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I am confused as to what is the relation between entropy and internal energy. Entropy is always presented as a measure of the randomness in a system. So when we supply heat to a well insulated system say ideal gas in a container with fixed boundary, the internal energy and temperature increase, which implies that the motion of gas particles increases and hence the system becomes more chaotic and thus entropy increases. But if we take the same system, and supply heat isothermally and reversibly, the defnition of entropy change ΔS=Q/T , says that the entropy of system would increase(at the cost of equal entropy drop in surroundings). But for an ideal gas, internal energy is only a function of temperature and so internal energy remains constant here,no change in average kinetic energy of gas particles takes place, so where does the chaos come from to increase entropy of the system.

How are the two related?

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  • $\begingroup$ Your starting hypotheses seem contradictory: supplying heat means transferring energy to the system, so the internal energy U of the system necessary increases, if it is an ideal gas (for which U depends only on T) then necessarily T will increase. Conclusion: you cannot supply heat isothermally to an ideal gas. $\endgroup$
    – The Quark
    Commented May 10, 2023 at 16:41

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But for an ideal gas, internal energy is only a function of temperature and so internal energy remains constant here,no change in average kinetic energy of gas particles takes place, so where does the chaos come from to increase entropy of the system.

'Chaos' is not a very well defined term in context of statistical physics. It is not necessary to use it to understand entropy in this theory.

One way to understand this entropy is the Boltzmann-Planck definition: entropy is logarithm of number of distinct possible states that are compatible with macroscopic variables like $V$,$U$ etc. No chaos is needed in this definition.

You seem to assume that in general, increase in internal energy is necessary to increase entropy. That is not the case, for there are systems where entropy can increase just by making sudden change to the system, without supplying any energy. Consider ideal gas in a chamber that is connected to another chamber via small pipe. At first, the pipe is blocked by a closed valve, so the gas stays in the first chamber. Then the valve is opened and the gas rushes to the second chamber. In the end, the volume is twice the original value.

Now that the gas has twice the original volume, the number of accessible states increased by factor of two. But no energy was supplied or lost, so internal energy stayed the same. According to the above definition, entropy increased by a factor $\ln 2$.

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  • $\begingroup$ Currently i am studying classical thermodynamics, so i am not very informed about the number of accessible states concept, but did get it from the unpredictability point of view, because the volume doubles, a gas particle can be present in either of the two chambers, whereas earlier the gas particle's unpredictability was confined to only 1 chamber. $\endgroup$ Commented Jun 26, 2016 at 11:29
  • $\begingroup$ What is striking about this system is that U = 3/2 nRT in a monoatomic gas, yet ST drops by nRT for every factor-of-e expansion in volume. In chemistry it's easy to think of delta G as negative... but here it seems free energy can be negative in an "absolute" sense - with ST potentially greater than all the kinetic (thermal) energy of all the particles. Still trying to wrap my head around this... $\endgroup$ Commented Aug 27, 2022 at 1:42
  • $\begingroup$ @MikeSerfas Why do you think $ST$ drops when the gas expands? Do you assume the expansion does work on the system environment? In this answer, I was describing expansion into vacuum. $\endgroup$ Commented Aug 27, 2022 at 21:32
  • $\begingroup$ @JánLalinský - Oh I flubbed that up in the writing! I meant ST is going up so the free energy is dropping - if we picture starting with "zero" entropy and let the gas expand by just e^2, we're already well into negative free energy. But per Sackur-Tetrode we're starting at anything but zero entropy. (I've since become aware that there are simple, interesting ways of describing Heimholtz free energy - Matthew Schwartz has some very readable explanations; there's a lot to go through. $\endgroup$ Commented Aug 28, 2022 at 16:50
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Entropy is the order of disorderness of a system, which means greater will be the irreversibility of a process.

Internal energy is the sum of kinetic and potential energies of particles.

Entropy increases only if there is enough energy in particles. Thus, if there is no internal energy there won't be any entropy.

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The chaos comes from by changing of volume or pressure of the system. The average kinetic energy doesn't change, but number of collisions increases (if pressure increase) or length of paths increases (if volume increases).

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    $\begingroup$ Hmm, the pressure part sounds right, but at a constant temperature how can pressure and volume increase simultaneously, that seems to contradict boyle's law. $\endgroup$ Commented Jun 26, 2016 at 9:40
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    $\begingroup$ You are right. I'll correct the answer. $\endgroup$
    – lucas
    Commented Jun 26, 2016 at 9:41

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