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How to understand the change in internal energy microscopically, (in terms of molecular energy levels/ states,)

  1. when the heat added into the system

  2. when work is done on the system?

Ultimately, I want to get clear on how the work (pressure) and heat are changing the energy levels and states?

Also, are the work and heat are same microscopically?

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The assumption made in statistical mechanics for quasi static changes is that a small quantity of work done on the system results in a small upward shift in energy levels, but a small amount of heat flowing into the system results in a small change in populations of (unchanged) energy levels.

A simple example: A particle (modelling an ideal gas) in a cuboidal box of side lengths π‘Ž, 𝑏, 𝑐 can only have energies given by

$$E=\frac{h^2}{8m}\left(\frac{n_x^2}{a^2}+\frac{n_y^2}{b^2}+\frac{n_z^2}{c^2}\right)$$

$n_x, n_y, n_z$ are integers. If we do work slowly on the gas by pushing in a piston we reduce π‘Ž, 𝑏 or 𝑐, so we increase the energy of any given level, as identified by a given set of integers $n_x, n_y, n_z$. We don't – according to a theorem of Ehrenfest – alter the populations of the levels.

But if we feed in heat we don't alter the shape of the box, so we don't alter the energy levels. But we do alter the distribution of particles in the various energy levels.

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  • $\begingroup$ Thanks Wood. In a closed system is there any chance to take all the molecules at the same level by adding heat? or will they follow any distribution ? $\endgroup$
    – mustang
    Commented Jun 4, 2018 at 4:49
  • $\begingroup$ Hello! I need your help to understand the part about the small change in populations of energy levels. For your answer, I am thinking of a gas undergoing pressure vs heating. What would be the difference? Would it be hard to add examples to your answer? All the best, thank you ;) $\endgroup$
    – Curious Watcher
    Commented Aug 18, 2021 at 21:23
  • $\begingroup$ A simple example: A particle (modelling an ideal gas) in a cuboidal box of side lengths $a$, $b$, $c$ can only have energies given by $E=\frac{h^2}{8m}\left(\frac{n_x^2}{a^2}+\frac{n_y^2}{b^2}+\frac{n_z^2}{c^2}\right)$. $n_x$, $n_y$, $n_z$ are integers. If we do work slowly on the gas by pushing in a piston we reduce $a$, $b$ or $c$ so we increase the energy of any given level, as identified by a given set of integers $n_x,\ n_y,\ n_z$. We don't alter the populations of the levels. But if we feed in heat we don't alter the shape of the box, so we don't alter the energy levels. $\endgroup$
    – Philip Wood
    Commented Aug 18, 2021 at 22:33
  • $\begingroup$ But we do alter the distribution of particles in the various energy levels. @Curious Watcher $\endgroup$
    – Philip Wood
    Commented Aug 20, 2021 at 10:58
  • $\begingroup$ Amazing, thank you for an update, definitely adds weight. And me, I am still working on understanding this, my internal video card is still processing the visualization on energy levels, are we talking of the energy levels as in internal energy levels, starting from 0 and marked as integers? I have doubts and no better way at the moment than to ask, thank you for your patience. $\endgroup$
    – Curious Watcher
    Commented Aug 23, 2021 at 19:15

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