Problem 62 - Chemistry and Chemical Reactivity

Question

How many orbitals correspond to each of the following designations? (a) \(3 p\) (b) \(4 p\) (c) \(4 p_{x}\) (d) \(6 d\) (e) \(5 d\) (f) \(5 f\) (g) \(n=5\) (h) \(7 s\)

Step-by-Step Solution

Answer

(a) 3; (b) 3; (c) 1; (d) 5; (e) 5; (f) 7; (g) 16; (h) 1.

Step 1: Understanding Quantum Numbers

To solve the exercise, we need to recall that orbitals are defined by the principal quantum number \(n\) and the azimuthal quantum number \(l\). The total number of orbitals in a subshell is given by \(2l+1\).

Step 2: Calculate Orbitals for 3p

For the designation \(3p\), the principal quantum number \(n=3\) and \(l=1\) for a \(p\) subshell. Therefore, the number of orbitals is \(2(1) + 1 = 3\).

Step 3: Calculate Orbitals for 4p

For the designation \(4p\), \(n=4\) and \(l=1\) for a \(p\) subshell. Thus, the number of orbitals is also \(2(1) + 1 = 3\).

Step 4: Calculate Orbitals for 4px

The designation \(4p_{x}\) specifies one orbital within the \(p\) subshell. So, there is exactly 1 orbital.

Step 5: Calculate Orbitals for 6d

For the designation \(6d\), \(n=6\) and \(l=2\) for a \(d\) subshell. The number of orbitals is \(2(2) + 1 = 5\).

Step 6: Calculate Orbitals for 5d

For the designation \(5d\), \(n=5\) and \(l=2\) for a \(d\) subshell. Thus, the number of orbitals is \(2(2) + 1 = 5\).

Step 7: Calculate Orbitals for 5f

For the designation \(5f\), \(n=5\) and \(l=3\) for an \(f\) subshell. Hence, the number of orbitals is \(2(3) + 1 = 7\).

Step 8: Calculate Orbitals for n=5

When \(n=5\), we consider all subshells: \(s\), \(p\), \(d\), and \(f\). There are: 1 orbital (\(s\)), 3 orbitals (\(p\)), 5 orbitals (\(d\)), and 7 orbitals (\(f\)). Total = \(1 + 3 + 5 + 7 = 16\) orbitals.

Step 9: Calculate Orbitals for 7s

For the designation \(7s\), \(n=7\) and \(l=0\) for an \(s\) subshell. Hence, the number of orbitals is \(2(0) + 1 = 1\).

Key Concepts

Orbitals

In the realm of quantum chemistry, "orbitals" are regions around an atom's nucleus where electrons are most likely to be found. Orbitals are not fixed paths but rather cloud-like regions that have certain shapes and orientations.

• Each orbital can hold two electrons with opposite spins.
• Orbitals are categorized as s, p, d, or f, each with distinct shapes.

S orbitals have a spherical shape, while p orbitals are dumbbell-shaped. D and f orbitals have more complex shapes. These shapes are determined by the mathematical equations of quantum mechanics, and they help predict chemical bonding behavior.
Understanding orbitals is crucial since they explain how elements form compounds and the arrangement of electrons influences an element's chemical properties.

Subshells

Subshells are subsets of electron energy levels, encompassing groups of orbitals that share the same principal and azimuthal quantum numbers. Each type of subshell – s, p, d, and f – is defined by a specific shape and number of orbitals.

• An s subshell contains one orbital.
• A p subshell contains three orbitals.
• A d subshell contains five orbitals.
• An f subshell contains seven orbitals.

The arrangement of subshells and the distribution of electrons among them follows the "Aufbau principle," which dictates that electrons fill lower energy subshells first before moving to higher energy ones. This structure explains elements' periodic table position, predicting reactivity and properties.

Principal Quantum Number

The principal quantum number, denoted as "n," indicates the main energy level of an electron within an atom.

• It defines the size and energy of an atomic orbital.
• Larger n values correspond to orbitals further from the nucleus and higher energy.

The principal quantum number can be a positive integer (1, 2, 3, ...). As n increases, the electron cloud becomes larger, meaning electrons are more spread out and at higher energy levels. This quantum number is a core aspect of the Bohr atomic model, elaborating on hydrogen-like atoms’ electron arrangements and energy levels.

Azimuthal Quantum Number

The azimuthal quantum number, "l," describes the shape of an electron's orbital and helps distinguish subshells within a principal energy level. It is sometimes called the angular momentum quantum number.

• Values of l range from 0 to n-1, where n is the principal quantum number.
• Each number corresponds to a specific subshell type: l = 0 (s), l = 1 (p), l = 2 (d), l = 3 (f).

This quantum number plays a significant role in determining the "shape" of orbitals, influencing interaction forces between electrons. It is key in quantum mechanics to understand molecular geometry, chemical bonding, and spectral lines produced by atoms.