Placed in equivalent freezers, would a liter of water or a liter of lava turn from liquid to solid first? (question from a 6 year old)
You've got the right idea — you want to simplify the problem — but I don't think you're using quite the right simplification.  Take a look at the book's set-up of the situation, and ask yourself how the water is stopping the lava. You'll see the idea is that we're using liquid water, not ice, to solidify the lava.
So your question should really be:  How much heat do we need to absorb from a liter of lava to turn it into a solid, and how much heat can a liter of water at room temperature absorb before it turns to steam?  If the latter is larger than the former, then a liter of water can cool a liter of lava to the point where it solidifies before the water all changes into steam.
[I'm using &quot;heat&quot; when I should really be using &quot;thermal energy&quot;, but this is for a 6 yo, so I'm keeping it simple.]
First, let's do the calculation for water.  Here (since it's for a 6 yo), I'm not going to show all the steps in the calculations:
Energy to heat 1 L liquid water at room temp (25C ) to 100C = 75 kcal
Energy to turn 1 L liquid water to steam at 100C = 533 kcal
Total = 608 kcal
According to https://en.wikipedia.org/wiki/Lava, lava is typically 700 to 1200 C, so let's call it 1000 C.
And, using this source (https://ocw.mit.edu/courses/earth-atmospheric-and-planetary-sciences/12-109-petrology-fall-2005/lecture-notes/Nov3notes.pdf), let's assume it melts at 900 C (there's actually a wide range of lava types, and thus a wide range of lava temps and melting points).
Density of lava = 3.1 g/cm^3, so 1 L lava weighs 3100 g
Energy released when 1 L lava cools from 1000 C to 900 C = 93 kcal
Energy released when 1 L lava solidifies at 900 C: 310 kcal
Total = 403 kcal
So, based on the above, 1 L of water is enough to solidify 1.5 L of lava.
Finally, why does the water win?  The main reason is that the water is going between a liquid and a gas, while the lava is going between a solid and a liquid.  And, in general, it takes much more energy to change liquids into gases than solids into liquids, because in the former case you are pulling the molecules completely apart from each other.
But I said it takes &quot;much more energy&quot; to change a liquid to a gas than a solid to a liquid, yet the difference here is only a factor of 1.5. The discrepancy is because  in this case you're not comparing masses of water and lava, you're comparing volumes (and for the same volume, the mass of lava is 3.1 x greater). Thus, if you do it on a mass basis, 1 kg of water is enough to solidify 6 kg of lava (because, while 1 L of water weighs 1 kg, 1.5 L of lava weighs 6 kg).
[For these quantities, mass is typically a more fundamental basis of comparison than volume, which is why heat capacities, heats of fusion, and heats of vaporization are usually quoted on a mass basis. When you see the term &quot;specific&quot;, as in &quot;specific heat capacity, that means &quot;per unit mass&quot;.  Of course, one can also use a number basis (moles).]