Understanding Quantum Numbers for Potassium's Valence Electron

Quantum numbers are a set of four numbers that describe the unique state of an electron in an atom. They provide information about the electron's energy level, shape of the orbital, orientation of the orbital in space, and its spin.

The question asks for the four quantum numbers of the valence electron of a potassium atom. To find this, we first need to determine the electronic configuration of potassium.

Electronic Configuration of Potassium (K)

Potassium (K) has an atomic number of 19, meaning it has 19 electrons. The electronic configuration follows the Aufbau principle, Hund's rule, and the Pauli exclusion principle:

The electronic configuration of potassium is $1s^2 2s^2 2p^6 3s^2 3p^6 4s^1$.

Identifying the Valence Electron

The valence electron is the electron in the outermost shell (the shell with the highest principal quantum number). In the electronic configuration $1s^2 2s^2 2p^6 3s^2 3p^6 4s^1$, the outermost shell is the $n=4$ shell, which contains one electron in the $4s$ orbital. Therefore, the valence electron is the electron in the $4s^1$ orbital.

Determining the Four Quantum Numbers ($n, l, m_l, m_s$)

Now, let's determine the four quantum numbers for the valence electron in the $4s^1$ orbital:

• Principal Quantum Number ($n$): This number indicates the main energy shell. For an electron in the $4s$ orbital, the principal quantum number is the shell number, which is 4. So, $n=4$.
• Azimuthal or Angular Momentum Quantum Number ($l$): This number describes the shape of the orbital. It depends on the value of $n$. For a given $n$, $l$ can have integer values from 0 to $n-1$. Different values of $l$ correspond to different subshells:
  • $l=0$ corresponds to the s subshell (spherical shape).
  • $l=1$ corresponds to the p subshell (dumbbell shape).
  • $l=2$ corresponds to the d subshell.
  • $l=3$ corresponds to the f subshell.
  For the $4s$ orbital, the subshell is 's', which corresponds to $l=0$. So, $l=0$.
• Magnetic Quantum Number ($m_l$): This number describes the orientation of the orbital in space. It depends on the value of $l$. For a given $l$, $m_l$ can have integer values from $-l$ to $+l$, including 0. For $l=0$ (the s subshell), $m_l$ can only be 0 (from $-0$ to $+0$). So, $m_l=0$.
• Spin Quantum Number ($m_s$): This number describes the intrinsic angular momentum of the electron, also called spin. An electron can spin in one of two directions, represented by $+1/2$ or $-1/2$. For a single electron in an orbital (like $4s^1$), the spin quantum number is conventionally assigned as $+1/2$. So, $m_s = +1/2$.

Summary of Quantum Numbers

For the valence electron of the potassium atom in the $4s^1$ orbital, the four quantum numbers are:

• $n = 4$
• $l = 0$
• $m_l = 0$
• $m_s = +1/2$

Therefore, the four quantum numbers are (4, 0, 0, 1/2).