Quantum Numbers of Nineteenth Electron in Chromium

To determine the correct set of four quantum numbers for the $19^{\text{th}}$ electron in Chromium ($\text{Cr}$, $Z = 24$) in accordance with the Aufbau principle, we consider the order of filling atomic orbitals based on increasing energy (the $(n+l)$ rule).

1. Sequence of Orbital Filling (Aufbau Principle)

The orbitals are filled in the following order: $1s \rightarrow 2s \rightarrow 2p \rightarrow 3s \rightarrow 3p \rightarrow 4s \rightarrow 3d \rightarrow \dots$

Let's count the electrons added sequentially to these orbitals:

1. $1s$ subshell: holds $2$ electrons $\implies \text{Electrons } 1 \text{ to } 2$
2. $2s$ subshell: holds $2$ electrons $\implies \text{Electrons } 3 \text{ to } 4$
3. $2p$ subshell: holds $6$ electrons $\implies \text{Electrons } 5 \text{ to } 10$
4. $3s$ subshell: holds $2$ electrons $\implies \text{Electrons } 11 \text{ to } 12$
5. $3p$ subshell: holds $6$ electrons $\implies \text{Electrons } 13 \text{ to } 18$
6. $4s$ subshell: holds the $19^{\text{th}}$ electron.

Thus, the $19^{\text{th}}$ electron enters the $4s$ orbital.

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2. Determining the Four Quantum Numbers for the $4s$ Orbital

• Principal Quantum Number ($n$): For the $4s$ orbital, $n = 4$.

• Azimuthal Quantum Number ($l$): For an $s$-subshell, $l = 0$.

• Magnetic Quantum Number ($m$): Since $l = 0$, the magnetic quantum number can only be $m = 0$.

• Spin Quantum Number ($s$): The first electron entering an orbital is assigned a spin quantum number of $s = +\frac{1}{2}$.

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Conclusion

The four quantum numbers for the $19^{\text{th}}$ electron are: $n = 4, \quad l = 0, \quad m = 0, \quad s = +\frac{1}{2}$

This matches Option D.