Animated Solution for Chemistry - Atomic Structure: The correct set of four quantum numbers for the valence electrons of rubidium atom $[Z=37]$ is

Imagine you are tasked with finding the exact "address" of the outermost electron in a Rubidium atom. In the quantum world, this address is given by a set of four quantum numbers. To find this address, we first need to know where the electron lives, which means writing down the electronic configuration of Rubidium.

Rubidium has an atomic number of $Z=37$. Writing the full configuration from scratch can be tedious, so we use the noble gas core method. The noble gas just before Rubidium is Krypton (Kr), which accounts for $36$ electrons.

This leaves us with just one electron to place. Following the Aufbau principle, after the $4p$ subshell is filled (which completes the Krypton core), the next available energy level is the $5s$ orbital. Therefore, the electronic configuration of Rubidium is:

Decoding the Quantum Numbers

Now that we know our valence electron resides in the $5s$ orbital, let's break down its quantum address step-by-step.

1. Principal Quantum Number ($n$) The principal quantum number tells us the main energy level or shell. Since our electron is in the $5s$ subshell, the shell number is $5$.

2. Azimuthal Quantum Number ($l$) The azimuthal quantum number defines the shape of the subshell. The values of $l$ are fixed for each type of orbital: $s=0$, $p=1$, $d=2$, and $f=3$. Since we are dealing with an $s$-orbital, the value is strictly zero.

3. Magnetic Quantum Number ($m$) The magnetic quantum number describes the spatial orientation of the orbital. Its value depends on $l$ and ranges from $-l$ to $+l$. Because $l=0$, the only possible value for $m$ is zero. This makes sense because a spherical $s$-orbital has only one orientation in space.

4. Spin Quantum Number ($s$) Finally, the spin quantum number describes the intrinsic spin of the electron. A single electron in an orbital can have a spin of either $+\frac{1}{2}$ (spin-up) or $-\frac{1}{2}$ (spin-down). Both are physically valid.

Bringing it all together, the complete set of four quantum numbers for the valence electron of Rubidium is:

Looking at our given options, this perfectly matches option (a). The beauty of quantum numbers lies in their strict, logical rules—once you know the orbital, the rest of the address naturally falls into place!