How does the mathematical definition of entropy (ΔS=Qrev/T) give us the degree of disorder? [closed]








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I think degree of disorder in thermodynamics means the number of possible locations that a molecule can take


How does the mathematical equation signify this?




Disorder is perhaps not the best term to use in describing what entropy measures. It is perhaps better described as a measure of what we don't know about a system. If the system is likely to be in one of a handful of states then the entropy is low and we have a pretty good idea as to what state, or which of the few states, the system is likely to be. If, OTOH, it can be in any of a multitude of states with roughly equal probability the entropy is high and we have less likelihood of being able to deduce in which one of those states it might be. Consider a stick of dynamite. If we ask, before detonation, "Where are the carbon atoms in the nitroglycerine?" we can confidently answer "They are all in that little cardboard tube". After the explosion our answer has to be "They are all over the place - maybe as far away as the next county by now." Entropy is related to the dispersion of available states. Distance of a carbon atom from the original location of the dynamite stick is a state variable but clearly not the only one. Nonetheless, it helps me, at least, to understand the concept. Originally the distances of carbon atoms from the center of the stick are in a tight distribution. For purposes of illustration lets assume that this is Gaussian with standard deviation 5 cm. After the explosion it is reasonable to suppose, continuing to assume a Gaussian distribution, that the standard deviation is now 10's of meters. The reason for picking the Gaussian for this ilustration is that the entropy of a Gaussian distribution varies directly as the standard deviation. This falls out of the definition of entropy: $S = -k\_BE{log p\_i} =-k\_B\sum\limits\_{i}^{N}p\_ilog p\_i$ where there are $N$ possible states numbered with subscript $i$. The probability the system is in state $i$ is $p\_i$. $k\_B$ is the Bolzman constant. $E$ is the expectation operator. If we have, for example, 4 possible states and one of them is much more likely to be occupied than the others then we might have $$ -(.01\*log(.01) + .01\*log(.01) + .01\*log(.01) + .97\*log(.97) ) = 0.0728314$$ If, on the other hand, each of the 4 was equally likely to be occupied then we'd have $$ - (.25\*log(.25) + .25\*log(.25) + .25\*log(.25) + .25\*log(.25) ) = 0.60206$$


We can no longer be assured that that one state is the one we would expect the system to be in. We know less about the system. Uncertainty has gone up. Entropy has increased.


I'm editing this to add the important fact that if all the possible states are equally likely then $$-\sum\limits\_{i}^{N}p\_ilog p\_i = -\sum\limits\_{i}^{N}(1/N)log (1/N) = -log(1/N) = log(N)$$ 


Thia gives the definition often found in the textbooks $S= k\_BlogN$ usually written as $$S = k\_Blog\Omega$$


in which thhey use $\Omega$ instead of $N$ to represent the number of available states. The important thing to realize when confronted with this formula is that the states have to be equally probable. In the Shannon based definition they don't.


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