Thermodynamics problem dealing with series

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missace31

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This is for my biothermodynamics class:
MOLECULAR WEIGHT DISTRIBUTION IN POLYMERS:
While the fraction of polymer chains having length k is P(k) = n/?_(k=1)^?n[sub:23j6xngx]k[/sub:23j6xngx]), the fraction of chains having molecular weight proportional to k is w=k*n/(?_(k=1)^?k*n[sub:23j6xngx]k[/sub:23j6xngx]).
Q: a. Show that w = k(1-p)*n
b. Compute the average molecular weight, <k> = ?_(k=1)^?k*w[sub:23j6xngx]k[/sub:23j6xngx]


A: a. Well... I can't figure out the answer yet despite the amount of time I have spent trying to figure this out. It is given in the book that n[sub:23j6xngx]k[/sub:23j6xngx]=(1-p)p^(k-1). I have attempted to plug this into the expression for w[sub:23j6xngx]k[/sub:23j6xngx] and manipulate the sum so as to solve for a closed form that would hopefully take a form that would simplify to the expression required to be proven. My attempts have failed thus far and I would appreciate any help I can get. I could easily simplify to a closed form if n[sub:23j6xngx]k[/sub:23j6xngx] were a steady constant, however the subscrip k tells me that although it is just a number, it will change with every value of k.

b. I know that the average can be computed like so: (?_(k=1)^?k*P(k)), but I might need a little help simplifying this expression to a closed expression as opposed to a summation.

Any help is greatly appreciate, thanks so much!!
 
Please straighten out your notation, which appears to be inconsistent, and define your terms.
 
Here is the definition of each term:
p: the probability that a monomer unit is reacted and connected in the chain.
n[sub:12b4o0f8]k[/sub:12b4o0f8]The most probable distribution. It is the statistical weight (unnormalized probability) for having a chain k monomers long. The factor of (1-p) arises in the definition for n because the terminal end unit must be unreacted for the molecule to be exactly k units long.
P(k): fraction of polymer chains having length k.
w[sub:12b4o0f8]k[/sub:12b4o0f8]: the fraction of chains having molecular weight proportional to k.
 
The probability that a polymer chain has length \(\displaystyle k\) is \(\displaystyle p^{k-1}(1-p)\). To find the mean, you can start with the geometric sum \(\displaystyle \sum_{k=0}^\infty p^k = \frac{1}{1-p}\).
Differentiate with respect to \(\displaystyle p\) to get
\(\displaystyle \sum_{k=1}^\infty kp^{k-1} = \frac{1}{(1-p)^2}\).
Now multiply by \(\displaystyle (1-p)\).

This is an example of the geometric distribution in statistics.
 
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