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I have come across many questions like: "if electron had 3 spins (-1/2,0,+12) then what change will be there in the periodic table?", also sometimes " if the capacity of each orbital becomes 5 then in what period or group will the particular element be?" So what is the relation between quantum no. and the periodic table?

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If electrons could have $m_s = -1/2, 0,$ and $1/2$, then the entire structure of the Periodic Table would be different. The s-block would have 3 elements instead of the 2 that we see now, because each s orbital can accommodate 3 electrons, and similarly the p-block would have 9 elements instead of 6.

It is not possible to say, for example, "sodium would be in so-and-so position in the new Periodic Table" because in such a universe, the element sodium would not even exist the way it does in our universe. The new Periodic Table would not have an element with the electronic configuration $\mathrm{(1s)^2(2s)^2(2p)^6(3s)^1}$. What I am basically saying is, there is no one-to-one correspondence between the elements in our universe and the elements in such a hypothetical universe (mathematically speaking, you can, but it would not make any chemical sense). Nevertheless, you could say that sodium is defined to be the element with 11 protons and 11 electrons. In that case, it would have a configuration of $\mathrm{(1s)^3(2s)^3(2p)^5}$ and it would be the fifth element in the Period 2 p-block.

Actually, there is no evidence that the aufbau principle should still hold true in such a universe: the large repulsion that arises from putting 3 electrons into the same orbital might well make filling the 2s orbital before the 1s orbital more favourable. So, our hypothetical "sodium" might have a weird electronic configuration of $\mathrm{(1s)^2(2s)^2(2p)^7}$. The Periodic Table is useful partly because it groups elements with similar electron configurations - which implies similar chemical properties - together. However, if these elements do not obey the aufbau principle, then there is no guarantee that the elements in the same group should have similar chemical properties, and therefore no guarantee that a Periodic Table in such a hypothetical universe would even be useful.

I should end off by saying that the laws of quantum mechanics stipulate that the different values of $m_s$ must increase in steps of $1$. Therefore, it is not actually possible that $m_s$ could ever take on values of $-1/2, 0$, and $1/2$, since these increase in steps of $1/2$. You might say: "Ah! But we could say, what happens if $m_s = -1, 0, 1$." Well, in that case, you have much more to worry about than just the orbital occupancy. Without going into too much detail, the Pauli exclusion principle would no longer apply to electrons, and you would have more problems than just "one orbital can fit 3 electrons".

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