Interchange of particles

The division of A/ 1 into two parts is true for other processes than those mentioned above. Consider, for example, the quantity 2wAb in the mixing of two liquids by successive interchange of particles, as discussed in Sec. 39. When two very dilute solutions have already been formed by the interchange of a few particles taken from the interiors of the two liquids, the work required to interchange q additional pairs of particles will bo 2qv>Au, provided that each particle is taken from a place distant from any particle that has already been substituted. Now the quantity 2qwAH is independent of the composition of the solutions and provides or contributes a unitary term thus AF will be of the form (60). 

There are two possible cases for the wavefunction of a system of identical fundamental particles such as electrons and photons. These are the symmetric and the antisymmetric cases. Experimental evidence shows that for fermions such as electrons and other particles of half integer spin the wavefunction must be anti-symmetric with respect to the interchange of particle labels. This... 

The early workers, when treating two electron systems, usually made the observation that singlet states spin functions are antisymmetric while triplet spin functions are symmetric with respect to the interchange of particles, t. e.,... 

The reason that kbOt, is higher than calculated from Eq. (6-12) may be explained qualitatively by three effects (1) splitting, coalescence, and rupture of bubbles (T18, T20) (2) direct contact of gas and particles in the transition zone from dense phase to dilute phase (F18) (3) the influence of the particle capacitance effect (M21, M22) as a result of a small steady interchange of particles between the bubble void and the emulsion. An example of this is the case where particles are raining through the bubble (D18, R8, Wl) and (4) asphericity of the bubbles (D18). If the particle capacitance effect (discussed in the next section) is responsible for high experimental values for kb b. such values should not be applied to the usual catalytic reactions, where m is on the order of unity and particle capacitance has little effect on kbOt,. For design purposes it is normally better to use experimental mass-transfer coefficients obtained by a properly sized fluid bed for the reaction system of interest. 

A function such as w+ that has the property (10.16) of being unchanged when particles 1 and 2 are interchanged is said to be symmetric with respect to interchange of particles 1 and 2. For eigenvalue -1, we have... 

Electrons belong to a greater group of particles called fermions, which change sign in Equation 1.85 for all possible interchanges of particle coordinates. The group of fermions also includes neutrons and protons. In the case of fermions, from Equation 1.85 we obtain... 

The function in (10.17) is antisymmetric with respect to interchange of particles 1 and 2, meaning that this interchange multiplies w by — 1. There is no necessity for an arbitrary function f qi, q2,..., q ) to be either synunetric or antisymmetric with respect to interchange of 1 and 2. 

In shear mixing, shear stresses give rise to slip zones and mixing takes place by interchange of particles between layers within the zone. Diffusive mixing occurs when particles roll down a sloping surface. Convective mixing is by deliberate bulk movement of packets of powder around the powder mass. 

If the molecules are actually identiced this must be dividedi by Nl because an interchange of particles does not give rise to a new quantum state of the system. However, for the present purposes, this is not im-... 

Some later models have been hybrids between these two concepts, considering both an interchange of particles across CS (due to centrifugation and turbulent dispersion) and particle migration to the cyclone wall. 

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