Number of particles

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. 

The quantity dnjdt is essentially the rate of disappearance of primary particles and hence the rate of decrease in the total number of particles. Integration gives... 

Then F( ) = S(/ -t d )- 2( ), and the density of states D E) = dS/d/ . A system containing a large number of particles N, or an indefinite number of particles but with a macroscopic size volume V, normally has the number of states S, which approaches asymptotically to... 

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... 

In an ideal Bose gas, at a certain transition temperature a remarkable effect occurs a macroscopic fraction of the total number of particles condenses into the lowest-energy single-particle state. This effect, which occurs when the Bose particles have non-zero mass, is called Bose-Einstein condensation, and the key to its understanding is the chemical potential. For an ideal gas of photons or phonons, which have zero mass, this effect does not occur. This is because their total number is arbitrary and the chemical potential is effectively zero for tire photon or phonon gas. 

The total number of particles in an ideal Bose gas at low temperaPires needs to be written such that the ground-state occupancy is separated from the excited-state occupancies ... 

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 closed systems with the same number of particles A and volume V (constant density) for each system at temperature T. The partition fiinction... 

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 ... 

It is a simple matter now to calculate number of particles per unit area, per unit time, that pass tln-ough a small hole in the wall of the vessel. This quantity is called the rate of effusion, denoted by n, and it governs the loss of particles in a container when there is a small hole in the wall separatmg the gas from a vacuum, say. This number is in fact obtained by integrating the quantity, 8 Af(v) over all possible velocities having the proper direction, and then dividing this number by A5f Thus we find... 

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... 

Our first result is now the average collision frequency obtained from the expression, (A3.1.10). by dividing it by the average number of particles per unit volume. Here it is convenient to consider the equilibrium case, and to use (A3.1.2) for f. Then we find that the average collision frequency, v, for the particles is... 

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. 

At the limit of extremely low particle densities, for example under the conditions prevalent in interstellar space, ion-molecule reactions become important (see chapter A3.51. At very high pressures gas-phase kinetics approach the limit of condensed phase kinetics where elementary reactions are less clearly defined due to the large number of particles involved (see chapter A3.6). 

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] ... 

A number of particles perfonn periodic rotations in a ring-shaped contamer with a small opening, through which some particles can escape. Two situations can now be distinguished. 

Case 1. The particles are statistically distributed around the ring. Then, the number of escaping particles will be proportional both to the time interval (opening time) dt and to the total number of particles in the container. The result is a first-order rate law. 

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