Question

Which of the following quantum numbers are correct for the outermost electron of sodium atom? (a) \(n=4, l=0, m=0, s=+1 / 2\) (b) \(n=3, I=0, m-0, s--1 / 2\) (c) \(n=3, l=1, m=+1, s=+1 / 2\) (d) \(n=3, l=2, m=-1, s=-1 / 2\)

Short answer

The correct quantum numbers for the outermost electron of a sodium atom are given in option (b) n=3, l=0, m=0, s=-1/2.

Step-by-step solution

Determine Sodium's Electron Configuration

The first step is to recall the electron configuration of sodium (Na), which has an atomic number of 11. The electron configuration ends with the 3s orbital having one electron. Thus, the quantum numbers for the outermost electron (the 3s electron) would have the principal quantum number () corresponding to the 3rd shell, the azimuthal (angular momentum) quantum number (l) for an s orbital which is 0, the magnetic quantum number (m) that can range from -l to l, for an s orbital this is 0, and the spin quantum number (s) which can be either +1/2 or -1/2.

Match with Given Options

Compare the correct quantum numbers for the outermost electron of sodium with the given options. The correct set of quantum numbers should be n=3 (the third energy level), l=0 (indicating an s orbital), m=0 (the only value for an s orbital), and s can be either +1/2 or -1/2. Therefore, any option that fits these criteria is correct.

Identify the Correct Option

Looking at the given options, the only one that exactly matches these quantum numbers is (b) n=3, l=0, m=0, s=-1/2. Note that even though s can be +1/2, option (a) is incorrect because it has n=4, which does not correspond to the outermost electron of sodium.

Key concepts

Electron Configuration

Understanding electron configuration is crucial when studying the properties of atoms. For sodium, which has 11 electrons, the configuration follows a specific pattern according to the periodic table. Most importantly, the electron configuration is written using the principle of lowest energy filling first.

The electron configuration starts at the lowest energy levels: 1s, 2s, then moves through the various p, d, and f orbitals corresponding to the increasing principal quantum number, n. Within each energy level, electrons fill the s orbital first, then p, and so forth, following the pattern outlined by the Aufbau principle. Moreover, each orbital can only hold a certain number of electrons (for instance, s orbitals can hold 2), which further dictates how these electrons are organized.

• 1s can hold two electrons
• 2s is next, also holding two
• When we reach 3s (where sodium’s outermost electron resides), it holds one electron

In the case of sodium’s outermost electron, the correct electron configuration is 3s¹, which directs us to the other quantum numbers for a full description.

Azimuthal Quantum Number

The azimuthal quantum number (), also known as the angular momentum quantum number, signifies the subshell or type of orbital where an electron can be found. It is integral to determining the shape of the orbital and has values that range from 0 to - 1. These numbers correspond to the s, p, d, and f orbitals, with s being 0, p being 1, d being 2, and f being 3.

For the sodium atom's outermost electron, which is in the 3s orbital, the azimuthal quantum number is 0, since s orbitals are spherical in shape and are the first in the sequence of orbital shapes within any principal energy level.

Magnetic Quantum Number

The magnetic quantum number (m) provides information about the orientation of an orbital within a subshell and indirectly relates to the magnetic field the electron may experience. It can take on any integral value between - and +, including zero. For s orbitals, however, because they are spherically symmetrical, this number is always 0, reflecting that the s orbital has only one orientation in space.

Thus, for sodium’s 3s electron, the magnetic quantum number is indeed 0. This value is critical for fully understanding and predicting the behavior of electrons within a magnetic field, which directly correlates with their spatial orientation.

Spin Quantum Number

Finally, the spin quantum number (s) describes the intrinsic angular momentum of an electron, essentially its 'spin'. Electrons act as if they are spinning on an axis and can have one of two possible spins: +1/2 or -1/2. This quantum number is fundamental in explaining the Pauli Exclusion Principle, which states that no two electrons in an atom can have the same set of four quantum numbers.

Since the 3s orbital in sodium has only one electron, it can exhibit either spin. In practice, this means that either +1/2 or -1/2 is acceptable when we are talking about a solitary electron in an orbital, as long as the electron does not share the same four quantum numbers as another electron within the atom.