Question

Designate the orbital that fits each of the following descriptions: (a) a spherical orbital in the sixth energy level (b) a dumbbell-shaped orbital in the third energy level

Short answer

(a) 6s orbital; (b) 3p orbital.

Step-by-step solution

Identify the Shape of Orbitals

Orbitals are different regions around a nucleus where electrons are most likely to be found. The possible shapes of orbitals are 'spherical' (s) and 'dumbbell-shaped' (p). For spherical orbitals, we use 's', and for dumbbell-shaped orbitals, we use 'p'.

Determine the Principal Quantum Number for Part (a)

In part (a), we're given a 'spherical orbital in the sixth energy level'. The principal quantum number, denoted by 'n', represents the energy level. So for the sixth energy level, we have n = 6.

Match the Orbital Shape for Part (a)

Since part (a) mentions a 'spherical orbital', this corresponds to an 's' orbital. Therefore, the orbital is 6s. An 's' orbital is always spherical, regardless of the principal quantum number.

Determine the Principal Quantum Number for Part (b)

In part (b), we need a 'dumbbell-shaped orbital in the third energy level'. The principal quantum number for the third energy level is n = 3.

Match the Orbital Shape for Part (b)

A 'dumbbell-shaped orbital' refers to a 'p' orbital. Therefore, the orbital is 3p. In this case, the 'p' orbital corresponds to an energy level of 3, making it 3p.

Key concepts

Quantum Numbers

Quantum numbers are essential in describing the position and properties of electrons in an atom. There are four distinct quantum numbers: principal (\( n \)), angular momentum (\( l \)), magnetic (\( m_l \)), and spin (\( m_s \)). Each number provides unique information about the electron's quantum state:

• The **principal quantum number** (\( n \)) indicates the energy level and size of the orbital. For instance, an orbital with \( n=1 \) is closer to the nucleus and lower in energy than one with \( n=2 \).
• The **angular momentum quantum number** (\( l \)) defines the shape of the orbital. It can have integer values from 0 to \( n-1 \). When \( l=0 \), the shape is spherical (s); when \( l=1 \), it's dumbbell-shaped (p).
• The **magnetic quantum number** (\( m_l \)) describes the orientation of the orbital in space, with possible values ranging from \( -l \) to \( +l \).
• The **spin quantum number** (\( m_s \)) accounts for the electron's spin and can be either +1/2 or -1/2, signifying two possible spin orientations.

Understanding quantum numbers is crucial for predicting the behavior and arrangement of electrons in atoms.

Orbital Shapes

Orbital shapes are integral to understanding the three-dimensional space around a nucleus where electrons are likely found. Different shapes indicate different types of orbitals:

• **S orbitals** are spherical. This means that no matter which way you turn an s orbital, it always looks the same. They occur in all energy levels starting from \( n = 1 \).
• **P orbitals**, on the other hand, are dumbbell-shaped, resembling two lobes on either side of the nucleus. Each energy level from \( n=2 \) and above has three p orbitals, oriented along three different axes.

The shape of an orbital affects the distribution of electron density and can influence chemical bonding and properties. For example, the spherical nature of an s orbital leads to different interactions when forming bonds compared to the directional nature of p orbitals.

Energy Levels

Energy levels denote the fixed or quantized distances of electrons from the nucleus. These levels are indicated by the principal quantum number \( n \). As \( n \) increases, the electron's distance from the nucleus increases as well, meaning the electron occupies a higher energy level:

• A **lower energy level** (like \( n=1 \) or \( n=2 \)) indicates electrons that are closer to the nucleus. These electrons are more strongly attracted to the nucleus and less energetic.
• **Higher energy levels** (such as \( n=3 \) and beyond) refer to electrons that are further from the nucleus and have higher energy. These electrons experience lower nuclear attraction and higher potential to involve in chemical bonding or excitation.

Energy levels are occupied according to the Aufbau principle, which guides electrons to fill lower energy orbitals first before moving to higher ones.Understanding energy levels is key to interpreting an atom's electron configuration, predicting chemical reactivity, and explaining the basis of the periodic table.