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If lava is nothing but magma breaking through the earth's surface, then why is basalt more dense than granite?

I understand that something happens during the cooling process, but cannot find any comprehensive answer.

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Magmas have a wide range of chemical (and mineralogical) compositions. Basalts come from mafic magmas (they used to be called "basic" magmas), while granites come from felsic magmas (used to be "acid"). (The reasons for such diversity is out of scope of the question ; you can look for "magmatic differentiation".) It is this difference in chemistry that creates the difference in density. Felsic magmas are enriched in silica, which is light, while mafic magmas contain more heavy oxides such as $\ce{FeO}$, $\ce{Fe2O3}$, $\ce{CaO}$... Here are two geochemical analyses (in wt%) from the USGS Geology, Geophysics, and Geochemistry Science Center. BHVO-2 is a basalt from Kilauea while G-3 is a granite from Rhode Island. Both are used worldwide as rock standards for analytic calibration.

Oxide BHVO-2 G-3
$\ce{SiO2}$ 49.60 68.66
$\ce{Al2O3}$ 13.44 15.92
$\ce{CaO}$ 11.40 1.830
$\ce{Fe2O3T}$ 12.39 2.866
$\ce{K2O}$ 0.5130 4.57
$\ce{MgO}$ 7.257 0.75
$\ce{MnO}$ 0.1690 0.0363
$\ce{Na2O}$ 2.219 3.968
$\ce{P2O5}$ 0.2685 0.136
$\ce{TiO2}$ 2.731 0.476

Of course these are just examples. There are basalts containing more potassium, or granites containing less aluminum, and these main rock types can be further divided depending on these criteria (e.g., a basalt can be alkali or subalkali; high-K, medium-K or low-K, etc., see Le Maitre et al. 2002 for a full classification). But overall the difference in silica content is always huge.

Note that mafic magmas can also stay in the crust and crystalize, like granite, yielding a rock called gabbro. Conversely, felsic magmas can erupt at the surface, yielding a rock called rhyolite.

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