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.

Short answer

(a) The experimental evidence for electron spin comes from experiments like the Stern-Gerlach experiment and spectroscopic measurements, such as hydrogen fine structure and Zeeman effect. (b) An energy-level diagram for 1s and 2s orbitals with two electrons in the 1s orbital can be drawn by depicting horizontal lines labeled as 1s and 2s and placing upward and downward pointing arrows above the 1s line. (c) Electron excitation from the 1s to the 2s orbital can be illustrated by drawing an arrow from one of the electrons in the 1s orbital to a position above the 2s orbital energy level line, with an upward or downward-spin arrow at the end of the excitation arrow.

Step-by-step solution

(a) Experimental Evidence for Electron Spin

The electron spin was first proposed theoretically by George Uhlenbeck and Samuel Goudsmit in 1925. Experimental evidence for electron spin came from various experiments like the Stern-Gerlach experiment and spectroscopic measurements. In the Stern-Gerlach experiment, atoms were sent through a magnetic field; it was expected that the atoms' magnetic moments would divide into two energy states (parallel and anti-parallel to the magnetic field). However, the observed splitting pattern was greater than expected, which led to the proposal that electrons have an intrinsic angular momentum called spin (with two possible values: +1/2 and -1/2). Other experiments, like the hydrogen fine structure and Zeeman effect in spectroscopy, connected electron spin with the observed spectral lines -- offering further evidence for electron spin.

(b) Energy-Level Diagram

To draw an energy-level diagram for 1s and 2s orbitals and put two electrons in the 1s orbital, follow these steps: 1. Draw a horizontal line and label it as 1s. This represents the energy level of the 1s orbital. 2. Draw an arrow pointing upwards above the 1s line. This represents an electron with an upward (positive) spin in the 1s orbital. 3. Draw an arrow pointing downwards above the 1s line, next to the previous arrow. This represents an electron with a downward (negative) spin in the 1s orbital. 4. Draw another horizontal line above the 1s line and label it as 2s. This represents the energy level of the 2s orbital. Now you have an energy-level diagram with two electrons in the 1s orbital.

(c) Electron Excitation

To show the excitation of an electron from the 1s to the 2s orbital: 1. Choose one of the electrons in the 1s orbital (either the upward-spin or downward-spin electron). 2. Draw an arrow from the chosen electron position in the 1s orbital to a position above the 2s orbital energy level line, representing the new energy level of the excited electron. 3. Draw an upward-spin or downward-spin arrow (matching the spin of the chosen electron) at the end of the excitation arrow, showing the electron now occupying the 2s orbital. The drawn arrow represents the electron's excitation from the 1s to the 2s orbital.