Problem 80 - Chemistry and Chemical Reactivity

Question

Suppose you live in a different universe where a different set of quantum numbers is required to describe the atoms of that universe. These quantum numbers have the following rules: \(N,\) principal \(\quad 1,2,3, \ldots, \infty\) \(L,\) orbital \(\quad=N\) \(M,\) magnetic \(\quad-1,0,+1\) How many orbitals are there altogether in the first three electron shells?

Step-by-Step Solution

Answer

There are 9 orbitals in the first three electron shells.

Step 1: Identify Principal Quantum Numbers

First, determine the principal quantum numbers \(N\) for the first three electron shells. These are \(N = 1, 2, 3\).

Step 2: Determine Orbital Quantum Numbers

According to the given rule, \(L = N\). So for each \(N\):- For \(N = 1, L = 1\).- For \(N = 2, L = 2\).- For \(N = 3, L = 3\).

Step 3: Determine Magnetic Quantum Numbers

The magnetic quantum number \(M\) can be \(-1, 0, +1\) regardless of \(N\) or \(L\). This means for each \(N\), each orbital can have 3 different \(M\) values.

Step 4: Calculate Total Orbitals for Each Shell

For each shell defined by \(N\), determine the number of orbitals.- For \(N = 1\), \(L = 1\), we have 3 orbitals corresponding to \(M = -1, 0, +1\).- For \(N = 2\), \(L = 2\), we also have 3 orbitals.- For \(N = 3\), \(L = 3\), we again have 3 orbitals.

Step 5: Sum the Orbitals

Add the number of orbitals from each shell:- Shell \(N = 1\): 3 orbitals- Shell \(N = 2\): 3 orbitals- Shell \(N = 3\): 3 orbitalsTotal = 3 + 3 + 3 = 9 orbitals.

Key Concepts

Principal Quantum Number

The concept of the principal quantum number is a key element in understanding how electrons are organized within an atom. In any universe, including the alternate one described in the exercise, the principal quantum number, denoted as \( N \), is the primary identifier of an electron shell. It represents the energy level or shell in which an electron resides, beginning from \( N = 1 \) and theoretically progressing to infinity.

• Each increase in \( N \) signifies a new, higher energy level.
• As \( N \) increases, the size of the electron shell (and, thus, the atom itself) increases, allowing more electrons to be held within that shell.
• In this fictional universe, each principal quantum number directly dictates the value of the orbital quantum number \( L \).

Recognizing the importance of \( N \) helps us understand how electron shells structure themselves around the nucleus, forming the basis upon which the whole atom is assembled.

Magnetic Quantum Number

The magnetic quantum number is another critical quantum number that specifies the orientation of electron orbitals within a given subshell. In the diverse universe of the exercise, the magnetic quantum number \( M \) can be \(-1, 0, +1\) each corresponding to a different orientation in space.

• Each value of \( M \) allows electrons to occupy different spatial orientations. This influences the magnetic and chemical properties of the atom.
• The variations in \( M \) do not depend on \( N \) or \( L \), which makes it a unique identifier of orientation within the same energy level and orbital.

Understanding \( M \) is pivotal, as it determines not only how electrons diversify within an energy level but also how an atom interacts with external magnetic fields.

Electron Orbitals

Electron orbitals define where an electron is likely to be found within an atom. These orbitals arise from the limitations imposed upon electrons by the quantum numbers \( N, \) \( L, \) and \( M \). Each orbital houses electrons in a cloud-like probability zone, indicating areas of high electron density.

• For a given \( N \), each \( L \) value corresponds to a unique shape of the orbital (like spherical, dumbbell, etc.).
• The variations in \( M \), as already noted, create differing orientations of these orbitals within the same subshell.
• In our hypothetical universe, three orbitals are possible within each \( N \) level, determined by the values of \( M \).

Thus, orbitals are fundamental constructs that dictate electron placement, atom shape, and how atoms engage with each other. The diversity in electron orbital configuration nurtures the complex behavior and interactions among atoms, forming the very building blocks of matter.