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Chapter 8: Problem 16

Can a molecule have zero vibrational energy? Zero rotational energy?

Short Answer

Expert verified
Molecules cannot have zero vibrational energy but can have zero rotational energy.

Step by step solution

01

Understanding Vibrational Energy

In quantum mechanics, molecules can only have certain discrete energy levels for vibration. The lowest possible vibrational energy level a molecule can have is called the zero-point energy, which is not zero. Therefore, a molecule cannot have zero vibrational energy.
02

Understanding Rotational Energy

Similar to vibrational energy, rotational energy levels of a molecule are also quantized. However, in the absolute ground state, which is the lowest rotational energy level (given by J=0 where J is the rotational quantum number), the rotational energy can indeed be zero or very close to it. For molecules in the ground rotational state, rotational energy is often negligible.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vibrational Energy
Molecules are like tiny dancers, moving at the atomic level. One key movement is vibration.
In quantum mechanics, this vibration is only allowed at specific energy levels, which is called quantization.
The vibrational energy levels are not like a sliding scale, but more like a set of stairs you can only stand on certain steps. When we talk about vibration, there's a special term called "zero-point energy." While it may sound like it should be nothing, it's actually not zero.
  • The zero-point energy is the lowest possible energy the vibrational mode can have.
  • This means that even at the absolute lowest state, a vibrational mode possesses some energy.
  • Because of this, a molecule can't completely "turn off" its vibrational energy.
Even at zero Kelvin, molecules vibrate with this smallest amount of energy, ensuring that vibrational energy is always present.
Rotational Energy
Rotational energy is another way molecules can display their energy. Just like vibrational energy, it too is quantized, meaning it only takes on specific values.
While vibrational energy never reaches zero, rotational energy behaves a bit differently.
In the quantum world, rotational motions are classified by a rotational quantum number denoted as \( J \).At \( J = 0 \), the rotational energy level is at its absolute lowest. In this ground state, the energy can be effectively zero, or nearly zero.
  • This means that as opposed to vibration, which never fully rests, the rotational movement can essentially pause.
  • For real life molecules, the rotational energy is almost negligible in this state.
  • Such states, where energy is minimal, are often reached under extremely low temperatures or special conditions.
The key takeaway is that molecules have more relaxation options with rotation than vibration.
Zero-point Energy
Zero-point energy is a fascinating concept in quantum mechanics. It describes the lowest energy level that a quantum mechanical system, such as a molecule, can possess.
Despite its name, this lowest level isn't actually zero, which is a tricky but essential point to grasp. Every atom or molecule inherently possesses zero-point energy. You can think of it as the residual energy left even when everything else is stripped away, or as trying to stop a swing and it never completely standing still.
  • Zero-point energy ensures that molecules are never completely at rest.
  • This concept showcases the fundamental energy that keeps molecules active, even at the coldest temperatures possible.
Understanding zero-point energy helps explain why in the quantum world, complete stillness is an impossibility.

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Most popular questions from this chapter

Find the frequencies of the \(J=1 \rightarrow J=2\) and \(J=2 \rightarrow J=3\) rotational absorption lines in NO, whose molecules have the moment of inertia \(1.65 \times 10^{-46} \mathrm{~kg} \cdot \mathrm{m}^{2}\)

The energy needed to detach the electron from a hydrogen atom is \(13.6 \mathrm{eV}\), but the energy needed to detach an electron from a hydrogen molecule is \(15.7 \mathrm{eV}\). Why do you think the latter energy is greater?

Microwave communication systems operate over long distances in the atmosphere. The same is true for radar, which locates objects such as ships and aircraft by means of microwave pulses they reflect. Molecular rotational spectra are in the microwave region. Can you think of the reason why atmospheric gases do not absorb microwaves to any great extent?

Assume that the \(\mathrm{H}_{2}\) molecule behaves exactly like a harmonic oscillator with a force constant of \(573 \mathrm{~N} / \mathrm{m}\). \((a)\) Find the energy (in \(\mathrm{eV}\) ) of its ground and first excited vibrational states. (b) Find the vibrational quantum number that approximately corresponds to its \(4.5-\mathrm{eV}\) dissociation energy.

\(\mathrm{A}^{200} \mathrm{Hg}^{35} \mathrm{Cl}\) molecule emits a 4.4-cm photon when it undergoes a rotational transition from \(J=1 t 0 J=0\). Find the interatomic distance in this molecule.
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