Please answer the last questions Candies Beads would work to

Please answer the last questions

Candies. (Beads would work too, but you don\'t get to eat that her wise identical each color in the table below. The number of the number of is roughly equal to the factorial of the number of each for example to as the \"degeneracy, \" 5 x 4 x 3 x 2 x 1= 720 waves in which these green a u example, if there are 6 green candies, there are 6! = 6 x set. Calculate the number of waves each set no i is can be arranged without changing the basic appearance of the number of ways each set of colors can be arranged and record it in the next column of your table. ND mental entropy, s, is defined as the natural logarithm of the degeneracy. Calculate the entropy for each color. Carefully place the candies by color, side by side, into a clear plastic sandwich bag as illustrated in the state is roughly the sum of the individual entropies. KEEP as many significant figures as you feel is n a\' put ta has no ambiguity; seven candies is EXACTLY 7.00000000000000000... candies. Give the bag a single shake. Comment on the result, and/or provide photo-documentation thereof. Repeat the shake and describe it again. Shake your bag of multicolored candies until any hint of the original ordering is long gone. How many shakes did it take to obliterate all semblance of order? Now that color doesn\'t figure into the arrangement (since the mechanical process of shaking doesn\'t favor one color over another), any ordering of candies is essentially the same as any other; thus, the degeneracy is simply the factorial of the total number of items in the bag. Calculate this degeneracy and its corresponding fundamental entropy. What was the change in the total entropy in the process of shaking? Explain why, or why not, one might ever expect random shaking of the bag to restore your original color -arrangement. Extra mile: repeat this experiment for a larger set of candies (a medium or large size bag) and observe the power of large

Solution

Note 1

We take a total of 50 candies of which 10 are red,5 are green,7are blue,12 are orange,8 are purple,and 8 are yellow.Each color\'s individual entropy is calculated and tabulated above.

Before shaking, we add the individual entropies to get the final entropy of the ordered system.From this entropy one can find out the degeneracy by taking the natural antilog of 69.6133.

After shaking,the entropy is simply the natural log of the degeneracy.Degeneracy here will be 50! since there is a random arrangement.

These results are also tabulated above.

Note 2

We would need at least 3-4 shakes to completely obliterate the orderly arrangement.Single shake would not be enough.After a single shake,we would see that the colors whose number of candies are less would be more scattered randomly while colors with large numbers of candies would still be in some sort of order.

Note 3

Change in total entropy in the process of shaking

=148.4777-69.6133=78.8644

which means that change in the total entropy is 78.8644.As we can see the entropy has increased in the process of shaking.This is as we would expect since the randomness has increased after shaking.

Note 4

Now, the more the degeneracy of an arrangement, the more probable it is.The degeneracy of the ordered arrangement was found to be 1.70 x 10³⁰.The degeneracy of the random arrangement,arrived after shaking,is 3.04 x 10⁶⁴.Since probability of an arrangement is directly proportional to its degeneracy,we find the the orderly arrangement is 10³⁴ times less-likely to occur than the random arrangement.Hence, shaking again and again once the random arrangement has been reached will NOT result in the ordered arrangment for overwhelming majority of the times.

Color	Number of members,n	Degeneracy,g=n!	Fundamental Entropy,s=ln g
Red	10	3628800	15.1044
Green	5	120	4.7874
Blue	7	5040	8.5251
Orange	12	479001600	19.9872
Purple	8	40320	10.6046
Yellow	8	40320	10.6046
All-Before Shaking	Precise arrangement	1.70 x 1030	69.6133
All-After Shaking	50	3.04 x 1064	148.4777