Problem 7 - Chemistry: Introducing Inorganic, Organic and Physical Chemistry

Question

How many orbitals are possible for \(n=5 ?\) Identify the orbital types giving the number of each. (Section 3.5 )

Step-by-Step Solution

Answer

For \(n=5\), there are 25 orbitals: 1 s, 3 p, 5 d, 7 f, and 9 g.

Step 1: Understand the Principal Quantum Number

The principal quantum number, denoted as \(n\), indicates the main energy level occupied by the electron in an atom. For any given value of \(n\), it can tell us about the shell of the atom where electrons reside.

Step 2: Calculate Total Number of Orbitals

The total number of orbitals for any principal quantum number \(n\) is given by \(n^2\). For \(n = 5\), the total number of orbitals is \(5^2 = 25\).

Step 3: List Possible Subshells

For a given \(n\), the possible subshells are determined by the azimuthal quantum number \(l\), where \(l\) ranges from \(0\) to \(n-1\). Thus, for \(n=5\), \(l = 0, 1, 2, 3, \) and \(4\) corresponding to s, p, d, f, and g subshells respectively.

Step 4: Determine Number of Orbitals per Subshell

The number of orbitals per subshell is determined by the formula \(2l + 1\). - For \(l = 0\), the s subshell has \(2(0) + 1 = 1\) orbital.- For \(l = 1\), the p subshell has \(2(1) + 1 = 3\) orbitals.- For \(l = 2\), the d subshell has \(2(2) + 1 = 5\) orbitals.- For \(l = 3\), the f subshell has \(2(3) + 1 = 7\) orbitals.- For \(l = 4\), the g subshell has \(2(4) + 1 = 9\) orbitals.

Step 5: Summarize the Results

For \(n = 5\), there are 25 orbitals split into the subshells as follows:- 1 s orbital- 3 p orbitals- 5 d orbitals- 7 f orbitals- 9 g orbitals

Key Concepts

Principal Quantum Number

The principal quantum number, symbolized by \( n \), is fundamental in understanding atomic structure. It describes the primary energy level that an electron occupies within an atom.
The principal quantum number can be thought of as the "floor" of a building where an electron resides, with the ground floor being \( n=1 \), and each higher floor representing higher energy levels.

• A higher \( n \) value indicates that the electron is further from the nucleus and in a higher energy state.
• The principal quantum number also helps determine the number of orbitals and the overall size of the electron's wave function.
• In mathematical terms, the number of orbitals present in a shell is given by the expression \( n^2 \). So, for \( n = 5 \), there are \( 5^2 = 25 \) orbitals.

Understanding \( n \) is key to navigating the quantum world where electrons roam.

Azimuthal Quantum Number

The azimuthal quantum number, commonly denoted as \( l \), plays a vital role in defining the shape and type of subshells within an energy level.
It determines the orbital angular momentum and describes the subshells present for a given \( n \). Each \( l \) value corresponds to a specific subshell type:

• \( l = 0 \) for s subshell
• \( l = 1 \) for p subshell
• \( l = 2 \) for d subshell
• \( l = 3 \) for f subshell
• \( l = 4 \) for g subshell

For a particular principal quantum number \( n \), \( l \) can take on values from \( 0 \) to \( n-1 \). This variety allows for different shapes and capacities in the shells, accommodating more electrons as \( n \) increases.

Orbital Shapes

Orbital shapes are defined by the azimuthal quantum number \( l \) and are crucial for visualizing where electrons are likely to be found in an atom.
Each type of subshell exhibits a unique shape:

• The s subshell (\( l = 0 \)) has a spherical shape, providing a uniform distribution around the nucleus.
• The p subshell (\( l = 1 \)) has a dumbbell shape, with lobes oriented along specific axes.
• The d subshell (\( l = 2 \)) features more complex shapes like cloverleafs or double dumbbells.
• The f subshell (\( l = 3 \)) is even more complex, having between 6 and 8 lobes.

Understanding these shapes is fundamental to predicting how atoms interact with each other during chemical bonding.

Subshells

Subshells, related to the azimuthal quantum number \( l \), dictate the structural layout within an atom's electron cloud. Each principal energy level can contain several subshells, adding layers of complexity.
For \( n = 5 \), there are five subshells: s, p, d, f, and g. Let's consider how these are distributed:

• The s subshell (\( l = 0 \)) consists of 1 orbital.
• The p subshell (\( l = 1 \)) consists of 3 orbitals.
• The d subshell (\( l = 2 \)) has 5 orbitals.
• The f subshell (\( l = 3 \)) has 7 orbitals.
• The g subshell (\( l = 4 \)) features 9 orbitals.

Each subshell can hold two electrons per orbital, which explains the arrangement of electrons and contributes to the atom's unique chemical properties. This tiered structure ensures electrons maintain organized spacing, adhering to Pauli’s exclusion principle and Hund’s Rule.