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In a volcanic eruption, magma rushes to the 'outside' of the Earth. Does this mean the size of Earth also increases? If not, how is the volume left after the magma rushed out refilled?

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This is actually a more complicated question than it seems on the surface (no pun intended). The short answer is that the volume vacated by the magma is eventually refilled by the very tectonic processes that filled it in the first place. Crust is subducted, molten, and then rises to fill magma chambers. The process goes on and on. In some cases the plate that was being subducted eventually disappears (this will happen to the Juan de Fuca plate in a few million years), and the volcanoes associated with become extinct, not dormant but extinct. But generally speaking, volcanism is an ongoing process; the Earth is constantly recycling crustal material

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In one word: No.

Let me explain with the help of a water filled balloon. Here water is a proxy for the magma in the earth, and the balloon surface afor the earth crust.

If you puncture the balloon, water will come out (similar to lava).

Whenever volcanism happens the total mass on the surface of earth does not change.
Since the molten rock is equivalent to a liquid, no cavity remains inside the earth (think of the water filled balloon).
Here the total mass is constant, which implies that the earth size should not increase or decrease.

However. the picture is not the same since volcanism causes heat loss as well!

Earth is constantly loosing heat by various means and volcanism is one of them. As you you know: the colder the material, the more it contracts. So we can say it helps in contracting.

Note: There are radiogenic materials (Uranium, thorium, etc) which cause energy production in the mantle and thus expand the material volume. Hence there is a kind of equilibrium maintained for the transient time. On an earth scale this time could be of tens of thousands years.

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    $\begingroup$ Molten rock is a liquid. It does not form the bulk of the mantle though. Most of the mantle is solid. $\endgroup$ – bon Jul 22 '16 at 11:00
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    $\begingroup$ Almost all of the mantle, some 66% of the mass of the Earth, is solid, albeit slightly viscous in places. Only a very tiny fraction of volcanic material is from the mantle - it is nearly all crustal, though possibly with a mix of fractionated mantle. $\endgroup$ – Gordon Stanger Jul 22 '16 at 12:31
  • $\begingroup$ While it is in principle correct that volcanism adds to the cooling of the Earth, it most likely does not contribute in any significant way. The loss of thermal energy to space is a function of the surface of the globe. The whole globe radiates at an average of 288 K over a surface of 510,072,000 km2. Let's say a big volcano covers 1,000 km2 in liquid lava at around 1,500 K. Before it cools off... how much do you think this will have increased the overall radiative loss? Pretty much not at all. Sure, there'll also be hot gas, but for how long... $\endgroup$ – tipavi Jul 24 '16 at 22:27

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