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Hello, I need help doing this assignment.

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One way of understanding entropy is to say that since a system with random motion moves through all microstates with equal probability, if most microstates correspond to a particular macrostate, that's the state that the system will move towards.This is basically the second law of thermodynamics.

Perhaps the simplest example of this that lets us work out the math of this is a set of objects that take two states. The microstate is the specification of the state of each object; the macrostate is the specification of how many of each state is present. A simple physical example of this is the flipping of a set of fair coins that can come up either heads (H) or tails (T).


Hello, I need help doing this assignment.

 

Here's the link for the relevant pictures and details: http://umdberg.pbworks.com/w/page/50457651/Polymers%20and

 

%20entropy

 

One way of understanding entropy is to say that since a system with random motion moves through all microstates

 

with equal probability, if most microstates correspond to a particular macrostate, that's the state that the system will

 

move towards.This is basically the second law of thermodynamics.

 

Perhaps the simplest example of this that lets us work out the math of this is a set of objects that take two states. The

 

microstate is the specification of the state of each object; the macrostate is the specification of how many of each

 

state is present. A simple physical example of this is the flipping of a set of fair coins that can come up either heads

 

(H) or tails (T).

 

Part 1: Coin tosses

 

Consider a set of N coins. If we toss each coin, each has two ways of coming down, H or T. Since the first coin can

 

come down 2 ways, and the second coin can come down 2 ways, etc., the number of different ways (microstates)

 

that the N coins can come down is 2 x 2 x ... (N times) = 2N. While this is interesting, this is not the number we want.

 

Rather, we want to know if we choose a particular macrostate (a given number of heads and tail) how many

 

microstates correspond to that macrostate. That is, how many different ways could you get a string of coin flips that

 

came up with that particular number of heads and tails?

 

A. For 4 coins, count explicitly how many different ways there are to get each of the following macrostates: 4H, 0T 3H, 1T 2H, 2T 1H, 3T 0H, 4T. B. Now suppose that you had N coins. Create an mathematical expression that would allow you to calculate how

 

many different ways you could create a string of flips that would give M heads and (N-M) tails. Consider a set of N

 

coins that have M heads and N-M tails showing. How many different ways could you choose a sequence of the

 

coins? (Hint: You could choose the first one in N different ways. You could then choose one of the remaining N-1 in N1 different ways; etc.) Since we don't care what order we get the heads or tails in, you have to divided by the number

 

of ways of permuting the heads and the tails. This result is called NCM, the number of ways of choosing M objects out

 

of a set of N without respect to order. (What you are to do for this part of the problem is justify the expression for the

 

number of combinations in terms of the relevant factorials by describing the choosing and arranging process.) C. Use a spreadsheet to draw bar graphs of the number of microstates of coin flipping to get M heads out of N flips, NCM, as a function of

 

and 30. (You probably want to use the FACT(N) function which gives the value of N factorial (N!). An example of such a bar graph for N=

 

right. We see that 3H, 3T is the most likely result and 6H or 6T only have one way of getting them. Once you have these bar graphs, fill out the following table that shows: the fraction of the total that correspond to the 50-50 macrostate;

 

the peak (about how far down you have to go on each side of the middle for the number to fall to half -- just eyeball it); and the ratio of th

 

compared to N. The values for 6 are given in the table below. N Total number of different

 

ways the result can come out Fraction of microstates

 

that correspond to 50-50

 

(the most common macrostate) Half width

 

(eyeball it) H 6 26 = 64 20/64 = 0.31 ~3 3 10 20 30 Does the peak get wider or narrower as the number of total flips goes up?

 

Part 2: Polymer folding

 

Consider a polymer like DNA. One very simple model of such a polymer is to assume that the polymer forms a onedimensional chain consisting of N >> 1 links, each having a particular length a. Each of the links in the chain may be

 

freely oriented to the right or left, with no energy difference between these two orientations. The likelihood that each

 

link in the chain orients to the left or the right is precisely 50/50, just like a coin toss.

 

Suppose that nR is the number of elements oriented to the right and nL is the number of elements oriented to the left,

 

such that N = nL + nR. A. Refer to the figure at the right, in which one possible conformation of polymer links is illustrated (but where the individual links have

 

distributed vertically for clarity). For the example drawn, what are the values of N, nR, and nL? For the example drawn, what is the valu

 

of the link length a? B. Write down a general expression for the end-to-end extension of such a chain,L,in terms of the parameters nR,nL, and a. Of course, f

 

particular configuration drawn, your general expression must reduce to L = 6a. C. Write down an expression for the number of arrangements W as a function of the total number of links N and the

 

number of links pointing left or right, nL and nR. Explain your reasoning. (Hint: Refer back to your analysis in part 1.)

 

D.What would the state of minimum and maximum entropy of this polymer look like?

 

E. Can you use your results from parts A-D of this problem (and the second law of thermodynamics) to predict what

 

you think the natural state of such a polymer would most likely look like?

 


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