Handbook of Solvents - George Wypych - ChemTech - Ventech!

4.4 Measurement of solvent activity 187
tation and diffusion coefficient yields virial coefficients of a polymer solution. The
so-called Svedberg equation reads:
⎛ D⎞
⎡ 1
2 ⎤
⎜ ⎟( 1−
ν2, spezρ1)
= RT ⎢ + 2Ac 2 2 + 3Ac
3 2 + � ⎥
[4.4.52]
⎝ s ⎠
⎣M2
⎦
where:
D diffusion coefficient
ν2,spez partial specific volume of the polymer
ρ1 density of the solvent
c2 mass by volume concentration c2 =m2/ν A2,A3, ... second, third, ... osmotic virial coefficients
M2 molar mass of the polymer
Equation [4.4.52] is strictly valid for monodisperse polymers, i.e., one single component
2. For polydisperse polymers, different averages were obtained for the sedimentation
and the diffusion coefficient, which depends on the applied measuring mode and the subsequent
calculations. The averages of M2 correspond with averages of D and s and are mixed
ones that have to be transformed into the desired common averages - for details please see
Refs. 35-37
Sedimentation-diffusion equilibrium in an ultracentrifuge gives also a virial series: 35
( 1−2,
spez 1)
2
ω h ν ρ
⎛ ∂lnc
⎜
⎝ ∂h
2
⎞ ⎡ 1
2 ⎤
⎟ = RT ⎢ + 2Ac 2 2 + 3Ac
3 2 + �⎥
⎠ ⎣M2
⎦
[4.4.53]
where:
h distance from the center of rotation
ω angular velocity
Equation [4.4.53] is again valid for monodisperse polymers only. Polydisperse polymers
lead to apparent molar mass averages and to averages of the virial coefficients which
have to be transformed into the desired common averages by appropriate calculation methods.
35-37
A somewhat different way of avoiding the virial expansion in Equation [4.4.53] was
developed by Scholte. 176,177 Without going into details, his final relation was:
where:
( 1−
2, spez 1)
2
ω h ν ρ
Mw 1 2 ∂ μ1
RT
w2 1 ∂w2
M
⎛ ⎞ ⎡ Δ ⎛ M
⎜
⎟
⎝ − ⎟
= ⎢ + ⎜
−
⎠ ⎣ ⎝MM
w2 mass fraction of the polymer
M1 molar mass of the solvent
Mn number average molar mass of the polymer
Mw mass average molar mass of the polymer
n refractive index of the solution
1 1
n w
⎞⎤
⎟
⎟⎥
⎠⎦
⎛ ⎞
⎜ ⎟
⎝ ⎠
⎛ dn dn ⎞
/ ⎜
⎟
dh ⎟
[4.4.54]
⎝dw2
⎠
Some assumptions were made for the derivation of Equation [4.4.54], especially the
partial specific volume, the refractive index, and the derivative dn/dw 2 must not depend on
the molar mass distribution of the polymer. If one further assumes that the Flory-Huggins