The possible correct set of quantum numbers for the unpaired electron of \(\mathrm{Cl}\) atom is : (a) \(2,0,0,+\frac{1}{2}\) (b) \(2,1,-1,+\frac{1}{2}\) (c) \(3,1,1, \pm \frac{1}{2}\) (d) \(3,0, \pm \frac{1}{2}\)

The principal quantum number, denoted as 'n', is one of the four key quantum numbers used to describe the unique quantum state of an electron in an atom. It defines the energy level on which an electron resides, and it can take any positive integer value starting from 1, 2, 3, and so forth. As 'n' increases, the electron's energy and its most probable distance from the nucleus also increase. This is crucial for understanding the electronic structure of atoms and the distribution of electrons across different energy levels.

For instance, in the case of a chlorine atom, when 'n=3', it indicates that the unpaired electron we are interested in is located in the third energy level, which is key to figuring out the atom's electron configuration and chemical properties.

The azimuthal quantum number, symbolized by 'l', determines the shape of the electron's orbital, which is essentially the cloud around the nucleus where an electron is likely to be found. The value of 'l' can range from 0 to (n-1), where 'n' is the principal quantum number. Each value of 'l' corresponds to a different type of orbital; 0 represents an s orbital, 1 represents a p orbital, 2 for a d orbital, and 3 for an f orbital.

In the exercise, a correct 'l' value for the unpaired electron in a chlorine atom should be associated with a p orbital, because we know the electron is in the '3p' orbital. Therefore, 'l' must be equal to 1.

The magnetic quantum number, labelled as 'ml', indicates the orientation of the electron's orbital around the nucleus and can have integer values between -l and +l, including zero. Each value of 'ml' stands for a specific orbital within a particular subshell (designated by 'l'). For example, when 'l=1' for a p orbital, 'ml' can be -1, 0, or +1, corresponding to the three different orientations the p orbital can have in three-dimensional space.

For a chlorine atom with an unpaired electron in the 3p orbital, we must choose a value for 'ml' that falls within the correct range for a p orbital, that is -1, 0, or +1.

Now let's delve into the spin quantum number denoted as 'ms'. This quantum number is related to the intrinsic angular momentum of an electron. Electron spin is a fundamental property much like charge or mass. There are only two possible values for 'ms': +1/2 or -1/2, which represent the two possible spin orientations of the electron ('up' or 'down'). This spin feature also plays a role in the Pauli exclusion principle, which states that no two electrons in an atom can have the same set of four quantum numbers.

Thus, for the unpaired electron in a chlorine atom, 'ms' can be either +1/2 or -1/2, which represents its two possible spin states.

Electron configuration provides a layout of how electrons are distributed among different orbitals around the nucleus of an atom. The 'building-up' (or 'Aufbau') principle guides us here, suggesting that lower energy orbitals are filled before higher energy ones. The notation follows a pattern where we list the principal quantum number and the orbital type (s, p, d, f), followed by a superscript indicating the number of electrons in those orbitals.

As in the given exercise, the chlorine atom has 17 electrons with an electron configuration of '1s2 2s2 2p6 3s2 3p5'. This tells us exactly where the electrons are and which orbitals still have space, information which is vital while predicting chemical reactions and bond formation.

Valence electrons are the electrons in the outermost shell of an atom; they play a key role in chemical reactivity and bonding. The number of valence electrons is determined by the electron configuration of an atom, especially the electrons occupying the s and p orbitals of the highest principal quantum number. For example, in chlorine ('1s2 2s2 2p6 3s2 3p5'), it's the 3s and 3p electrons that count as valence electrons, being responsible for the typical properties of chlorine including its reactivity and the types of bonds it forms.

Understanding quantum numbers and electron configuration isn't just a topic of interest for chemistry enthusiasts but is also a critical aspect of Physical Chemistry for the Joint Entrance Examination (JEE) in India. This highly competitive exam assesses knowledge across various chemistry concepts, and a firm grip on quantum numbers can bolster a student's ability to solve complex problems. Integration of these ideas, like identifying unpaired electrons and determining electron configurations, is essential not only for textbook exercises but also for succeeding in exams like JEE, which demands a deep and intuitive understanding of chemistry fundamentals.