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Posted by on Nov 21, 2011 in Physical Chemistry | 0 comments

Entropy (part 3): Randomness by rolling four dice

The difficulty students have in understanding that entropy is a measure of randomness can be illustrated by rolling dice. Two and three dice are treated in the first two entries,

Entropy (Part 1): Randomness by rolling two dice

Entropy (Part 2): Randomness by rolling three dice

In this entry four dice are rolled.

The basic idea is that a physical system has many different arrangements (states) of particles which are consistent with some macroscopic quantity, like the temperature. Boltzmann found that out of all possible ways those particles can be arranged, only those that are consistent with the actual temperature need be considered.  The chance of any other arrangements is negligible in comparison.  Rolling dice illustrates this nicely.

One die is considered to be a particle with 6 states, so n dice have have 6n possible arrangements: thus two dice have 36, three have 216 and four dice have 1,296 arrangements. Four dice have outcomes of 4 to 24 and the most probable roll is the one with the most randomness (different ways of forming the same outcome), just like Boltzmann realized for a physical system.

Figure 1 Entropy

Figure 1

Figure 2 Entropy

Figure 2

Figure 3 Entropy

Figure 3

Even so, rolling four dice does not allow us to ignore the ordered states. Certainly you can roll four dice and the values of 4 and 24 do occur from time to time, but the most probable outcome is 14. Clearly there are many more ways of rolling four dice to give a 14 than any other outcome. The outcome 14 is the most random state for four dice.

For four dice the distribution is even sharper than for three. The plot in Figure 1 is for 10,000 rolls of 4 dice and figures 2 and 3 give the probability distributions for two and three dice respectively. Although this is not quite statistical they are pretty good. Note again the same conclusions but now there are 64=1,296 arrangements and the chance of a 4 or 24 outcome is very small, less than 0.08%. Rolling a 14 uses up most of the arrangements and is the most probable. Compare the chance of rolling a two in Figure 3 and a three in Figure 2. For more dice, these ordered states become less probable relative to the most random state.

In the next entry I jump to 10 dice with over 60 million arrangements. Still 610 is a long way from Avogadro’s number of dice with 61023 particles.

None-the-less we can begin to see the most probable outcome corresponds to the most number of arrangements: this is the state with the most randomness.

 

The interactive software used in this video is part of the General Chemistry Tutorial and General Physics Tutorial, from MCH Multimedia. These cover most of the topics found in AP (Advanced Programs in High School), and college level Chemistry and Physics courses.

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