Q-moment

Full solution of Eq. 6.10 for all values of q, given a set of initial conditions, yields a mathematical description of the distribution of floccule size in a suspension as time passes. From the point of view of experimentation, information about the temporal evolution of floccule size often is obtained by measurement of the q-moments u... 

The q-moments that correspond to Eq. 6.18 can be calculated recursively after inserting this latter equation into Eq. 6.11 and taking the time derivative of both sides ... 

Once M0(t) is known, all other q-moments can be calculated using Eq. 6.19. Equation 6.13a shows that M0(t) satisfies the differential equation... 

The second q-moment follows readily from Eq. 6.19 after incorporation of Eq. 6.21 ... 

In particular, the first three q-moments satisfy the differential equations (cf. Eq. 6.13) ... 

Direct particle counting of an initially monodisperse suspension was used to measure the time dependence of the q-moment M0, as given in the following table. Examine these data for conformity to either transport- or reaction-controlled flocculation kinetics and estimate the characteristic time scale, 2/kn p0, wherekn = kmn for m n 1. (Answer k n= 3.05 x 10 22 m3 s"1 = 2KD/Wmn, corresponding to Wmn = 4.07 X 104 for all m, n.)... 

The magnitude of the quadrupole splitting is proportional to the electric field gradient q) tensor which interacts with the quadrupole [Q) moment of the nucleus. While the 5 measures the s electron charge density of the nucleus the electric field gradient tensor measures the distortion from cubic symmetry of the electron distribution and ligands around the Mossbauer atom. 

Figure 13. Nuclear magnetic dipole (fi) and electric quadrupole (Q ) moments in the sequence of 5/2" ground states of...

There is a set of moment averages for each moment distribution. The general form for the p moment average of the q moment distribution is... 

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 ... 

The rotational energy of a rigid molecule is given by 7(7 + l)h /S-n- IkT, where 7 is the quantum number and 7 is the moment of inertia, but if the energy level spacing is small compared to kT, integration can replace summation in the evaluation of Q t, which becomes... 

The long-range interactions between a pair of molecules are detemiined by electric multipole moments and polarizabilities of the individual molecules. MuJtipoJe moments are measures that describe the non-sphericity of the charge distribution of a molecule. The zeroth-order moment is the total charge of the molecule Q = Yfi- where q- is the charge of particle and the sum is over all electrons and nuclei in tlie molecule. The first-order moment is the dipole moment vector with Cartesian components given by... 

These moments are related to many physical properties. The Thomas-Kulm-Reiche sum rule says that. S (0) equals the number of electrons in the molecule. Other sum rules [36] relate S(2),, S (1) and. S (-l) to ground state expectation values. The mean static dipole polarizability is md = e-S(-2)/m,.J Q Cauchy expansion... 

In addition, there could be a mechanical or electromagnetic interaction of a system with an external entity which may do work on an otherwise isolated system. Such a contact with a work source can be represented by the Hamiltonian U p, q, x) where x is the coordinate (for example, the position of a piston in a box containing a gas, or the magnetic moment if an external magnetic field is present, or the electric dipole moment in the presence of an external electric field) describing the interaction between the system and the external work source. Then the force, canonically conjugate to x, which the system exerts on the outside world is... 

Here each < ) (0 is a vibrational wavefiinction, a fiinction of the nuclear coordinates Q, in first approximation usually a product of hamionic oscillator wavefimctions for the various nomial coordinates. Each j (x,Q) is the electronic wavefimctioii describing how the electrons are distributed in the molecule. However, it has the nuclear coordinates within it as parameters because the electrons are always distributed around the nuclei and follow those nuclei whatever their position during a vibration. The integration of equation (Bl.1.1) can be carried out in two steps—first an integration over the electronic coordinates v, and then integration over the nuclear coordinates 0. We define an electronic transition moment integral which is a fimctioii of nuclear position ... 

The electronic transition moment of equation (B1.1.5) is related to the intensity that the transition would have if the nuclei were fixed in configuration Q, but its value may vary with that configuration. It is often usefiil to expand Pi CQ) as a power series in the nonnal coordinates, Q. ... 

There are cases where the variation of the electtonic ttansition moment with nuclear configuration caimot be neglected. Then it is necessary to work with equation (B 1.1.6) keeping the dependence of on Q and integrating it over the vibrational wavefiinctions. In most such cases it is adequate to use only the tenns up to first-order in equation (B 1.1.7). This results in modified Franck-Condon factors for the vibrational intensities [12]. 

For certain values of q and a harmonic potential, the distribution pq (F) can have infinite variance and higher moments. This fact has motivated the use of the g-expectation value to compute the average of an observable A... 

Only if the total charge on the system (q) equals zero will the dipole moment be unchanged Similar arguments can be used to show that if both the charge and the dipole moment ar zero then the quadrupole moment is independent of the choice of origin. For convenience the origin is often taken to be the centre of mass of the charge distribution. 

I is the bond length. The experimental quadrupole moment is consistent with a charge, q, of approximately 0.5e. In fact, a better representation of the electrostatic potential around the nitrogen molecule is obtained using the five-charge model shown in Figure 4.20. 

Qualitatively, the selection rule for IR absorption for a given mode is that the symmetry of q T[ must be the same as q l j . Quantitatively, the transition dipole moment is proportional to the dipole derivative with respect to a given normal mode dp/dq. 

Nuclide Natural abundance, % Spin I Sensitivity at constant field relative to NMR frequency for a 1-kG field, Mffz Magnetic moment J-T-1 Electric quadrupole moment Q, 10 m ... 

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