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I'm aware that andesite consists of about 60 percent silica and as for minerals, pretty sure it's usually pyroxene and plagioclase. What form would these take on if they were to be separated? Especially interesting, I think, is what would happen to the silica: would it be in its "commercial" powdered form?

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By definition andesite is mainly plagioclase (mostly variety andesine),and pyroxene with either quartz or olivine (not both), and minor amounts of magnetite, ilmenite, zircon, apatite, possibly biotite mica, and maybe a few traces of rarer minerals. The pyroxene is probably a mixture of orthopyroxene and augite. In addition to plagioclase there may be small quantities of other, more alkaline feldspars.

Andesite is chemically about 60+/-3% silica, SiO2, but the amount of free silica is very small, if present at all - at most 2 or 3 percent. So no, you won't extract commercial powdered silica. Better than 99.9% pure silica is found very abundantly in some quartzites, a common sedimentary rock. So nobody is going to try to extract silica from other silicates.

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I'll add a bit to Gordon's answer. Andesite usually contains several large crystals (that can be pyroxene, plagioclase, olivine, or any combination of these). Their sizes can be up to several mm. These crystals will be set in a groundmass (also called matrix) of either very fine crystals of all of the other minerals Gordon mentioned, volcanic glass, or both.

When you break it mechanically, it will tend to break along mineral grains. Depending on the final size you break the grains to, you might end up with individual grains of each mineral. There is a good chance that some of the groundmass will stick to the larger grains, or that there will be inclusions of other minerals inside the larger minerals. The finer you break it down, the higher chances that each individual grain will be only one mineral (or glass). If required, there are then methods to separate the minerals to have an almost pure fractions of each mineral. For example, magnetic separation will separate oxides from silicates, and it will separate iron-bearing silicates from iron-absent silicates. This is a very time consuming process. Heavy liquids is another methods for separating the minerals.

As Gordon said, the rock is around 60% SiO2, but this this is not in the form of quartz (the mineral with the formula SiO2), but rather bound in silicates, such as plagioclase (CaAl2Si2O8) or clinopyroxene (CaMgSi2O6). Separating the chemical SiO2 from these minerals to produce commercial grade SiO2 is very hard, expensive, and pointless. It is much easier to get SiO2 from sandstones, beach sand, and quartzites, most of them are almost pure SiO2 in the form of quartz. This is the case because the chemical grade SiO2 we use in our own lab has small but higher than expected amounts of ZrO2 in it. This basically means that it has a tiny bit of zircon (ZrSiO4) in it. This is a mineral that's extremely resistant to chemical attack, and it accumulates in sedimentary detrital rocks, for example sandstone. Gives you an idea on where the SiO2 used to make it came from...

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