Question

1\. As noted in Example 10.2 . the HD molecule differs from \(\mathrm{H}_{2}\) in that a deuterium atom replaces a hydrogen atom. (a) What effect. if any, does the replacement have on the bond length and force constant? Explain. (b) What effect does it have on the rotational energy levels? (c) And what effect does it have on the vibrational energy levels?

Short answer

Replacing one hydrogen atom in a hydrogen molecule with a deuterium atom tends to decrease the bond's force constant due to the increased mass but it does not noticeably affect the bond length. The inclusion of Deuterium decreases the spacing between both rotational and vibrational energy levels because of an increase in the reduced mass of the molecule.

Step-by-step solution

Effect on Bond Length and Force Constant

Replacing a hydrogen atom with a deuterium atom in the H2 molecule increases the mass, but the bond length is generally not affected. This is due to the electronegativity of the atoms remaining the same, as it is a property intrinsic to the atom and not dependant on its isotopes. Thus, the bond length between the atoms remains unaffected. However, the force constant, which depends on the mass of the atoms, does change. The force constant decreases with the increase in mass of the molecule. This is because the force constant is inversely proportional to the square root of the reduced mass of the molecule. Since Deuterium has a higher mass than Hydrogen, the reduced mass increases and thus the force constant decreases.

Effect on Rotational Energy Levels

Rotational energy levels depend on the moment of inertia of the molecule, which in turn depends on the reduced mass of the molecule. When a Hydrogen atom is replaced by a Deuterium atom, the reduced mass of the molecule increases. This leads to an increase in moment of inertia. The energy of rotational levels is inversely proportional to the moment of inertia, and so the spacing between the rotational energy levels decreases with the inclusion of Deuterium.

Effect on Vibrational Energy Levels

Vibrational energy levels depend on the force constant and the reduced mass of the molecule. As discussed in Step 1, the force constant decreases with the inclusion of Deuterium due to an increase in reduced mass. As both these factors decrease, the vibrational energy levels (which are directly proportional to the square root of the force constant and inversely proportional to the square root of the reduced mass) also decrease. This means that the gap between vibrational energy levels decreases when Hydrogen is replaced with Deuterium.

Key concepts

Bond Length

When discussing molecular physics, bond length refers to the average distance between the nuclei of two bonded atoms. It's an essential concept because it gives insight into the structural stability of a molecule. In the context of the HD molecule (Hydrogen-Deuterium), replacing a hydrogen atom with a deuterium atom doesn't significantly affect the bond length. This might seem surprising at first. However, the reason lies in the atomic properties. Both hydrogen and deuterium are isotopes of the same element, hydrogen.

Isotopes have the same electron configuration, which means they have the same electronegativity. Electronegativity is a key factor in determining bond length because it dictates how strongly the atoms pull on their shared electrons. Since electronegativity remains unchanged in this atomic swap, the bond length remains consistent. This stable bond length is crucial for maintaining the geometry and properties of the molecule.

Force Constant

The force constant, or bond stiffness, is a measure of the strength of the chemical bond between two atoms in a molecule. It is crucial in determining how resistant a molecule is to deformation. There's a direct relationship between the force constant and the energy required to stretch or compress a bond.

When a hydrogen atom is replaced by a deuterium atom, as in our problem, the mass of the bonded atoms increases. The force constant is indirectly related to the reduced mass of the bonding atoms. More precisely, it is inversely proportional to the square root of the reduced mass.

• Increased mass of deuterium leads to a higher reduced mass.
• A higher reduced mass means that the force constant decreases.

Thus, with a heavier atom like deuterium, the bonds become "softer" or less stiff, resulting in a decreased force constant. Decreasing force constant plays a role in how the molecule vibrates.

Rotational Energy Levels

The concept of rotational energy levels is important in understanding how molecules rotate. These levels depend largely on the moment of inertia, which is determined by the mass distribution in the molecule. When discussing molecules like HD, changing a hydrogen atom for a deuterium atom increases the mass of the molecule and, consequently, the moment of inertia.

The rotational energy of a molecule is inversely proportional to its moment of inertia, meaning:

• The increase in moment of inertia from adding deuterium lowers the rotational energy levels.
• The energy spacing between these rotational levels becomes smaller.

Thus, heavier isotopes result in smaller energy gaps between rotational energy levels, impacting how molecules absorb and emit energy under rotational transitions. Understanding these changes can be helpful in spectroscopic studies and in gaining insights into molecular dynamics.

Vibrational Energy Levels

Vibrational energy levels describe the quantized states of vibration of the molecules. They reflect the "quantum jumps" that a molecule's vibrational state can take. These energy levels are influenced by the force constant and the reduced mass of the molecule, both of which shift when a hydrogen atom is swapped for a deuterium atom.

As we have seen, the force constant decreases due to heavier deuterium atoms, and so does the vibrational energy levels:

• Vibrational energy levels decrease as the force constant decreases.
• The spacing between these levels also decreases due to increased reduced mass.

Thus, when deuterium replaces hydrogen, the vibrational transitions of the molecule require less energy. This change can affect how molecules interact with electromagnetic radiation and is vital for understanding molecular spectra and reactions.