The dissociation energy of the hydrogen molecule (i.e., the...

The dissociation energy of the hydrogen molecule (i.e., the energy required to separate the two atoms) is $4.48 \mathrm{eV}$. In the gas phase (treated as an ideal gas), at what temperature is the average translational kinetic energy of a molecule equal to this energy?

Key Concepts

Component Comparison in Angular Momentum

Comparing different angular momentum components involves examining how the fixed spin value (or its z-component) relates to the variable orbital contributions. In problems where one is asked to compare, for example, the spin’s z-component to the total orbital angular momentum, one uses the quantized values of each – reflecting the separate origins and magnitudes of these two forms of angular momentum in quantum systems.

Quantum Numbers

Angular momentum in quantum systems is described by quantum numbers. The orbital quantum number l defines the magnitude and allowed projections of the orbital angular momentum, while the magnetic quantum number m? specifies its component along a chosen axis (typically the z-axis). Similarly, the spin quantum number s and its projection m? characterize the intrinsic spin angular momentum of particles.

Orbital Angular Momentum

This is the angular momentum associated with the motion of a particle in an orbital, characterized by the quantum number l. Its magnitude in quantum mechanics is given by ?[l(l+1)] times ?, and the z-component is quantized with values m?? where m? ranges from –l to +l. It plays a key role in determining the shape and energy of electron orbitals in atoms.

Spin Angular Momentum

Spin is an intrinsic form of angular momentum carried by particles such as electrons. Unlike orbital angular momentum, spin does not arise from any spatial motion but is a fundamental quantum property with fixed magnitude ?[s(s+1)] times ? where s is the spin quantum number (1/2 for electrons). Its z-component is given by m??, with m? typically being +1/2 or –1/2.

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