Problem 53 - University Physics with Modern Physics

Question

Weird Universe. In another universe, the electron is a \(\operatorname{spin} \frac{3}{2}\) rather than a spin- \(\frac{1}{2}\) particle, but all other physics are the same as in our universe. In this universe, (a) what are the atomic numbers of the lightest two inert gases? (b) What is the ground-state electron configuration of sodium?

Step-by-Step Solution

Answer

(a) Lightest inert gases are at atomic numbers 4 and 20. (b) Sodium's configuration: \(1s^4\ 2s^4\ 2p^3\).

Step 1: Understand Spin and Its Impact

In our universe, electrons have a spin of \(\frac{1}{2}\), which means they can occupy "up" or "down" spin states. This influences the atomic structure as electron configurations follow the Pauli exclusion principle, filling in sublevels accordingly. In this alternate universe, each electron possesses a spin of \(\frac{3}{2}\), which means there are four possible spin states (\(-\frac{3}{2}, -\frac{1}{2}, \frac{1}{2}, \frac{3}{2}\)). This affects how many electrons can occupy each sublevel.

Step 2: Calculate Electron Capacity in Sublevels

For a spin-\(\frac{3}{2}\) electron, each orbital can accommodate four electrons (since each has four spin states). Calculate the maximum number of electrons for each shell and subshell using the formula \(2n^2\), where \(n\) is the shell number, adjusted for the new spin states. Thus, each s, p, d, and f subshell can hold 4, 12, 20, and 28 electrons respectively.

Step 3: Determine the First Inert Gas Atomic Number

An inert gas, or noble gas, has full electron shells. Calculate this for the first few potential noble gases using the modified electron capacities: \(1s^4\) results in 4 electrons needed for the first full shell.

Step 4: Determine the Second Inert Gas Atomic Number

For the next noble gas after filling the 1s and 2s shells, fully occupy the 2p subshell as well: \(1s^4\ 2s^4\ 2p^{12}\), totaling 4 + 4 + 12 = 20 electrons. Hence, the second inert gas corresponds to an atomic number of 20.

Step 5: Ground-State Configuration of Sodium

For sodium with atomic number 11, fill the electron configuration with modified sublevel capacities: \(1s^4\ 2s^4\ 2p^3\). Thus, the first eight electrons fill the 1s and 2s shells, leaving 3 electrons in the 2p.

Key Concepts

Atomic Structure

Atomic structure refers to the arrangement of electrons around the nucleus in atoms. In traditional quantum mechanics, electrons are arranged in shells and subshells, obeying the Pauli exclusion principle. This principle states that no two electrons can have the same set of quantum numbers within an atom.
In our universe, electrons have a spin of \( \frac{1}{2} \). They can have two possible spin states: "up" (\(+\frac{1}{2}\)) or "down" (\(-\frac{1}{2}\)). Each orbital, or region around the nucleus where an electron is likely to be found, can hold two electrons of opposite spins.
In the hypothetical universe where electrons have a spin of \( \frac{3}{2} \), this means that each electron can exist in one of four spin states: \(-\frac{3}{2}, -\frac{1}{2}, +\frac{1}{2}, +\frac{3}{2}\). This dramatically changes the atomic structure:

• Each orbital can hold four electrons instead of two.
• The subshells are filled differently, resulting in different electron capacities for each shell.

Understanding these changes is essential for determining how atoms form in this alternate universe.

Electron Configuration

Electron configuration is the distribution of electrons among the various atomic orbitals. With a spin of \(\frac{3}{2}\), electrons in our fictional universe occupy orbitals in a unique configuration.
In our universe, each shell can be found with specific subshells:

• s subshell can hold 2 electrons.
• p subshell holds 6 electrons.
• d subshell holds 10 electrons.
• f subshell holds 14 electrons.

With spin-\(\frac{3}{2}\) electrons:

• s subshell can hold 4 electrons.
• p subshell holds 12 electrons.
• d subshell holds 20 electrons.
• f subshell holds 28 electrons.

This means, for example, the first inert gas in this universe is at atomic number 4, because the \(1s^4\) shell is full. The second inert gas is at atomic number 20, filled with \(1s^4 2s^4 2p^{12}\). These differences in structure affect how atoms stabilize themselves, impacting the chemical properties in this universe.

Quantum States

Quantum states refer to the specific discrete configurations or arrangements available to an electron within an atom. These configurations are defined by a set of quantum numbers:

• Principal quantum number \(n\): indicates the energy level.
• Angular momentum quantum number \(l\): defines the shape of the orbital.
• Magnetic quantum number \(m\): describes orientation of the orbital in space.
• Spin quantum number \(s\): denotes the direction of the electron's spin.

In a universe where electrons are \(\frac{3}{2}\) spin particles, the set of spin quantum numbers is larger: \(-\frac{3}{2}, -\frac{1}{2}, \frac{1}{2}, \frac{3}{2}\).
This implies that the possible quantum states for each electron significantly increase, influencing not only the electron's own behavior but also the behavior of the atom.
Each additional state allows an electron more ways to avoid violating the Pauli exclusion principle, potentially enabling new electron configurations and altering the atom's properties. Understanding how these quantum states interact sheds light on the underlying mechanisms of atomic interactions in this hypothetical universe.