All the chemical elements discovered or created so far follow a regular pattern in how their electrons are arranged: the *n*th shell contains up to 2*n* – 1 suborbitals that each contain up to two electrons. For a given atomic number, you can determine how its electrons are distributed into shells and suborbitals using the Aufbau principle.

The Aufbau principle is a good approximation, but not exact. For this post we’ll assume it is exact, and that everything in the preceding paragraph generalizes to an arbitrary number of shells and electrons.

Under those assumptions, what would the periodic table look like if more elements are discovered or created?

D. Weiss worked out the recurrence relations that the periods of the table satisfy and found their solutions.

The number of elements in *n*th period works out to

and the atomic numbers of the elements at the end of the *n*th period (the noble gases) are

We can verify that these formulas give the right values for the actual periodic table as follows.

>>> def p(n): return ((-1)**n*(2*n+3) + 2*n*n + 6*n +5)/4 >>> def z(n): return ((-1)**n*(3*n+6) + 2*n**3 + 12*n**2 + 25*n - 6)/12 >>> [p(n) for n in range(1, 8)] [2.0, 8.0, 8.0, 18.0, 18.0, 32.0, 32.0] >>> [z(n) for n in range(1, 8)] [2.0, 10.0, 18.0, 36.0, 54.0, 86.0, 118.0]

So, hypothetically, if there were an 8th row to the periodic table, it would contain 50 elements, and the last element of this row would have atomic number 168.

## Related

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Thank you for sharing. While in high school I thought why couldn’t we add more rows to the periodic table, for which I learned later in high school they wouldn’t be stable in the world we live in (temperature, pressure, etc.). I always wondered if these higher number elements would be of special use in some special situations. May be somebody will find out in the future.

There is a predicted there would be an island of stability if we can keep creating new elements.