Gyrators

VSTR = O Connell characteristic volume parameter, cm /g-mol ZRA = Rackett equation parameter RD = mean radius of gyration, A DM = dipole moment, D R = UNIQUAC r Q = UNIQUAC q QP = UNIQUAC q ... 

MEAN RADIUS OF GYRATION, ANGSTROMS DIPCLF MOMENT, DEBYES... 

MEAN RADIUS OF GYRATION OF COMPONENT I I A I. CRITICAL TEMPERATURE OF COMPONENT I (DEGREES K). TEMPERATURE OF MIXTURE (DEGREES Kl. 

Umesi, N.O. (1980), Diffusion coefficients of dissoived gases in iiquids -Radius of gyration of solvent and solute . M.S. Thesis, The Pennsylvania State University, PA. 

Fig. XI-6. Polymer segment volume fraction profiles for N = 10, = 0-5, and Xi = 1, on a semilogarithinic plot against distance from the surface scaled on the polymer radius of gyration showing contributions from loops and tails. The inset shows the overall profile on a linear scale, from Ref. 65.

Estimate the thickness of a polymer layer from the loop profile in Eq. XI-20. Assume x = 0,Xs = 2.,= 0.01, and N = Ifr. Calculate the second moment of this profile (this is often measured by ellipsometry) and compare this thickness to the radius of gyration of the coil Rg = VN/6. 

For free particles, the mean square radius of gyration is essentially the thennal wavelength to within a numerical factor, and for a ID hamionic oscillator in the P ca limit. 

A graphical method, proposed by Zimm (thus tenned the Zinnn plot), can be used to perfomi this double extrapolation to detemiine the molecular weight, the radius of gyration and the second virial coefficient. An example of a Zinnn plot is shown in figure Bl.9.6 where the light scattering data from a solution of poly... 

One of the most important fiinctions in the application of light scattering is the ability to estimate the object dimensions. As we have discussed earlier for dilute solutions containing large molecules, equation (B 1.9.38) can be used to calculate tire radius of gyration , R, which is defined as the mean square distance from the centre of gravity [12]. The combined use of equation (B 1.9.3 8) equation (B 1.9.39) and equation (B 1.9.40) (tlie Zimm plot) will yield infonnation on R, A2 and molecular weight. 

The radius of gyration of tire whole particle, R can be obtained from the distance distribution fimction p(r) as... 

The above radius of gyration is for an isotropic system. If the system is anisotropic, the mean square radius of gyration is equal to... 

Thus the slope of the I q) versus q plot is related to the values of two radii of gyration. 

In dilute polymer solutions, hydrodynamic interactions lead to a concerted motion of tire whole polymer chain and tire surrounding solvent. The folded chains can essentially be considered as impenneable objects whose hydrodynamic radius is / / is tire gyration radius defined as... 

Figure C2.5.6. Thennodynamic functions computed for the sequence whose native state is shown in figure C2.5.7. (a) Specific heat (dotted curve) and derivative of the radius of gyration with respect to temperature dR /dT (broken curve) as a function of temperature. The collapse temperature Tg is detennined from the peak of and found to be 0.83. Tf, is very close to the temperature at which d (R )/d T becomes maximum (0.86). This illustrates...

Altliough tire tlieories of colloid stability and aggregation kinetics were developed several decades ago, tire actual stmcture of aggregates has only been studied more recently. To describe tire stmcture, we start witli tire relationship between tire size of an aggregate (linear dimension), expressed as its radius of gyration and its mass m ... 

For a continuous distribution, summation may be replaced by integration and by assuming a Gaussian distribution of size, Stoeckli arrives at a somewhat complicated expression (not given here) which enables the total micropore volume IFo, a structural constant Bq and the spread A of size distribution to be obtained from the isotherm. He suggests that Bq may be related to the radius of gyration of the micropores by the expression... 

For a body that consists of n masses mj, each separated by a distance rj from the axis of rotation of the array, the radius of gyration is defined... 

We may therefore think of r as the weight average value of r, by analogy with Eq. (1.12). As a reminder of how the radius of gyration comes to be defined this way, recall that the moment of inertia I of this same body is given by... 

There exists some radial distance from the axis of rotation at which all of the mass could be concentrated to produce the same moment of inertia that the actual distribution of mass possesses. This distance is defined to be the radius of gyration. According to this definition,... 

As should be expected, both (fg ) and r show the same dependence on the degree of polymerization or molecular weight. Since the radius of gyration can be determined experimentally through the measurement of viscosity or light scattering, it is through this quantity that we shall approach the evaluation of 1. 

The summation in Eq. (2.52) corresponds to n times the square of a two-dimensional radius of gyration r 2d there is no rotation in the z direction. 

The three-dimensional radius of gyration of a random coil was discussed in Sec. 1.10 and found to equal one-sixth the mean-square end-to-end distance of the polymer [Eq. (1.59)]. What we need now is a connection between two-and three-dimensional radii of gyration. Since the molecule has spherical symmetry r, r> = V + r + r, = 3r . If only two of these contributions are present, we obtain (2/3)rg 3 = rg2o- this result and Eq. (1.59) to... 

At first glance, the contents of Chap. 9 read like a catchall for unrelated topics. In it we examine the intrinsic viscosity of polymer solutions, the diffusion coefficient, the sedimentation coefficient, sedimentation equilibrium, and gel permeation chromatography. While all of these techniques can be related in one way or another to the molecular weight of the polymer, the more fundamental unifying principle which connects these topics is their common dependence on the spatial extension of the molecules. The radius of gyration is the parameter of interest in this context, and the intrinsic viscosity in particular can be interpreted to give a value for this important quantity. The experimental techniques discussed in Chap. 9 have been used extensively in the study of biopolymers. 

We saw in Chap. 1 that the random coil is characterized by a spherical domain for which the radius of gyration is a convenient size measure. As a tentative approach to extending the excluded volume concept to random coils, therefore, we write for the volume of the coil domain (subscript d) = (4/3) n r, and combining this result with Eq. (8.90), we obtain... 

Both the intrinsic viscosity and GPC behavior of random coils are related to the radius of gyration as the appropriate size parameter. We shall see how the radius of gyration can be determined from solution viscosity data for these... 

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