The Number of Particles

The Number of Particles.—We have thus far considered the number of polymer particles in the emulsion as an independent variable. Ordinarily, the number of polymer particles will depend on conditions prevailing at the outset when they are being created. Specifically, their ultimate number will depend on the emulsifier and its initial concentration, and also on the rate at which primary radicals are generated. Since the number N of particles plays such an important part in determining both the rate and the degree of polymerization, it is desirable to consider briefly Smith and Ewart s theoretical estimation of it. 

The combined area zAt of all particles present at time t is given by the sums of the areas of all particles formed from time = 0 to r. Or, since pdr particles are generated in the interval r to r+dr 

Soap is assumed to form a continuous monolayer over the polymer particles at the point of exhaustion of micellar soap. If we let represent the area thus occupied by a gram of soap, the total area of the particles in one cc. at the time h when this occurs will be CsUs, where Cs is the soap concentration in grams per cc. Equating at t = t to this quantity, we have from Eq. (32) 

The number thus calculated is too large on account of the assumption 

The increase in iV, and therefore in the rate as well, with initial soap concentration is thus explained. Quantitative results agree approximately with the predicted three-fifths power dependence. The prediction of an increase in polymerization rate with also has been confirmed by experiments at variable initiator concentrations.t Most important of all, the actual number of particles N calculated from Eq. (35) agrees within a factor of two with that observed. It is thus apparent that the theory of emulsion polymerization developed by Harkins and by Smith and Ewart has enjoyed spectacular success in accounting for the unique features of the emulsion polymerization process. 

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Avogadro s number, L The number of particles (atoms or molecules) in one mole of any pure substance. L = 6 023 x 10. It has been determined by many methods including measurements of Brownian movement, electronic charge and the counting of a-particles. 

The excess heat of solution of sample A of finely divided sodium chloride is 18 cal/g, and that of sample B is 12 cal/g. The area is estimated by making a microscopic count of the number of particles in a known weight of sample, and it is found that sample A contains 22 times more particles per gram than does sample B. Are the specific surface energies the same for the two samples If not, calculate their ratio. 

Its ratio to the first temi can be seen to be (5 J / 5 Ef) E HT. Since E is proportional to the number of particles in the system A and Ej, is proportional to the number of particles in the composite system N + N, the ratio of the second-order temi to tire first-order temi is proportional to N N + N. Since the reservoir is assumed to be much bigger than the system, (i.e. N) this ratio is negligible, and the truncation of the... 

In the grand canonical ensemble, the number of particles flucPiates. By differentiating log E, equation (A2.2.121) with respect to Pp at fixed V and p, one obtains... 

When the temperaPire is high and the density is low, one expects to recover the classical ideal gas limit. The number of particles is still given by N = Thus the average number of particles is given by equation... 

Fluctuations of observables from their average values, unless the observables are constants of motion, are especially important, since they are related to the response fiinctions of the system. For example, the constant volume specific heat of a fluid is a response function related to the fluctuations in the energy of a system at constant N, V and T, where A is the number of particles in a volume V at temperature T. Similarly, fluctuations in the number density (p = N/V) of an open system at constant p, V and T, where p is the chemical potential, are related to the isothemial compressibility iCp which is another response fiinction. Temperature-dependent fluctuations characterize the dynamic equilibrium of themiodynamic systems, in contrast to the equilibrium of purely mechanical bodies in which fluctuations are absent. 

This is a collection of systems at constant p, Vand T in which the number of particles can flucPiate. It is of particular use in the sPidy of open systems. The PF... 

The grand canonical ensemble is a collection of open systems of given chemical potential p, volume V and temperature T, in which the number of particles or the density in each system can fluctuate. It leads to an important expression for the compressibility Kj, of a one-component fluid ... 

The result, (A3.1.7), can be viewed also as the number of particles per unit area per unit time colliding from... 

Now, again, we use a probabilistic argument to say that the number of particles with velocity in this total volume is given by the product of the total volume and the number of particles per unit volume with velocity Vp that is, 8v(v,Uj) To complete the calculation, we suppose that the gas is so dilute that each of the... 

There are four mechanisms that change the number of particles in this region. The particles can ... 

We again assume that there is a time interval 5/which is long compared with the duration of a binary collision but is too short for particles to cross a cell of size 5r. Then the change in the number of particles in 8r8v in time 8/ can be written as... 

The free streaming tenn can be written as the difference between the number of particles entering and leaving the small region in time 5t. Consider, for example, a cubic cell and look at the faces perpendicular to the v-... 

The number of (v, v)-collision cylinders in the region 8r8v is equal to the number of particles with velocity v in this region,/(r,v,0 r5v. 

Stosszahlansatz. The total number of (Vj, v)-collisions taking place in bt equals the total volume of the (Vj, v)-collision cylinders times the number of particles with velocity per unit volume. 

Elementary reactions are characterized by their moiecuiarity, to be clearly distinguished from the reaction order. We distinguish uni- (or mono-), hi-, and trimoiecuiar reactions depending on the number of particles involved in the essential step of the reaction. There is some looseness in what is to be considered essential but in gas kinetics the definitions usually are clearcut through the number of particles involved in a reactive collision plus, perhaps, an additional convention as is customary in iinimolecular reactions. 

The unimolecular rate law can be justified by a probabilistic argument. The number (A Vdc x dc) of particles which react in a time dt is proportional both to this same time interval dt and to the number of particles present (A Vc x c). However, this probabilistic argument need not always be valid, as illustrated in figure A3.4.2 for a sunple model [20] ... 

The first tenn, P(q), represents the interferences within particles and its contribution is proportional to the number of particle, N. The second tenn, Q(q), involves interparticle interferences and is proportional to the... 

The number of particles scattered per unit time by the field particle and detected per unit time is then... 

In the Lab frame, the projectile is scattered by 0 and the target, originally at rest, recoils tlirough angle 9g. The number of particles scattered into each solid angle in each frame remains the same, the relative speed v is now and j- = N v in each frame. Hence... 

The differential cross section da.j-/ dj for — /transitions from any one of the g-initial states is defined as [dR-j-/ / gjj, the transition frequency per unit incident current. Since current is the number of particles... 

Finally, by considering increasing the number of particles by one in the canonical ensemble (looking at the excess, non-ideal, part), it is easy to derive the Widom [34] test-particle fomuila... 

Perram J W, Petersen H G and DeLeeuw S W 1988 An algorithm for the simulation of condensed matter which grows as the 3/2 power of the number of particles Moi Phys. 65 875-93... 

Ewald s formalism reduces the infinite lattice sum to a serial complexity of in the number of particles n, which has been reduced to n logn in more recent formulations. A review of variants on Ewald summation methods which includes a more complete derivation of the basic method is in [3]. 

Molecular dynamics simulations can produce trajectories (a time series of structural snapshots) which correspond to different statistical ensembles. In the simplest case, when the number of particles N (atoms in the system), the volume V,... 

In Ihc canonical, microcanonical and isothermal-isobaric ensembles the number of particles is constant but in a grand canonical simulation the composition can change (i.e. the number of particles can increase or decrease). The equilibrium states of each of these ensembles are cha racterised as follows ... 

The correct treatment of boundaries and boundary effects is crucial to simulation methods because it enables macroscopic properties to be calculated from simulations using relatively small numbers of particles. The importance of boundary effects can be illustrated by considering the following simple example. Suppose we have a cube of volume 1 litre which is filled with water at room temperature. The cube contains approximately 3.3 X 10 molecules. Interactions with the walls can extend up to 10 molecular diameters into the fluid. The diameter of the water molecule is approximately 2.8 A and so the number of water molecules that are interacting with the boundary is about 2 x 10. So only about one in 1.5 million water molecules is influenced by interactions with the walls of the container. The number of particles in a Monte Carlo or molecular dynamics simulation is far fewer than 10 -10 and is frequently less than 1000. In a system of 1000 water molecules most, if not all of them, would be within the influence of the walls of the boundary. Clecirly, a simulation of 1000 water molecules in a vessel would not be an appropriate way to derive bulk properties. The alternative is to dispense with the container altogether. Now, approximately three-quarters of the molecules would be at the surface of the sample rather than being in the bulk. Such a situation would be relevcUit to studies of liquid drops, but not to studies of bulk phenomena. 

The Boltzmann distribution gives the number of particles n, in each energy level e, as ... 

Just as one may wish to specify the temperature in a molecular dynamics simulation, so may be desired to maintain the system at a constant pressure. This enables the behavior of the system to be explored as a function of the pressure, enabling one to study phenomer such as the onset of pressure-induced phase transitions. Many experimental measuremen are made under conditions of constant temperature and pressure, and so simulations in tl isothermal-isobaric ensemble are most directly relevant to experimental data. Certai structural rearrangements may be achieved more easily in an isobaric simulation than i a simulation at constant volume. Constant pressure conditions may also be importai when the number of particles in the system changes (as in some of the test particle methoc for calculating free energies and chemical potentials see Section 8.9). 

If we think in terms of the particulate nature of light (wave-particle duality), the number of particles of light or other electi omagnetic radiation (photons) in a unit of frequency space constitutes a number density. The blackbody radiation curve in Fig. 1-1, a plot of radiation energy density p on the vertical axis as a function of frequency v on the horizontal axis, is essentially a plot of the number densities of light particles in small intervals of frequency space. 

An interesting historical application of the Boltzmann equation involves examination of the number density of very small spherical globules of latex suspended in water. The particles are dishibuted in the potential gradient of the gravitational field. If an arbitrary point in the suspension is selected, the number of particles N at height h pm (1 pm= 10 m) above the reference point can be counted with a magnifying lens. In one series of measurements, the number of particles per unit volume of the suspension as a function of h was as shown in Table 3-3. 

The number of C H2n+2 iso mers has been calculated for values of n from 1 to 400 and the comment made that the number of isomers of C167H336 exceeds the number of particles in the known universe (10 °) These obser vations and the historical background of isomer calcu lation are described in a pa per in the April 1989 issue of the Journal of Chemical Edu cat/on (pp 278-281)... 

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