Question

Few electrons have following quantum numbers, (i) \(n=4, l=1\) (ii) \(n=4, l=0\) (ii) \(n=3, l=2\) (iv) \(n=3, l=1\) Arrange them in the order of increasing energy from lowest to highest. (a) (iv) \(<\) (ii) \(<(\mathrm{iii})<(\mathrm{i})\) (b) (ii) \(<(\mathrm{iv})<(\mathrm{i})<\) (iii) (c) \((\mathrm{i})<(\mathrm{iii})<(\mathrm{ii})<(\mathrm{iv})\) (d) (iii) \(<(\mathrm{i})<(\mathrm{iv})<(\mathrm{ii})\)

Short answer

(b) (ii) < (iv) < (i) < (iii)

Step-by-step solution

Recall the Quantum Numbers

Quantum numbers describe the properties of atomic orbitals and the electrons in those orbitals. The principal quantum number (n) specifies the energy level of the electron. The azimuthal or angular momentum quantum number (l) defines the shape of the orbital and influences energy within a given n-level, where for a given n, lower l values typically correspond to lower energy sublevels.

Understand the Energy Order for Different Orbitals

Generally, the energy of an electron in an atom increases with increasing n and l values. However, the n value has a greater influence on energy than l. Within the same n value, the energy increases with l. For electrons in different n levels, the one with the lower n value will have lower energy, regardless of l.

Arrange the Given Electrons in Order of Increasing Energy

Following the general rule: (ii) with n=4, l=0 has the lowest energy due to the smallest l within the same n level. Next is (iv) with n=3, l=1, because even though it has a smaller n value, the increased l-value still makes it lower energy than i and iii. (i) with n=4, l=1 has higher energy than ii because of the higher l value. Finally, (iii) with n=3, l=2 has the highest energy among these because it has the largest l value. Therefore, the electrons should be arranged as (ii) < (iv) < (i) < (iii).

Key concepts

Principal Quantum Number

The principal quantum number, represented by the symbol 'n,' is a fundamental concept in quantum mechanics, particularly in the context of atomic physics. It specifies the energy level of an electron within an atom and determines the average distance of an electron from the nucleus. The larger the value of 'n,' the higher the energy level, and consequently, the farther the electron is likely to be from the nucleus.

Quantum numbers are like addresses that give the location of electrons in an atom. If you think of an atom as a multi-story building, the principal quantum number would tell you the floor on which the electron 'lives.' Each increase in the principal quantum number signifies a new 'shell' or energy level, and each shell can hold more electrons than the one before it. As 'n' increases, electrons inhabit spaces that allow them to have more energy; this often makes them less tightly bound to the atom's nucleus.

In the aforementioned exercise, electrons are compared based on this number. The comparison becomes crucial in understanding how electrons fill the atomic orbitals and in predicting the chemical and physical properties of elements. Understanding 'n' is the first step in unraveling the complex behavior of electrons within an atom.

Azimuthal Quantum Number

The azimuthal quantum number, denoted by 'l,' is also known as the angular momentum quantum number. It determines the shape of the atomic orbital and has significance for the energy within a principal quantum level ('n'). The values of 'l' range from 0 to (n-1) for each principal quantum number. Consequently, for every 'n,' there are 'n' possible values for 'l' (0 to n-1).

Within an energy level, the azimuthal quantum number divides the shells into subshells labeled as 's' (when l=0), 'p' (l=1), 'd' (l=2), and 'f' (l=3). These subshells are like different types of apartments on the same floor of our building analogy - each with a unique layout (orbital shape). The 's' orbitals are spherical, while 'p' orbitals are dumbbell-shaped, and so on.

The energy of an electron is affected by both 'n' and 'l' with energy typically increasing as these numbers increase. However, within the same energy level (same 'n'), orbitals with lower 'l' values (like 's' orbitals) will have lower energy compared to those with higher 'l' values (like 'p' or 'd' orbitals). Thus, considering the values of 'l' alongside 'n' helps to explain the arrangement of atomic orbitals and predict electron configurations in atoms.

Atomic Orbital Energy Levels

Atomic orbital energy levels represent the different states in which an electron within an atom can exist, having unique amounts of energy. Each electron in an atom occupies an orbital, and these orbitals are organized into different energy levels and sublevels based on the principal and azimuthal quantum numbers.

Understanding the concept of atomic orbital energy levels is like understanding a complex hierarchy within a building, where not only the floor level ('n') matters but also the particular room ('l') an electron occupies. Within a principal energy level, sublevels of orbitals exist that are associated with specific azimuthal quantum numbers. Electrons in lower energy orbitals (like the 1s orbital) are more tightly bound to the nucleus compared to those in higher energy orbitals (like the 2p orbital).
In the context of the given exercise, the energy levels of electrons determine their placement in the order of increasing energy. This arrangement is crucial, as it directly influences the atom’s chemical behavior, such as emission spectra and chemical bonding. Understanding the hierarchy of atomic orbital energy levels can help students predict the probable location of electrons and the types of chemical bonds an element might form.