0.20 Equilibrium and the second law of thermodynamics  (Page 3/11)

Interestingly, however, when we set up the real ink and water experiment, we did not randomly distribute theink molecules. Rather, we began initially with a drop of ink in which the dye molecules were already congregated. We know that,according to our kinetic theory, the molecules are in constant random motion. Therefore, they must be constantly rearrangingthemselves. Since these random motions do not energetically favor any one arrangement over any other one arrangement, we can assumethat all possible arrangements are equally probable. Since most of the arrangements do not correspond to a drop of ink, then most of the time we will not observe a drop. In the case above with five blue marbles in 500 boxes, we expect tosee a drop only once in every 500 million times we look at the "glass". In a real glass of water with a real drop of ink, thechances are very much smaller than this.

We draw two very important conclusions from our model. First, the random motions of molecules make everypossible arrangement of these molecules equally probable. Second, mixing occurs spontaneously simply because there are vastly manymore arrangements which are mixed than which are not. The first conclusion tells us "how" mixing occurs, and the second tells us"why." On the basis of these observations, we deduce the following preliminary generalization: a spontaneous process occurs because itproduces the most probable final state.

Probability and entropy

There is a subtlety in our conclusion to be considered in more detail. We have concluded that all possiblearrangements of molecules are equally probable. We have further concluded that mixing occurs because the final mixed state isoverwhelmingly probable. Placed together, these statements appear to be openly contradictory. To see why they are not, we mustanalyze the statements carefully. By an "arrangement" of the molecules, we mean a specification of the location of each andevery molecule. We have assumed that, due to random molecular motion, each such arrangement is equally probable. In what sense,then, is the final state "overwhelmingly probable"?

Recall the system illustrated in [link] , where we placed three identical blue marbles into ten spaces. We calculated before that there are120 unique ways to do this. If we ask for the probability of the arrangement in [link] , the answer is thus $\frac{1}{120}$ . This is also the probability for each of the other possiblearrangements, according to our model. However, if we now ask instead for the probability of observing a "mixed" state (with nodrop), the answer is $\frac{112}{120}$ , whereas the probability of observing an "unmixed" state (with adrop) is only $\frac{8}{120}$ . Clearly, the probabilities are not the same when considering theless specific characteristics "mixed" and "unmixed".

In chemistry we are virtually never concerned with microscopic details, such as the locations of specific individual molecules. Rather, we are interested in more generalcharacteristics, such as whether a system is mixed or not, or what the temperature or pressure is. These properties of interest to usare macroscopic . As such, we refer to a specific arrangement of the molecules as a microstate , and each general state (mixed or unmixed, for example) as a macrostate . All microstates have the same probability of occurring, according to our model. As such, the macrostates havewidely differing probabilities.