Question

A 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 \(1,2,3, \ldots, \infty\) \(L,\) orbital \(\quad=N\) \(M,\) magnetic \(\quad-1,0,+1\) How many orbitals are there altogether in the first three electron shells?

Short answer

There are 9 orbitals in the first three electron shells.

Step-by-step solution

Understanding Quantum Numbers

In this exercise, we are given different quantum numbers to describe orbitals in a hypothetical universe. The principal quantum number, \(N\), can be any positive integer starting from 1. The orbital quantum number, \(L\), equals \(N\) in this scenario, and the magnetic quantum number \(M\) can take values of -1, 0, and +1.

Identifying Orbits for Each Shell

For each value of \(N\), \(L = N\), there is only one value of \(L\), simplifying the orbital structure. For each \(N\), since \(M\) can be -1, 0, or +1, we have three possible orbitals per shell.

Counting Orbitals for Shells 1, 2, and 3

Since each shell \(N\) has exactly three orbitals (because \(M = -1, 0, +1\)), the number of orbitals per shell is constant:- For \(N=1\), there are 3 orbitals.- For \(N=2\), there are 3 orbitals.- For \(N=3\), there are 3 orbitals.

Calculating Total Orbitals for Three Shells

Adding the orbitals from each shell, we calculate the total number:- Total orbitals = (Orbitals for \(N=1\)) + (Orbitals for \(N=2\)) + (Orbitals for \(N=3\))- Total orbitals = 3 + 3 + 3 = 9.

Key concepts

Principal Quantum Number

In quantum mechanics, the principal quantum number is one of the four quantum numbers assigned to each electron in an atom to describe its state. In the given universe of the exercise, the principal quantum number, denoted as \(N\), represents the main energy level occupied by an electron. It is always a positive integer, starting from 1 and going up to infinity. This means the value of \(N\) determines the size and energy of the orbital—the larger \(N\), the larger and more energetic the orbital.

Within an atom, electrons occupy energy levels or shells that correspond to different principal quantum numbers:

• Level 1: \(N=1\)
• Level 2: \(N=2\)
• Level 3: \(N=3\)

Each shell can accommodate electrons, and generally, a higher principal quantum number indicates an electron is further from the nucleus, with higher energy.

It is important to notice that the principal quantum number directly impacts the other quantum numbers, because they all derive from the choice of \(N\).

Orbital Quantum Number

The orbital quantum number describes the shape of the electron's orbital and is denoted as \(L\). Typically, \(L\) can take on any integer value from 0 up to \(N-1\). However, in the specialized scenario of this imaginary universe, the orbital quantum number is equal to the principal quantum number: \(L = N\).

This specific rule implies that for any given principal quantum number value, there is only one corresponding orbital shape, greatly simplifying the possible configurations of electrons in this universe compared to our usual quantum mechanical reality.

In classical quantum theory, different \(L\) values correspond to different shapes:

• \(L=0\): s-orbital (spherical shape)
• \(L=1\): p-orbital (dumbbell shape)
• \(L=2\): d-orbital (cloverleaf shape)

However, in the exercise scenario, since \(L=N\), there isn't such diversity for each \(N\); the shape of each electron's path through space remains uniform across that shell.

Magnetic Quantum Number

The magnetic quantum number, represented by \(M\), is critical in defining the orientation of an orbital in space. In this specifically designed quantum universe, \(M\) can take on three specific values: -1, 0, and +1.

This constraint results in three orientations for any given orbital, regardless of the principal quantum number \(N\).

• \(M=-1\): The orbital is oriented in one direction
• \(M=0\): The orbital is oriented in a neutral or central direction
• \(M=+1\): The orbital is oriented in the opposite direction from \(M=-1\)

The magnetic quantum number interacts with both the principal quantum number and the orbital quantum number to define a specific electron's behavior.

In the context of this exercise, each energy level \(N\) offers three possible magnetic orientations due to these \(M\) values, leading to a fixed number of orbitals per shell. For instance, regardless of being in shell 1, 2, or 3, each shell will contain exactly 3 orbitals due to the three possible values of \(M\), emphasizing the simplicity of this alternate quantum setting.