Iron is common as dirt, because all stars eventually burn their fuel into iron. Iron nuclei are at the lowest energy level attainable, so they won’t release any more energy by nuclear reactions.
Edit: all stars don’t, smaller ones stop earlier.
Gold is much more rare, only formed in cataclysmic explosions of large stars when they die.
So while there’s gold, lead, iridium etc in the Earth’s core, just by virtue of them being heavy, it’s vastly overshadowed by the amount of iron.
There’s quite a lot of gold, actually, compared to what we’ve got up here: https://www.abc.net.au/news/2006-06-15/researcher-calculates-gold-within-earths-core/1778918
A big factor here is the actual physical size of the atom. More compact atoms will tend to sink while larger ones will tend to “float” on top, just because it’s easier for a small atom to slip under a big atom than vice versa.
The phenomenon known as desert pavement is thought to be a reflection of this–if you go out to a hardscrabble kind of desert (as opposed to a sandy one) and dig, you’re likely to find the top layer consists of a lot of bigger rocks while deeper down you’re finding mostly smaller ones. For a more everyday example, if you open a new bag of chips/crisps that has been sitting with a consistent “up” direction, the top will have a lot of unbroken chips while the tiny fragments will have settled to the bottom.
The gold that was on the planet when it formed mostly did end up in the core. Iron is just the most common heavy element, so the core is mostly made of it.
The reason iron is so common is that nuclear reactions want to turn light elements heavier and heavy elements lighter, and iron is the lightest element that doesn’t want to turn heavier. So stars, which start with light elements like hydrogen and helium, can build iron or anything lighter than it without spending extra energy. Whereas anything heavier than iron costs a star energy to make.
The center of the Earth is not molten iron. It is solid mostly-iron. There are many different elements present there – including both heavier and lighter ones – but iron is by far the most common one, so “iron core” is used as a common descriptor.
There is a layer of molten outer core wrapping the solid inner core. The solid part is about a thousand km thick and the liquid part is about another two thousand km thick.
There is a lot of gold and platinum and other heavy rare metals in the core, too. In fact, all the heavy metals that were on the earth when it first formed are there. The gold we can find now was deposited on earth later. A lot came from when a mars-sized planetoid crashed into earth and blasted a bunch of material off, some of which formed the moon. More was deposited by other meteors. In those days there were a lot more rocks flying around so there were a lot more impacts.
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