Physical Chemistry Third Edition

28.8 Polymers in Solution 1199

whereη 0 is the viscosity of the pure solvent. Equation (28.8-3) can be written in the

form

ηsp

η

η 0

− 1 ηr− 1 

5

2

φ (28.8-4)

whereηspis called thespecific viscosityandηris called therelative viscosity. Since the

volume of the spherical particles is proportional to their number, the specific viscosity

of a dilute suspension of spheres is proportional to the concentration of the spheres.

We denote the mass concentration of the polymer byc, usually expressed in grams per

deciliter. The specific viscosity divided bycis independent of the concentration:

1

c

ηsp

5

2

1

c

v

V



5

2

1

M

4

3

πr^3 (28.8-5)

whereMandrare the mass and the radius of one of the spheres.

Exercise 28.14

Verify Eq. (28.8-5).

The behavior of a dilute solution of polymer molecules becomes more and more

like that of a suspension of spheres as the concentration is made smaller, because the

polymer molecules become more distant from each other and interfere less with each

other. We define theintrinsic viscosity,[η], also called thelimiting viscosity number,by

[η]lim

c→ 0

(

1

c

ηsp

)

lim

c→ 0

[

1

c

(

η

η 0

− 1

)]

(28.8-6)

We assume that the radius of the sphere occupied by a polymer molecule and associated

solvent molecules is proportional to the root-mean-square end-to-end distance and is

thus proportional to the square root of the molecular mass, denoted byM. The volume

of the sphere is proportional toM^3 /^2 and the intrinsic viscosity is proportional toM^1 /^2 :

[η]K′M^1 /^2 (28.8-7)

whereK′depends on temperature and on the identities of the solvent and polymer, but

not onM. Equation (28.8-7) is called theMark–Houwink equation. It is considered to

be valid for a polymer in a theta solvent. For other solvents we can write a modified

equation:

[η]KMa (28.8-8)

where the constantKand the exponentaare determined by experiment. Flory and

Leutner prepared monodisperse samples (samples with molecules of nearly the same

molecular mass) of polyvinyl alcohol (PVA) and found that for aqueous solutions of

PVA at 25◦C^41

[η](2. 0 × 10 −^4 dL g−^1 )(M/1 amu)^0.^76 (28.8-9)

so thata 0 .76 for PVA.

For a polydisperse sample of a single polymer, one can apply Eq. (28.8-8) to each

molecular mass that is present, multiply each equation byWi, the mass fraction for

(^41) P. J. Flory and F. S. Leutner,J. Polymer. Sci., 5 , 267 (1950).