Question #2d40e

You are asked to find the number of electrons located in a sodium atom that can have the spin quantum number equal to #+1/2#.

As you know, the spin quantum number, #m_s#, or simply #s#, denotes the spin of an electron inside an orbital and can take two possible values

• # m_s = +1/2 -># the electron has spin-up
• #m_s = -1/2 -># the electrons has spin-down

Now, an orbital can hold a maximum of #2# electrons of opposite spins, as stated by the Pauli Exclusion Principle. In other words, a fully-occupied orbital must hold an electron that has #m_s = +1/2# and an electron that has #m_s = -1/2#.

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This means that a starting point here would be to write the electron configuration for a neutral sodium atom and look for orbitals that are completely filled, since you know that

#color(red)(ul(color(black)("1 fully-occupied orbital" -> "1 electron with m"_s = +1/2)))#

The electron configuration for a neutral sodium atom looks like this -- for neutral atoms, the atomic number, which gives you the number pf protons located in the nucleus, also gives you the number of electrons that surround the nucleus

#"Na: " 1s^2 2s^2 2p^6 3s^1#

Here you have #Z = 11#, which means that the electron configuration must account for a total of #11# electrons

The electron configuration for a neutral sodium atom features

• #1s^2 -># the 1s orbital is completely filled
• #2s^2 -># the 2s orbital is completely filled
• #2p^6 = {(2p_x^2), (2p_y^2), (2p_z^2) :} -># all three 2p orbitals are completely filled
• #3s^1 -># the 3s orbital is only partially filled

So, you know that you have

#"5 fully-occupied orbitals" -> "5 electrons that have m"_s = +1/2#

Now, the #3s# orbital is only partially filled, meaning that it contains #1# electron. By convention, we designate #m_s = +1/2# for electrons that occupy an empty orbital, so the number of electrons that meet the required criterion becomes

#color(darkgreen)(ul(color(black)("5 e"^(-) + "1 e"^(-) = "6 e"^(-))))#

This is how the electron diagram for a neutral sodium atom would look like.

![https://commons.wikimedia.org]()

Here the electrons that have #m_s = +1/2# are shown using up arrows, #uarr#, and the electrons that have #m_s = -1/2# are shown using down arrows, #darr#.

As you can see, if we follow the aforementioned convention, we find that #6# electrons have spin-up in a neutral sodium atom.