A Toy Block Analogy to Help Understand Accessible Microstates This story is abou

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A Toy Block Analogy to Help Understand Accessible Microstates This story is about how things become disordered, which is not quite the same as increasing entropy. It is useful, however, because it will help us to get a better idea of what we mean by states and microstates, how probabilities enter the picture, and what we mean by accessible microstates. The story of the boy and his toys goes like this. Once there was a little boy who had some toys. Mostly, the toys stayed in the boy’s room. Every now and then his mother would clean up his room and put the toys away where “they belonged.” But as soon as she finished picking up and left the little boy alone in his room, he began to interact with the toys. Soon they were again scattered all over the room. And no matter how much the mother wished it would be so, it never happened that after the boy and the toys interacted, they ended up back “where they belonged.” That is, after the boy and the toys “came to equilibrium,” the toys were always scattered all over the room. An interesting question arises here: “How long does it take for the boy and toys to “come to equilibrium?” It depends on details of the interaction: how vigorously the boy plays with his toys, his attention span, etc. These questions require much more detailed information to answer. This will also be true in thermodynamics. The “how long” kinds of questions are generally not answerable from a focus on initial and final states of a system. Now we are going to simplify the analogy so the math won’t be too complicated, but the basic ideas are all still here. The toys can be in many different locations; the toy box, which is where “they belong” is one of the possible locations. For the other locations, let’s divide the floor up into one-­foot squares. Suppose the child’s room is 12 ft by 12 ft, so there is a total of 144 sq ft of floor space. Further, suppose the bed, dresser, etc., take up 45 sq ft, so there are 99 sq ft of bare floor on which toys can be scattered. A particular toy can be on any of these 99 one-­foot squares. A toy can also be in the toy box. Thus, there are 100 actual places a particular toy can be located. Let’s assume we have only two toys, a red and a green wooden block. A particular microstate of this system might be the red block in the toy box and the green block on the floor at square number 57. This is one of many possible microstates. Immediately after the mother has picked up the room, both toys will be in the toy box. This is another one of the many possible microstates of the system. How many accessible microstates are there in this system? By accessible, we mean microstates that could actually occur consistent with any constraints imposed on the system. For example, if the mother puts the blocks in the toy box and closes the lid and sits on it, the microstates consisting of blocks on particular squares of the floor are not accessible. But suppose the mother leaves the room. Now the boy can open the box. From the imposed constraints (the toys must be in the room in one of the allowed places) the toys must be either in the toy box or on the floor on one of the squares. Note that we have imposed the constraint that the toys cannot be under or on the bed, on the dresser, etc. The red block could be in one of 100 positions. For any position of the red block, the green block could also be in any of 100 positions. The total number of accessible microstates is 100 x 100 = 10,000. Note that the number of accessible microstates is determined by the constraints that are imposed on the system. For example, suppose we removed the constraint that the toys had to stay in the room. If the window were opened, the little boy might throw one of the toys outside. We could divide

Explanation / Answer

a)    accessible states are there for this system = 23 = 8

b)    total number of accessible microstates for this system   =   256

c)     probability of finding the red block in the toy box = 3/8

         the blue box on square #15   = 1/2

           the green block on square #75 = 1/8

d)    the probability of finding the red block in the toy box   = 1/16

e)   Probability if we asked for it after the boy had been playing for only 15 seconds =   1/4

f)     condition must be satisfied   =    microstates that could actually occur consistent with any constraint microstates

                                                         must have more red balls than green balls .

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