Question

Discuss the differences between the rotational and vibrational energy levels of the deuterium ("heavy hydrogen") molecule ${\mathrm{D}}_2$ and those of the ordinary hydrogen molecule ${\mathrm{H}}_2$. A deuterium atom has twice the mass of an ordinary hydrogen atom.

Short answer

The hydrogen molecule will have a greater rotational and vibrational energies compared to a deuterium molecule because the reduced mass is inversely proportional for both rotational and vibrational energies.

Step-by-step solution

Definition of vibration energy

According to experts in vibrational energy, some moods and thinking patterns (like pleasure, peace, and acceptance) produce high-frequency vibrations. In contrast, others (like rage, despair, and fear) produce lower frequency vibrations.

Determine the differences between the rotational and vibrational energy levels

Now, the combined rotational and vibrational energy of a diatomic molecule is calculated using

$E_{nl}=I(I+1)\frac{h^2}{2I}+\left(n+\frac{1}{2}\right)h\sqrt{\frac{k}{m_r}}$

We are given that the mass of a deuterium molecule is twice that of an ordinary hydrogen molecule, which thus means that the reduced mass of deuterium will be greater than the reduced mass of a hydrogen molecule. Thus, $m_r$ will inversely affect both the rotational and vibrational energies.

Therefore, the hydrogen molecule has higher rotational and vibrational energies compared to a deuterium molecule.