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I remember from my chemistry classes that (after the initial irregularities) a new block of elements starts every two periods. After the initial s-block and p-block following it shortly, we have d-block starting at period IV, and f-block starting at period VI.

Now that Element 118 has been discovered, we're about to open period VIII and we're two elements short of a new block.

What block will it be? How many groups, what name etc?

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As you noted, this is a very appropriate question in light of the IUPAC announcement that we have just finished filling Period 7!

The names of the subshells s, p, d, and f are named after the old spectroscopic terms sharp, principal, diffuse, and fundamental. We ran out of fancy names after that, so the subsequent subshells are named in alphabetical order - g, h, and so on - which means that after the 8s block is filled, we would theoretically have a 5g block.

The orbitals in the g subshell would be labelled with the quantum number $l = 4$, so $m_l$ would take integer values between $-4$ and $4$ (inclusive) giving a total of nine g-orbitals. Each g-orbital could hold two electrons with opposite spins, so the g-block would have $18$ electrons.

However, it is worth noting that the electronic configurations may or may not obey the aufbau principle fully. Whether the 5g orbitals will actually be filled or not will certainly not be easy to determine, considering how short the half-lives of those elements are likely to be.

Wikipedia has an article which talks about it.

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