Can lava that has cooled to a hardened state be returned to a molten state by, say, a new river of molten lava flowing over it - like re-smelting iron? If you threw a lava rock into the mouth of a volcano, would it remain in its solid state or would it melt and become part of the cauldron of magma?
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Yes, at least partially.
It is important to remember that a "rock" usually does not have a single melting point. It melts over an interval. Igneous (magmatic) rocks are made of combination of minerals (for example, quartz, olivine, pyroxene, feldspar) that each will melt at a different point. When a rock is heated up, the minerals with the lower melting point melt first, followed by the higher melting point minerals with progressive heating.
Some rocks are composed mostly of easier-to-melt material (for example the stuff that came out of Mt. St Helens, mostly granitic composition) and others are made of harder-to-melt material (for example Hawaiian basalts). Taking a a granite and throwing it into a basalt will melt most of it for sure, if not all. Not the opposite though.
Even if you do not completely melt the rock, you will disintegrate it enough that the remaining harder-to-melt minerals will just drift away and mix with the lava, making it impossible to distinguish the original rock any more.
Likewise, when hot lava flows on easy-to-melt material (for example clay-rich soil), some of it will melt or vitrify (turn to glass).
I will finish by emphasising that all lavas, every single one of them, were originally solid rocks that melted at depth by some kind of process. So yes, it is possible to melt rocks.
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$\begingroup$ Great explanation! However, your last statement begs the question (at least to me) - isn't the molten core the origin of the earth with the crust cooling on the outside? You seem to be saying that the earth is/was a solid rock that somehow melted, developing into it's molten center - or am I misreading/twisting your words? $\endgroup$ Aug 4, 2018 at 4:44
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1$\begingroup$ The Earth was once all molten, about 4.5 billion years ago. It segregated into a metallic (mostly iron and nickel) core, and an eventually solidified rocky mantle. The core is about 2900 km deep. The core is still partly molten. But, the mantle is overwhelmingly solid. Lavas that we see on Earth's surface form by partial melting of the solid material in the uppermost part of the mantle and the crust (usually top 200 km, commonly shallower). Magmas that form deeper are extremely rare (for example, kimberlites), and even then it's never more than a few hundreds of kilometres. $\endgroup$– GimelistAug 4, 2018 at 4:57
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1$\begingroup$ @Gimelist: "The Earth was once all molten..." but before that, at least according to my understanding of planetary formation theories, it was a lot of small solid bodies that accreted, with the energy of impact heating everything to the melting point. $\endgroup$– jamesqfAug 5, 2018 at 18:08
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$\begingroup$ As an extreme example, with some volcanoes, a large proportion of the lava erupted is formed from melted country rock (the solid-or-at-least-previously-solid rock that the volcano erupts through); this is how, for instance, a basaltic hotspot can produce explosively-erupting rhyolitic melts. $\endgroup$– VikkiFeb 3, 2022 at 0:23