Statistical mechanics is about systems with a very large number of
particles. As the number of particles becomes very large, discrete
particles (atoms, monomers, molecules, etc.) begin to take on what
appears to be a continuous character. Hence, if one were able to count
the number of particles possessing a certain state of the system at
some specific time, they would find that the number of particles
having this state to within some range of tolerance is
roughly equal to the product of the total number of possible
configurations times the probability.
The value of in (3) represents a measurable
thermodynamic state variable. For the state
, there will be
polymers with a rms end-to-end separation-distance of
. For a
large ensemble,
. We let
and we perturb the system such that the parameters in
(3) reflect the character of distorted chain (
hence
. Because of the inverse relationships between
and
, the overall character of (3) remains
unchanged (one can check this by substitution). We approximate the
number of molecules having this state as
and use these relationships in the Maxwell-Boltzmann
expression
Now, we take the logarithm of both sides and apply Stirling's approximation at zeroth order yielding
where we have further simplified by using
. Using the definition
for
and
, we have
To find the elastic FE for swelling or contracting the coil,
we must integrate over all the microstates of . Changing the
summation to integration, the Helmholtz FE as a function of
for the elastic contribution to the FE becomes
and the result simplifies to
Now, differentiating this expression yields the optimal value for