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Posted by on Nov 14, 2011 in Physical Chemistry | 0 comments

Entropy (Part 2): Randomness by rolling three dice

In blog entry Entropy (Part 1): Randomness by rolling two dice, it is suggested the difficulty students have in understanding that entropy is a measure of randomness can be approached by rolling dice. In the first entry two dice were rolled but in that case there are only 36 arrangements and 10 outcomes (rolls from 2 to 12). This does not show that the most random state dominates (i.e. the one with most number of arrangements consistent with a roll of 7) . To show that more dice need be rolled.  In this entry three dice are shown to have more randomness in the outcomes (3 to 18).

For three dice we see that the probability of the least ordered states, 3 and 18, have a much lower chance of occurring than the most probable outcomes which are 10 and 11.

For three dice there are 63=216 possible arrangements, so the chance of a 3 or an 18 outcome is 1/216.  But there are 27 arrangements that give a 10 or 11, with a probability of 27 times greater than rolling a 3 or 18.

Figure 1 - Physical Chemistry, Entropy - 3 Dice

Figure 1

Figure 2 - Physical Chemistry, Entropy - 3 Dice

Figure 2

Figure 1 shows a plot of the outcomes relative to the the probability.  In a series of 3,000 rolls, the number of times a given outcome occurs is recorded, and this is divided by the total number of rolls.  For enough rolls the distribution becomes Gaussian and each column is the probability of getting an outcome for a given roll.

Figure 2 (click to enlarge) shows the distribution of possible arrangements for the different outcomes. Clearly in comparison to two dice (whence six arrangements gives a 7 and only one gives a 2 or 12), the distribution is more peaked in the center than the distribution of two dice (compare with Figure 3).

The number of accessible states for two dice is 36 and for three it is 216, but this is a long way from rolling Avogadro’s constant of dice. In that case the distribution is so peaked, only the random states need be retained. To move towards that limit, in the next entry four dice are rolled followed by ten.

Figure 3 - Physical Chemistry, Entropy - 3 Dice

Figure 3

 


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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