15.7 Statistical interpretation of entropy and the second law of  (Page 2/8)

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The two most orderly possibilities are 5 heads or 5 tails. (They are more structured than the others.) They are also the least likely, only 2 out of 32 possibilities. The most disorderly possibilities are 3 heads and 2 tails and its reverse. (They are the least structured.) The most disorderly possibilities are also the most likely, with 20 out of 32 possibilities for the 3 heads and 2 tails and its reverse. If we start with an orderly array like 5 heads and toss the coins, it is very likely that we will get a less orderly array as a result, since 30 out of the 32 possibilities are less orderly. So even if you start with an orderly state, there is a strong tendency to go from order to disorder, from low entropy to high entropy. The reverse can happen, but it is unlikely.

100-coin toss
Macrostate Number of microstates
100 0 1
99 1 $1\text{.}0×{\text{10}}^{2}$
95 5 $7\text{.}5×{\text{10}}^{7}$
90 10 $1\text{.}7×{\text{10}}^{\text{13}}$
75 25 $2\text{.}4×{\text{10}}^{\text{23}}$
60 40 $1\text{.}4×{\text{10}}^{\text{28}}$
55 45 $6\text{.}1×{\text{10}}^{\text{28}}$
51 49 $9\text{.}9×{\text{10}}^{\text{28}}$
50 50 $1\text{.}0×{\text{10}}^{\text{29}}$
49 51 $9\text{.}9×{\text{10}}^{\text{28}}$
45 55 $6\text{.}1×{\text{10}}^{\text{28}}$
40 60 $1\text{.}4×{\text{10}}^{\text{28}}$
25 75 $2\text{.}4×{\text{10}}^{\text{23}}$
10 90 $1\text{.}7×{\text{10}}^{\text{13}}$
5 95 $7\text{.}5×{\text{10}}^{7}$
1 99 $1\text{.}0×{\text{10}}^{2}$
0 100 1
Total: $1\text{.}\text{27}×{\text{10}}^{\text{30}}$

This result becomes dramatic for larger systems. Consider what happens if you have 100 coins instead of just 5. The most orderly arrangements (most structured) are 100 heads or 100 tails. The least orderly (least structured) is that of 50 heads and 50 tails. There is only 1 way (1 microstate) to get the most orderly arrangement of 100 heads. There are 100 ways (100 microstates) to get the next most orderly arrangement of 99 heads and 1 tail (also 100 to get its reverse). And there are $1.0×{\text{10}}^{\text{29}}$ ways to get 50 heads and 50 tails, the least orderly arrangement. [link] is an abbreviated list of the various macrostates and the number of microstates for each macrostate. The total number of microstates—the total number of different ways 100 coins can be tossed—is an impressively large $1\text{.}\text{27}×{\text{10}}^{\text{30}}$ . Now, if we start with an orderly macrostate like 100 heads and toss the coins, there is a virtual certainty that we will get a less orderly macrostate. If we keep tossing the coins, it is possible, but exceedingly unlikely, that we will ever get back to the most orderly macrostate. If you tossed the coins once each second, you could expect to get either 100 heads or 100 tails once in $2×{\text{10}}^{\text{22}}$ years! This period is 1 trillion ( ${\text{10}}^{\text{12}}$ ) times longer than the age of the universe, and so the chances are essentially zero. In contrast, there is an 8% chance of getting 50 heads, a 73% chance of getting from 45 to 55 heads, and a 96% chance of getting from 40 to 60 heads. Disorder is highly likely.

Disorder in a gas

The fantastic growth in the odds favoring disorder that we see in going from 5 to 100 coins continues as the number of entities in the system increases. Let us now imagine applying this approach to perhaps a small sample of gas. Because counting microstates and macrostates involves statistics, this is called statistical analysis    . The macrostates of a gas correspond to its macroscopic properties, such as volume, temperature, and pressure; and its microstates correspond to the detailed description of the positions and velocities of its atoms. Even a small amount of gas has a huge number of atoms: of an ideal gas at 1.0 atm and $0º C$ has $2\text{.}7×{\text{10}}^{\text{19}}$ atoms. So each macrostate has an immense number of microstates. In plain language, this means that there are an immense number of ways in which the atoms in a gas can be arranged, while still having the same pressure, temperature, and so on.

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