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Most Random Systems
Published in Philipp Kornreich, Mathematical Models of Information and Stochastic Systems, 2018
Because λ has the dimension of length, as expected, the probability density has dimensions of reciprocal length. Substituting Equation 8.60 into the constraint Equation 8.72 to evaluate the exponent containing the Lagrange multiplier constant β: () a)1=1λe−1−β∫0adxorb)1λe−1−β=1a
Fundamentals of Polymer Solutions
Published in E. D. Goddard, K. P. Ananthapadmanabhan, Interactions of Surfactants with Polymers and Proteins, 2018
Neutron scattering has become an increasingly important tool in the study of polymer solutions, especially nondilute solutions, owing to the possibility to create contrast among different constituents of the solution by selective isotopic labeling, usually with deuterium. Essentially all the phemomenology of scattering introduced above for light scattering applies as well to neutron scattering,100 except for the mechanism of generating scattering contrast. Whereas electromagnetic waves interact with the electrons in a scattering sample, neutrons are scattered by the atomic nuclei. The quantity that plays the role of (∂n/∂c) in a neutron-scattering experiment is the neutron-scattering contrast length density, (b/v)neut, having the dimension of reciprocal length squared. The data on which these contrast lengths are based, the scattering length densities for individual nuclei, (b/v)n, are tabulated for every nucleus and can be calculated for any type of solvent or polymer segment. To make the analogy complete, the contrast length density for light scattering, (b/v)light,
Plastics Properties and Testing
Published in Manas Chanda, Plastics Technology Handbook, 2017
Most resins by nature are clear and transparent. They can be colored by dyes and will become opaque as pigments or fillers are added. Polystyrene and poly (methyl methacrylate) are well known for their optical clarity, which even exceeds that of most glass. Optical clarity is a measure of the light transmitting ability of the material. It depends, among other things, on the length of the light path, which can be quantitatively expressed by the Lambert-Beer law, or log (I/I0) = −AL, where I/I0 is the fraction of light transmitted, L is the path length, and A is the absorptivity of the material at that wavelength. The absorptivity describes the effect of path length and has the dimension of reciprocal length.
Thermal gaussian wave packets
Published in Molecular Physics, 2021
A free particle of mass m is considered, the initial state of which is given by with coefficients where () and d are quantities with the dimension of a length; () and () are quantities with the dimension of a reciprocal length.
Mechanism of hydrolyzable metal ions effect on the zeta potential of fine quartz particles
Published in Journal of Dispersion Science and Technology, 2018
Chunfu Liu, Fanfei Min, Lingyun Liu, Jun Chen, Jia Du
In terms of the Gouy–Chapman approach, the Debye–Hükel reciprocal length parameter (κ) can be written aswhere ε is the dielectric constant of the medium, ci is the concentration of each type of ion in solution, zi is the valence of the ion of type i in the medium, and e is the magnitude of the charge on a mole of electrons (e = 96485C).[26]
The electric conductance of dilute sulfuric acid in water: a new theoretical interpretation
Published in Molecular Physics, 2021
In extending the DHO theory to DHO–SiS, first ω is multiplied by a factor p*, defined as This affords ω’ (‘omega prime’), In Equation (9), ν+ and ν− are, as above, the ion multiplicities of the positive and negative ions, respectively, and t+0 (= λ+0/Λ0) and t−0 (= λ−0/Λ0) are the limiting values of the corresponding transport numbers (ti’s). A corrected Equation (5) (so, corrected limiting law, LL) is thus obtained by replacing ω by ω’: To extend Equation (5a) to finite electrolyte concentration well above the LL, κ is replaced by −Ψ* of the DH–SiS theory (Smaller-ion Shell treatment [7]), where Ψ* = Ψ/δ, Ψ being the dimensionless total extra-electrostatic potential energy of ion–ion interaction. Since δ (as defined above) is a characteristic distance (here, in Å), −Ψ* is a reciprocal length; at low concentration, it ‘collapses’ to κ/(1+κa) of the DH extended equation, DHEE, a being the counterion distance of closest approach; and at infinite dilution, it asymptotically approaches κ, as in the DH theory, thus giving a LL. In the DH–SiS theory, this LL is identical to DHLL. At finite concentration, Ψ* is different for the cation and anion, being Ψ+* and Ψ−*, respectively [7], so, The Ψi* functions include the three ISPs of a binary electrolyte system (here used in Å units): a, b+ and b−; a is the counterion distance of closest approach, and the bi’s are the co-ion distances of closest approach of the positive (subscript +) and negative (subscript −) ions, so being their corresponding effective diameters. A non-additivity factor d/2 is defined as a - (b+ + b−)/2. For the ISP condition b+ ≤ a ≤ b− (of an electrolyte type called ‘Type I’), Ψ+* and Ψ−* are given [7,23] in terms of κ and the ISPs as Finally, to obtain the complete DHO–SiS expressions, a correction factor ω† (‘omega dagger’) is introduced, defined as so, The equivalent conductivity of the entire binary electrolyte solution is