Dear ChE 530 students,

I don't think I ever had followed up about some nuances from class concerning light scattering and osmometry.

*** (1) Light scattering and value of K for Zimm plot

The discrepancy in class was that I wrote a value

K = 2 pi^2 (n_0)^2 (dn/dc)^2 / [ N_A (lambda_0)^2 ]

while Hiemenz and Lodge write a value that is 2 times bigger: (see p 302 eq 8.4.22)

K = 4 pi^2 (n_0)^2 (dn/dc)^2 / [ N_A (lambda_0)^2 ]

The reason for the difference was that my left-hand side of the Zimm plot equation included an extra term,

K c (1 + cos^2 theta) / R_theta

while they only had (eq 8.4.24a, p 302)

K c / R_theta

The two discrepancies are related. If we multiply and divide my left side by 2, then we reach

(2K) c [ (1 + cos^2 theta)/2 ] / R_theta

My value of 2K now equals their value of K. The term (1 + cos^2 theta)/2 equals 1 at an angle of theta=zero. Hiemenz and Lodge suggest that this angle theta is related to the light polarization, and if we use vertically polarized light and measure in the horizontal plane, then this term always equals 1. See p 296 and the paragraph in between eqs 8.3.4 and 8.3.5. Note that in the rest of the chapter they assume vertically polarized light is in use, so this factor goes away.

In total, both methods are correct, as long as they are used self-consistently. I am correcting my notes to ensure that this confusion doesn't arise in the future.

*** (2) osmometry and y-axis for interpreting data

The most typical way to interpret osmometry data is implied by eq 7.4.7a and 7.4.7b (pp 260-261 in Hiemenz and Lodge), i.e.

pi/RTc = 1/M + [ beta (V_1/M)^2 ] c + ... = 1/M + B c + B_3 c^2 + ...

The term in square brackets is the second virial coefficient B. V_1 is the partial molar volume of solvent.

An alternate derivation in other texts leads to a similar equation, except the partial molar volume and RT are accounted for a little differently,

pi/c = RT/M [ 1 + B* V_1/M c + + C* (V_1/M)^2 c^2 + ... ]

Here's what wasn't clear in class: the similar M and V_1 dependence of the second and third terms lead to a suggestion that works in some systems. Say that C* = B*/4, which is about the right order of magnitude. Then the three terms on the right form a perfect square, and

(pi/c)^1/2 = (RT/M)^1/2 [ 1 + Gamma/2 c]

and Gamma = B* V_1 / M is another form of the second virial coefficient.

The short summary for applying osmometry is to plot pi/c vs c. If it isn't linear, then try plotting (pi/c)^1/2 vs c. If you find that that isn't linear either (and here I'm imagining a real-world case, not a homework problem], then go to the Polymer Science book by Elias and read the osmometry section that discusses osmometry and scaling theory. Fig 6-9 in the osmometry handout comes from that book.

See everyone in class tonight.

Prof. Greenfield

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