few grams of sand.
Adding other material to silica lowers the softening point. Soda-lime contains alakali metal oxides, $\ce{CaO}$, etc. Rather than using the caustic oxides, cheaper and safer carbonates (or bicarbonates, e.g. $\ce{NaHCO3}$) can be used, though water of hydration and $\ce{CO2}$ must be driven off by heat. You could try a small amount of very fine sand mixed with baking soda, which would eventually melt to make a lower-temperature glass... but be patient, because the glass is so viscous that it takes a long while to dissolve together.
Lead oxide is used to make lead "crystal", with a lower softening point and higher index of refraction than soda-lime glass, but working with lead in a home lab is definitely not advised!
If you want to completely liquefy the glass, you are going to be in the neighborhood of 1400$\ce{^oC}$ to 1700$\ce{^oC}$. A propane torch (not the kitchen sized ones) would allow you to melt small amounts of glass or soften larger pieces for bending, scoring, or cut. Sand to glass conversion is going to occur above 1700$\ce{^oC}$, with higher temperatures needed with the varying composition of the sand.
See here for discussion on heat treated glass vs tempered glass. In essence, the degree of strengthening imparted on the glass that is being treated post manufacture depends on the rate at which the heat is removed after the initial heating. The strength manifests as higher (flat) surface compression strength and edge compression strength. The treated glass may be more brittle however, that's the trade off for making it more rigid (i.e. higher compression strength).
