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I have read in many credible texts ( including "a brief history of time" ) that nature always increases entropy, that is, the entropy in the universe increases. But I learnt that atoms and molecules form bonds to attain stability by filling their orbitals.

So doesn't this violate that entropy should always increase. I mean if all the atoms are trying to attain stability ( order ) then why does entropy ( chaos ) increase?

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The formation of chemical bonds releases energy, which heats the universe, which increases its total entropy more than enough to compensate.

As an example, consider the oxidation of aluminum (Al) in air, which occurs essentially immediately. Every chunk of aluminum you've ever seen has been coated with an oxidized layer (alumina, $\mathrm{Al}_2\mathrm{O}_3$) with a thickness of at least a few nanometers. But, as you imply, why would the oxidation reaction

$$4\mathrm{Al}+3\mathrm{O}_2\to 2\mathrm{Al}_2\mathrm{O}_3,$$ occur spontaneously? After all, it involves the conversion of oxygen gas to a solid, which requires a notable decrease in entropy (630 J/K per mole of product; gases carry a lot of entropy because the molecules are free to move around with a variety of positions and speeds).

The resolution is that the reaction is an exothermic one that heats things up (the sample and the surrounding environment). That is, the bonds release energy as they form. The formation of alumina releases about 1700 kJ for the same mole of product, which heats the sample and its surroundings, which in turn increases their entropy (by around 5600 J/K) because the molecules can assume a wider range of positions and speeds. As a result, the general rule of total entropy maximization for all spontaneous processes still holds true.

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  • $\begingroup$ how do you define the entropy of the universe? $\endgroup$
    – GiorgioP
    Nov 29 '18 at 17:13
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    $\begingroup$ Some quantitative analysis, even of a single example, would help. $\endgroup$
    – DanielSank
    Nov 29 '18 at 17:54
  • $\begingroup$ @GiorgioP I’m sure there are aspects of that definition that would trip me up, so read “universe” as “arbitrarily large region”. To a single chemical bond, a cubic meter would look like a whole universe, and the thermodynamic entropy within a cubic meter of stuff can generally be well defined and quantified. $\endgroup$ Nov 30 '18 at 0:56
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All the statements about the entropy of the universe share a common problem: there is no way to measure such a quantity (to check this statement, try to list all the known experimental methods to measure entropy differences and you can decide if they could be applied to the entire universe). Even the theoretical definition is not very easy. There are many definitions of entropy, from thermodynamics, from statistical mechanics, from information theory, from theory of complexity, from dynamical system theory, ... and they are not all equivalent or they become equivalent only under additional conditions. Which one could be applied to the universe? And one is speeking about the equilibrium entropy or some non-equilibrium extension of the concept? The former looks questionable, unless one is convinced that we live in a universe at equilibrium. The latter goes back to the problem of which definition to use.

In thermodynamics people are used to speak about a "thermodynamical" universe which is the union of a particular system plus the surrounding environment, provided such environment could be well characterized from the thermodynamic point of view. For example, system+environment could be an isolated system. Such thermodynamic "universe" is less problematic as the whole known universe.

Much more on the earth, the presence of ordered structures in interacting systems is not in contradiction with the principle of increase of entropy, once one remembers that such principle is valid only for an isolated system at fixed energy, volume and number of particles. Under such constraints it is possible to find macrostates where the majority of the underlying microstates corresponds to ordered configurations.

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    $\begingroup$ Though I haven't studied dynamical systems or "theory of complexity" well enough to say for those, it is well known that the definitions of entropy in thermodynamics, statistical mechanics, and information theory are equivalent up to an arbitrary scaling constant. Can you elaborate on how "they are not all equivalent or they become equivalent only under additional conditions"? $\endgroup$ Nov 29 '18 at 16:22
  • $\begingroup$ It is well known that: i) information entropy is equivalent to statistical mechanics only if the probability densities of the equilibrium ensembles are used. But information entropy is much more general, and it is valid even for probability distributions which do not have any relation with energy or hamiltonians. So, they are not equivalent. ii) statistical mechanics formulae do not imply necessarily the thermodynamic behavior unless the thermodynamic limit is taken (provided it exists). For instance, stat. mechanics formulae do not guarantee extensivity or convexity without therm.limit. $\endgroup$
    – GiorgioP
    Nov 29 '18 at 17:10
  • $\begingroup$ This definition of "equivalent" doesn't seem to support your point, though. If the only difference between the definitions is that they have differing degrees of generality, and they otherwise agree completely whenever they are actually applicable, then there's no problem. You apply any definition which produces an answer other than "not applicable" for the entropy of the universe; whenever more than one definition is applicable in the first place, the definitions agree, so it doesn't matter which one you choose past that point. $\endgroup$ Nov 29 '18 at 17:30
  • $\begingroup$ This is an interesting subject. Can you add something to address the question? Perhaps you could answer the question from the perspectives of each form of entry you mention. $\endgroup$
    – DanielSank
    Nov 29 '18 at 17:56
  • $\begingroup$ @probably_someone: It's a little annoying to discuss about general themes as small comments to a question or to an answer. Even chat rooms are not the best tool. So, I can try to do my best, within the limits of this method. I cannot see your point. "Equivalent", for me, but I don't think to be alone, means that two definitions are fully interchangeable. Nobody would say that a harmonic force is equivalent to have no force, even though no force is the special case of a harmonic oscillator with a force constant equal to zero. Rather it is a particolar case. $\endgroup$
    – GiorgioP
    Nov 29 '18 at 22:35

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