Thermodynamics problem dealing with series
This is for my biothermodynamics class:
MOLECULAR WEIGHT DISTRIBUTION IN POLYMERS:
NOTE: nk=n(subscript k) wk=w(subscript k). NOT multiplication.
While the fraction of polymer chains having length k is P(k) = n/∑_(k=1)^∞nk), the fraction of chains having molecular weight proportional to k is w=k*n/(∑_(k=1)^∞k*nk).
Q: a. Show that w = k(1-p)*n
b. Compute the average molecular weight, <k> = ∑_(k=1)^∞k*wk
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 nk=(1-p)p^(k-1). I have attempted to plug this into the expression for wk 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 nk 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!!
Just for clerification, here are all the term definitions:
p: the probability that a monomer unit is reacted and connected in the chain.
nkThe 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.
wk: the fraction of chains having molecular weight proportional to k.