an atom has 2 electrons in the first energy level, 8 electrons in the second energy level, and 1 electron in the third energy level. What is the number of core electrons

Considering the definition of electronic configuration and valence shell, the number of core electrons is 1, because the valance shell is the third energy level.

The Electronic Configuration of the elements is the arrangement of all electrons of an element in energy levels and sub-levels (orbitals).  

There are 7 energy levels, numbered from 1 to 7, and in which electrons are distributed, logically in order according to their energy level. Electrons with less energy will be spinning at level 1.

Each level is divided into sub-levels. These sub-levels into which each level is divided can be up to 4. These 4 sub-levels are called: s, p, d, f.

Aufbau's principle is a principle of atomic physics, which explains how electrons are accommodated in their orbits around the nucleus of the atom.

This principle states that the filling of these orbitals occurs in increasing order of energy, that is, from the orbitals of lower energy to those of higher energy.

The simplest of the sub-levels is 1s. If you fill the 1s you can start filling the 2s. After the 2s sub-level is satisfied, the 2p can begin to fill and so on.  

The valence shell is the one where the outermost electrons are found, those electrons that are in the last level. Because it is the outermost shell, it is also the farthest from the atomic nucleus.

The valence shell is one whose electrons are responsible for the chemical properties of an element.

In this case, an atom has 2 electrons in the first energy level, 8 electrons in the second energy level, and 1 electron in the third energy level. So, the number of core electrons is 1, because the valance shell is the third energy level.

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