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N) in the solution, and vi be the speciﬁc volume of each component. 78) Show that the chemical potentials of the minor components (i = 1, 2, . . , n) are given by Fig. 79) j=1 and that the chemical potential of the major component is i=0 Assuming that vi are all constant, answer the following questions. (a) Show that the Gibbs free energy of the solution is given by G(N0 , . . , Nn , T , P ) = P V + V f (φ1 , φ2 , . . 73) where φi = vi Ni /V is the volume fraction of the component i. 5) For the osmotic pressure given by the lattice model (eq.

In a very dilute solution (Fig. 9(a)), polymer chains are well separated from each other: each chain occupies a spherical region of radius Rg . In such situations, the density of polymer segments is not homogeneous. The density is high (of the order of N /Rg3 ) inside the region of the polymer coil, but is zero outside the region. 68) The polymer coils start to overlap each other when the left-hand side of this equation exceeds one. 69) c∗ = 4π 3 3 Rg is called the overlap concentration. Below the overlap concentration c∗ , or near c∗ , the distribution of segments is strongly correlated: the segment density around a given segment is higher than the average value.

Eq. 33) is only valid for the situation of |r |/N b 1, or f b/kB T The exact relation between f and r can be calculated as follows. Suppose that an external force f is applied at the chain end. Then the average of the end-to-end vector r is not zero. We shall calculate the average of r in such a situation, and obtain the relation between f and r . 34) n The probability of ﬁnding the chain in the conﬁguration {b n } = (b 1 , b 2 , . . 35) where β = 1/kB T . 36) The subscript |b| = b attached to the integral means that the integral is taken over the surface of the sphere of radius b.