Problem 97 - Chemistry The Molecular Nature of Matter

Question

What are the values of \(n\) and \(\ell\) for the subshells: (a) \(3 s\) (b) \(5 d ?\)

Step-by-Step Solution

Answer

For (a) '3s', the values of 'n' and '\(\ell\)' are 3 and 0, respectively. For (b) '5d', the values of 'n' and '\(\ell\)' are 5 and 2, respectively.

Step 1: Identify the Principal Quantum Number

The principal quantum number, denoted as 'n', represents the energy level or shell of an electron in an atom. It is directly indicated by the number that precedes the letter representing the subshell. For subshell (a) '3s', the '3' represents the value of 'n'. Similarly, for subshell (b) '5d', the '5' represents the value of 'n'.

Step 2: Identify the Azimuthal (Angular Momentum) Quantum Number

The azimuthal quantum number, denoted as '\(\ell\)', represents the subshell or orbital type. The value of '\(\ell\)' is determined by the letter of the subshell: s, p, d or f, which correspond to values 0, 1, 2, and 3, respectively. For subshell (a) 's', '\(\ell\)' is 0. For subshell (b) 'd', '\(\ell\)' is 2.

Key Concepts

Principal Quantum Number

The principal quantum number, symbolized by 'n', is fundamental to understanding the structure of an atom. It indicates the relative size and energy of atomic orbitals and is one of the four quantum numbers needed to completely describe the state of an electron.

Put simply, think of the principal quantum number as telling us which floor an electron lives on in the 'hotel' of an atom. A higher 'n' means the electron is in a higher energy level, further away from the nucleus, similar to being on a higher floor. Electrons in lower 'n' values (lower floors) are more tightly bound to the nucleus, and as 'n' increases, so does the energy of the electron.

For the subshell '3s', the 3 directly shows us that 'n' is 3, meaning the electron is in the third energy level. Likewise, '5d' places the electron in the fifth energy level, where 'n' is 5. It's crucial to note that as 'n' increases, the electron's energy and distance from the nucleus both increase.

Azimuthal Quantum Number

Following the principal quantum number, we have the azimuthal (also called the angular momentum) quantum number, designated by 'l'. This quantum number provides information about the shape of an electron's orbital, and consequently, the type of subshell in which the electron resides.

To make sense of it, imagine each floor of our 'atomic hotel' has different rooms, each with distinct shapes. These 'rooms' are subshells, and they're labeled with charming letters: s, p, d, and f. Each letter correlates to an azimuthal quantum number value: s corresponds to 0, p to 1, d to 2, and f to 3. These numbers and letters reveal the orbital's shape where the s is spherical, p is dumbbell-shaped, and so on.

For our example, the 's' in '3s' tells us that 'l' is 0, indicating a spherical shape. The 'd' in '5d' means 'l' is 2, which signifies a more complex, clover-shaped orbital. The value of 'l' also dictates the number of angular nodes in the orbital – regions where the probability of finding an electron is zero.

Subshell Notation

Subshell notation is the language we use to describe the address of electrons within an atom. It's a combination of numbers and letters that precisely locates electrons by naming the energy level and the shape of their orbitals.

Every subshell notation starts with a number, representing the principal quantum number, 'n', that we've discussed. This is followed by a letter that indicates the azimuthal quantum number, 'l', revealing the type of subshell. As mentioned, these are s (0), p (1), d (2), or f (3).

Decoding the Notation

If you're given a subshell notation such as '4p', you can easily decode it: '4' is the principal quantum number suggesting the electron is on the fourth floor, and 'p' tells you it's in a room shaped like a dumbbell. These notations are critical because they allow chemists and physicists to discuss complex electron behaviors and interactions in a simple, standardized manner.

Atomic Orbitals

Atomic orbitals are regions around the nucleus of an atom where there is a high probability of finding an electron. Each orbital can be uniquely identified using the principal and azimuthal quantum numbers we've explored.

Analogous to clouds with different shapes, atomic orbitals come in various forms, including spherical (s), dumbbell (p), clover (d), and more complex (f). These shapes aren't just for looks; they have profound implications for the way an atom bonds and interacts with other atoms.

Orbitals and Their Capacities

It's important to understand that each type of atomic orbital has a specific capacity for electrons. An 's' orbital can hold up to 2 electrons, 'p' can hold 6, 'd' can hold 10, and 'f' can hold 14. The distribution of electrons among these orbitals explains the periodic table's structure and the foundation of chemical bonding. Additionally, the complex nature of orbital hybridization and interactions leads to the rich diversity of chemical compounds observed in nature.