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  1. Why various molecules in an ideal gas at a particular temperature can have only quantized energies? Why can't they have the energies distributed in a continuous fashion?

Following is an image taken from the mentioned reference, where it shows that various molecules can take only certain energies and not the energies in between those levels. What is the reason behind such quantization of energy in gas molecules?

The same idea Prof. Peter Atkins also talks about in his book "Very short introduction to Thermodynamics", but in the book, he did not explain the reason behind it.

  1. How do we know that such energy levels are there?

  2. What do we lose if we assume the distribution of energy to be continuous?

enter image description here

Reference: https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Thermodynamics/Energies_and_Potentials/Entropy/The_Molecular_Basis_for_Understanding_Simple_Entropy_Change

Devansh Mittal
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2 Answers2

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To answere your first question:

Due to the assumption that the ideal gas is in a closed body (mostly a cube with edge length L for easier calculation) the wave function of e.g electrons, considering a fermi-gas, need to fullfill certain boundary conditions. For the energy you then get that it needs to be quantized since there are different modes/quantum numbers for the solution of the Schroedinger equation.

You can look at the wavefunction of the particles analogous to a guitar string with fixed ends.

Tera
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  • https://physics.stackexchange.com/questions/520184/fermi-energy-definition/520196#520196 In this post I posted a calculation for my above statement, though it is a slightly different context. – Tera Dec 28 '19 at 17:58
  • Thanks a lot. Inspired by your response, I have shared my understanding on the topic in the other response here by Jeffery.

    Could you kindly confirm my understanding there? Thanks in advance.

     – Devansh Mittal Dec 28 '19 at 18:06
  • Kindly confirm the following understanding. The energy states are there due to quantum mechanics. The molecules will have their wave functions, due to their wave natures. Now as we see in the case of a particle in the box analogy, the particle is allowed to take only certain energy states, similarly, the molecules are also allowed to take only certain energy states in a collection of molecules in a gas. – Devansh Mittal Dec 28 '19 at 18:10
  • the translation energy levels are very close by, then rotational energy levels come at higher temperatures and they are relatively widespread and vibrational energy levels come at even higher temperatures and they are even more widely spread. This is also a reason why the heat capacity of gas is variable with temperature – Devansh Mittal Dec 28 '19 at 18:10
  • At lower temperatures, a small amount of heat is sufficient to take the gas molecules to higher translational energy levels, very easily, because the energy levels are very near by, so heat capacity is very less in the beginning. As we grow the temperature, the heat capacity also grows, because now more amout of heat will be required to increase the temperature of gas by unity, as the energy levels are far apart. – Devansh Mittal Dec 28 '19 at 18:11
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The picture and the explanation do not require that the energy levels are discreet. The picture is often shown / described this way for simplicity. The separation can just as well be infinitesimal (zero).

  • The three modes of storing internal energy in a gas are translation, rotation, and vibration. An ideal gas particle has zero volume and therefore has only translation. In an unconfined system, the separation between translation levels is infinitesimal (zero). In a finite-sized system where the collisions between particles dominates the collisions to the walls, the gas particles still behave classically. At the point where the gas particles collide with the walls more often than each other, the levels can become discreet (quantized). This is analogous to the particle-in-a-box state of quantum mechanics. See this link for further insight about molecular speed distributions.

  • Rotation and vibration levels appear in molecules. Their energies are quantized. The energy distribution that you show applies.

  • We loose nothing to translate from discreet to continuous. We translate the math from summations to integrals. We obtain a continuous distribution function rather than a discreet set of levels. Here for example is the continuous distribution plot of speed and kinetic energy for molecular oxygen at 150 °C.

distribution plots of molecular oxygen

Jeffrey J Weimer
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  • Thanks. My intention behind asking this question was to know the reason behind the discreteness of energy levels, let's say behind diatomic molecules. The key word there is "Discreteness". The reason behind that is pointed to some extent by Tera, in the other response. – Devansh Mittal Dec 28 '19 at 18:02
  • Kindly confirm the following understanding.

    The energy states are there due to quantum mechanics. The molecules will have their wave functions, due to their wave natures. Now as we see in the case of a particle in the box analogy, the particle is allowed to take only certain energy states, similarly, the molecules are also allowed to take only certain energy states in a collection of molecules in a gas.

     – Devansh Mittal Dec 28 '19 at 18:03
  • The amazing thing is, the translation energy levels are very close by, then rotational energy levels come at higher temperatures and they are relatively widespread and vibrational energy levels come at even higher temperatures and they are even more widely spread.

    This is also a reason why the heat capacity of gas is variable with temperature.

     – Devansh Mittal Dec 28 '19 at 18:04
  • At lower temperatures, a small amount of heat is sufficient to take the gas molecules to higher translational energy levels, very easily, because the energy levels are very near by, so heat capacity is very less in the beginning.

    As we grow the temperature, the heat capacity also grows, because now more amout of heat will be required to increase the temperature of gas by unity, as the energy levels are far apart.

     – Devansh Mittal Dec 28 '19 at 18:04
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    @DevanshMittal The translational modes of particles can be derived solely from classical mechanics and probability distributions. See my update for information. – Jeffrey J Weimer Dec 28 '19 at 19:39