Problem 63 - Chemistry The Central Science

Question

(a) What experimental evidence is there for the electron having a "spin"? (b) Draw an energy-level diagram that shows the relative energetic positions of a \(1 s\) orbital and a \(2 s\) orbital. Put two electrons in the \(1 s\) orbital. (c) Draw an arrow showing the excitation of an electron from the \(1 s\) to the \(2 s\) orbital.

Step-by-Step Solution

Answer

(a) The Stern-Gerlach experiment provides evidence for electron spin, as it showed silver atoms splitting into two groups due to their electron spins (+1/2 and -1/2). (b) In the energy-level diagram, the \(1s\) orbital is represented by a horizontal line below the higher energy \(2s\) orbital. Two arrows within the \(1s\) level symbolize the two electrons in spin-up and spin-down states. (c) A curved arrow from the \(1s\) to the \(2s\) level illustrates the excitation of an electron from the \(1s\) to the \(2s\) orbital.

Step 1: (a) Experimental Evidence for Electron Spin

The Stern-Gerlach experiment provides experimental evidence for the electron spin. In this experiment, silver atoms were passed through a magnetic field, which caused them to split into two groups. This splitting was a direct result of the electron's intrinsic angular momentum known as "spin". The two different spin states of electrons (+1/2 and -1/2) explained the observed behavior.

Step 2: (b) Drawing Energy-Level Diagram

To draw an energy-level diagram, begin by drawing two horizontal lines. Label the lower line as "1s" and the higher line as "2s". These lines represent the energy levels of the \(1s\) and \(2s\) orbitals, respectively. The \(1s\) orbital has lower energy and is therefore closer to the bottom of the diagram. Now, we need to place two electrons in the \(1s\) orbital. To do this, draw two arrows pointing upward (representing the spin-up state) and downward (representing the spin-down state) within the \(1s\) level.

Step 3: (c) Arrow for Electron Excitation

To show the excitation of an electron from the \(1s\) to the \(2s\) orbital, draw a curved arrow starting from the one of the arrows within the \(1s\) level and ending at the \(2s\) level. This arrow represents the process of an electron being excited from the lower energy \(1s\) orbital to the higher energy \(2s\) orbital.

Key Concepts

Stern-Gerlach Experiment

The Stern-Gerlach experiment is a groundbreaking investigation that offered clear evidence of the electron's spin, a fundamental property of electrons. In this experiment, silver atoms were sent through a non-uniform magnetic field. As the atoms passed through the magnetic field, they were deflected into two distinct paths rather than a single path or a continuous distribution of paths.
This unexpected behavior was due to the intrinsic angular momentum of electrons, known as "spin," which can be visualized as electrons having an inherent rotation.

• The experiment distinguished two possible spin states of the electron: \(+\frac{1}{2}\) and \(-\frac{1}{2}\).
• These two spin states result in the splitting of the silver atom stream, creating two separate bands when detected, rather than a smear.
• It provided a visual and experimental confirmation that electrons have a quantum property not accounted for by classical physics.

This discovery was instrumental in the development of quantum mechanics by providing insights into the non-classical behavior of particles.

Energy-Level Diagram

An energy-level diagram is a visual representation of the possible energy states electrons can occupy in an atom. It helps chemists and physicists understand electron configurations and behaviors. Here’s how you can draw a simple energy-level diagram for a hydrogen atom:
Begin by drawing two horizontal lines, with the lower one indicating the \(1s\) orbital and the higher one indicating the \(2s\) orbital. The position on the diagram corresponds to the energy level of these orbitals, where the \(1s\) is lower in energy compared to the \(2s\).

• Electrons are added as arrows within these lines to indicate the specific filled sublevels, either pointing upward for "spin-up" or downward for "spin-down."
• In the \(1s\) orbital, both "spin-up" and "spin-down" arrows are drawn, showing two electrons can exist in this level with opposite spins, due to the Pauli Exclusion Principle.
• The "2s" level remains open for additional electrons, which can be added following the same rules.

Such diagrams are crucial in determining electron arrangements and understanding the subsequent behavior of atoms in reactions and interactions.

Electron Excitation

Electron excitation is a key concept in atomic and molecular physics, where an electron is moved from a lower energy orbital to a higher energy orbital. This process is central in understanding how atoms absorb and emit energy. Picture an energy-level diagram showing various orbitals, and an electron residing in the \(1s \) level.
When energy is supplied, such as through heat or light, the electron may gain enough energy to "jump" to a higher level like the \(2s\). This jump is known as excitation.

• An arrow is usually drawn on an energy-level diagram to illustrate this transition, curving from the lower \(1s\) orbital state to the higher \(2s\) orbital state.
• The energy required for excitation equals the difference in energy between the \(1s\) and \(2s\) orbitals.
• Once excited, the electron doesn't stay in the higher energy level long and will eventually return to its original state, releasing energy in the form of light or heat.

This concept aids in comprehending phenomena such as light emission in flames, colors in fireworks, and the working of lasers and fluorescent lights.