From the most probable distribution, we derived the following expression that relates the
extent of reaction (p) to the number average degree of polymerization (DPn):
DPn=1/(1-p)
The derivative of the following function was used in the derivation.

p
(1)
So   p i 
1 p
i 1
PROVE IT BY SUBTRACTION METHOD (write first few terms of So
; write pSo , subtract & simplify)
A general form of S function and its partial derivative againest p is as follows
Sn = ∑(inpi) (n = 0,1,2…)
(2)
n+1
i-1
n+1
i
∂Sn/∂p = i ∑p = i 1/p∑p
(3)
Rearrange (3) into
p(∂Sn/∂p) = in+1∑pi = ∑(in+1pi) = ∑(in+1pi) = Sn+1
(4)
when n = 0
S1 = p(∂S0/∂p) = ∑(ipi)
When n = 1
S2 = p(∂S1/∂p) = ∑(i2pi)
When n = 2
S3 =p(∂S2/∂p) = ∑(i3pi)
.
.
.
Derive expressions for the weight average (DPw) and the z average degree of
polymerization (DPz) as a function of p and prove that DPw/DPn, DPz/DPw and DPz/DPn
approach 2, 3/2 and 3 as a full conversion is reached (p~1)
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