Question

Question #2d40e

Answer 1

`"6 e"^(-)`

Explanation:

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`.

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.

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.