(a) What is the lowest possible energy (in electronvolts) of...

(a) What is the lowest possible energy (in electronvolts) of an electron in hydrogen if its orbital angular momentum is $\sqrt{12 \hbar ?}$ (b) What are the largest and smallest values of the $z$ component of the orbital angular momentum (in terms of $\hbar$ ) for the electron in part (a)? (c) What are the largest and smallest values of the spin angular momentum (in terms of $\hbar$ ) for the electron in part (a)?

Key Concepts

Average Translational Kinetic Energy

This concept refers to the energy associated with the motion of particles in a substance. In thermodynamics, the average translational kinetic energy of molecules is related to temperature by the equipartition theorem, which for a monatomic gas gives (3/2)kBT, where kB is the Boltzmann constant. In the context of this question, equating the van der Waals bond energy to this kinetic energy helps determine the temperature at which molecular motion overcomes the weak bonding forces.

Boltzmann Constant and Thermal Energy

The Boltzmann constant (kB) is a fundamental physical constant that relates the average kinetic energy of particles in a gas with the temperature of the gas. It serves as a bridge between microscopic energy scales (like electronvolts) and macroscopic temperature scales (Kelvin). Using kB allows one to convert an energy value (e.g., 0.50 eV) into a corresponding temperature, providing insight into the thermal energy that molecules have at a given temperature.

Phase Behavior and Energy Comparison

This concept involves understanding how the kinetic energy of molecules compares to intermolecular bonding energies to determine the phase (solid, liquid, or gas) of a substance. When the average translational kinetic energy is high compared to the strength of intermolecular forces like van der Waals bonds, the substance tends to be in the gaseous state, as molecules have enough energy to overcome attractive interactions. Conversely, lower kinetic energy relative to the bonding energy favors a liquid or solid phase.

van der Waals Forces

Van der Waals forces are weak, short-range attractions between molecules that arise from transient electrical polarization. Although these forces are much weaker than strong chemical bonds, they play a role in determining the physical properties of materials, such as boiling points and condensation, especially in nonpolar substances. In water, while they exist, they are overshadowed by the much stronger hydrogen bonding; however, van der Waals forces still contribute subtly to the overall intermolecular interactions.

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