Consider the chemical equilibrium of a solution of linear polymers made up of identical units. The basic reaction step is monomer + Nmer = (N + 1)mer. Let K_N denote the equilibrium constant for this reaction.

(a) Show from the law of mass action that the concentrations [∙∙∙] satisfy

[N + 1] = [1] ^N+1 /K₁ K₂ K₃ … K_N

(b) Show from the theory of reactions that for ideal gas conditions (an ideal solution):

Here

n_Q(N) = (2πh²/M_Nτ)^-3/2,

Where M_N is the mass of the N mer molecule, and F_N is the free energy of one N mer molecule.

(c) Assume N >> 1, so that n_Q(N) ≈ n_Q(N + 1). Find the concentration ration [N + 1]/[N] at room temperature if there is zero free energy change in the basic reaction step: that is, if ∆F = F_N+1 – F_N – F₁ = 0. Assume [1] = 10²⁰cm^-3, as for amino acid molecules in a bacterial cell. The molecular weight of the monomer is 200.

(d) Show that for the reaction to go in the direction of long molecules we need ∆F < – 0.4cV, approximately. This condition is not satisfied in Nature, but an ingenious pathway is followed that simulates the condition. An elementary discussion is given by C, Kittel.  

Transcribed Image Text:

[FN+1 - FN - F1)] exp no (N)nq (1) no (N+ 1) KN