0.6 Equilibrium and the second law of thermodynamics  (Page 4/11)

We come to an important result: the probability of observing a particular macrostate ( e.g. , a mixed state) is proportional to the number of microstates with that macroscopic property. For example,from [link] , there are 112 arrangements (microstates) with the "mixed" macroscopic property.As we have discussed, the probability of observing a mixed state is $\frac{112}{120}$ , which is obviously proportional to 112. Thus, one way to measurethe relative probability of a particular macrostate is by the number of microstates $W$ corresponding to that macrostate. $W$ stands for "ways", i.e. , there are 112 "ways" to get a mixed state in [link] .

Now we recall our conclusion that a spontaneous process always produces the outcome with greatestprobability. Since $W$ measures this probability for any substance or system of interest, we couldpredict, using $W$ , whether the process leading from a given initial state to a given finalstate was spontaneous by simply comparing probabilities for the initial and final states. For reasons described below, we insteaddefine a function of $W$ ,

called the entropy , which can be used to make such predictions about spontaneity. (The $k$ is a proportionality constant which gives $S$ appropriate units for our calculations.) Notice that the more microstates thereare, the greater the entropy is. Therefore, a macrostate with a high probability ( e.g. a mixed state) has a large entropy. We now modify our previous deduction to say that a spontaneous processproduces the final state of greatest entropy. (Following modifications added below, this statement forms the Second Law of Thermodynamics .)

It would seem that we could use $W$ for our calculations and that the definition of the new function $S$ is unnecessary. However, the following reasoning shows that $W$ is not a convenient function for calculations. We consider two identicalglasses of water at the same temperature. We expect that the value of any physical property for the water in two glasses is twice thevalue of that property for a single glass. For example, if the enthalpy of the water in each glass is $H_1$ , then it follows that the total enthalpy of the water in the twoglasses together is $H_{\mathrm{total}}=2H_1$ . Thus, the enthalpy of a system is proportional to the quantity ofmaterial in the system: if we double the amount of water, we double the enthalpy. In direct contrast, we consider the calculationinvolving $W$ for these two glasses of water. The number of microstates of the macroscopic state of one glass of water is $W_1$ , and likewise the number of microstates in the second glass of wateris $W_1$ . However, if we combine the two glasses of water, the number ofmicrostates of the total system is found from the product $W_{\mathrm{total}}=\times (W_1, W_1)$ , which does not equal $2W_1$ . In other words, $W$ is not proportional to the quantity of material in the system. This isinconvenient, since the value of $W$ thus depends on whether the two systems are combined or not. (If it isnot clear that we should multiply the $W$ values, consider the simple example of rolling dice. The number of statesfor a single die is 6, but for two dice the number is $\times (6, 6)=36$ , not $6+6=12$ .)