Number of collisions

Then it follows that the total number of collisions per unit time suffered by particles with all velocities is... 

The number of collisions z that a molecule in the gas phase makes per unit time, when only one species is present, is given by... 

Increasing either the gas velocity or the liquid droplet velocity in a scrubber will increase the efficiency because of the greater number of collisions per unit time. The ultimate scrubber in this respect is the venturi scrubber, which operates at extremely high gas and liquid velocities with a very high pressure drop across the venturi throat. Figure 29-8 illustrates a commercial venturi scrubber unit... 

The coalescence-redispersion (CRD) model was originally proposed by Curl (1963). It is based on imagining a chemical reactor as a number population of droplets that behave as individual batch reactors. These droplets coalesce (mix) in pairs at random, homogenize their concentration and redisperse. The mixing parameter in this model is the average number of collisions that a droplet undergoes. 

Suppose particle A moves through space with average speed v A will collide with a B particle if their center-to-center distance is less than or equal to ta -t- rg. Thus, particle A sweeps out an area irlrA + rB) v in which it can collide with B, and the corresponding volume swept out per second is irfrA -t- rg fv. If the concentration of B is B molecules cm , the number of collisions of B particles by this single A particle, per second, is 7r(rA -t- rgfngv. However, the volume also... 

The dependence of reaction rate on concentration is readily explained. Ordinarily, reactions occur as the result of collisions between reactant molecules. The higher the concentration of molecules, the greater the number of collisions in unit time and hence the faster the reaction. As reactants are consumed, their concentrations drop, collisions occur less frequently, and reaction rate decreases. This explains the common observation that reaction rate drops off with time, eventually going to zero when the limiting reactant is consumed. 

Fig. 8-1. The number of collisions per second depends upon concentration.

An increase in temperature of I0°C rarely doubles the kinetic energy of particles and hence the number of collisions is not doubled. Yet, this temperature increase may be enough to double the rate of a slow reaction. How can this be explained ... 

In deriving this relation it has been assumed that the distribution function does not change significantly in a distance b, so that the distribution function describing the number of v2 particles is evaluated at the same point in space as that for the vx particles. Since the number of vx particles is f(r,v1,t)drdv1, the number of collisions between particles of velocity vx and v2 in At is... 

Second Derivation of the Boltzmann Equation.—The derivation of the Boltzmann equation given in the first sections of this chapter suffers from the obvious defect that it is in no way connected with the fundamental law of statistical mechanics, i.e., LiouviUe s equation. As discussed in Section 12.6of The Mathematics of Physics and Chemistry, 2nd Ed.,22 the behavior of all systems of particles should be compatible with this equation, and, thus, one should be able to derive the Boltzmann equation from it. This has been avoided in the previous derivation by implicitly making statistical assumptions about the behavior of colliding particles that the number of collisions between particles of velocities v1 and v2 is taken proportional to /(v.i)/(v2) implies that there has been no previous relation between the particles (statistical independence) before collision. As noted previously, in a... 

Here Zj = gq/gj and ZE = energy space respectively. 

The important fact is that the number of collisions Zr increases with temperature. It may be attributed to the effect of attraction forces. They accelerate the molecule motion along the classical trajectories favouring more effective R-T relaxation. This effect becomes relatively weaker with increase of temperature. As a result the effective cross-section decreases monotonically [199], as was predicted for the quantum J-diffusion model in [186] (solid line) but by classical trajectory calculations (dotted and broken lines) as well. At temperatures above 300 K both theoretical approaches are in satisfactory mutual agreement whereas some other approaches used in [224, 225] as well as SCS with attraction forces neglected [191] were shown to have the opposite temperature dependence for Zr [191]. Thus SCS results with a... 

The average number of collisions with the wall during the interval At is half the number in the volume AvxAt ... 

We have calculated the momentum of one molecule and the number of collisions during the interval At. Now we bring the parts of the calculation together. The total momentum change in that interval is the change 2mvx that an individual molecule undergoes multiplied by the total number of collisions ... 

To set up a quantitative theory based on this qualitative picture, we need to know the rate at which molecules collide and the fraction of those collisions that have at least the energy Emin required for reaction to occur. The collision frequency (the number of collisions per second) between A and B molecules in a gas at a temperature T can be calculated from the kinetic model of a gas (Section... 

The time constant r, appearing in the simplest frequency equation for the velocity and absorption of sound, is related to the transition probabilities for vibrational exchanges by 1/r = Pe — Pd, where Pe is the probability of collisional excitation, and Pd is the probability of collisional de-excitation per molecule per second. Dividing Pd by the number of collisions which one molecule undergoes per second gives the transition probability per collision P, given by Equation 4 or 5. The reciprocal of this quantity is the number of collisions Z required to de-excite a quantum of vibrational energy e = hv. This number can be explicitly calculated from Equation 4 since Z = 1/P, and it can be experimentally derived from the measured relaxation times. 

First, consider increasing the amount of the gas while keeping the temperature and volume fixed. Figure 5-10 shows that doubling the amount of gas in a fixed volume doubles the number of collisions with the walls. Thus, pressure is directly proportional to the amount of gas. This agrees with the ideal gas equation. 

Molecular speed affects pressure in two ways that are illustrated in Figure 5-12. First, faster-moving molecules hit the walls more often than slower-moving molecules. The number of collisions each molecule makes with the wall is proportional to the molecule s speed. Second, the force exerted when a molecule strikes the wall depends on the molecule s speed. A fast-moving molecule exerts a larger force than the same molecule moving slower. Force per collision increases with speed, and number of collisions increase with speed, so the total effect of a single molecule on the pressure of a gas is proportional to the square of its speed. 

For a better understanding of the effect of changing concentrations on the rate of a chemical reaction, it helps to visualize the reaction at the molecular level. In this one-step bimolecular reaction, a collision between molecules that are in the proper orientation leads to the transfer of an oxygen atom from O3 to NO. As with the formation of N2 O4, the rate of this bimolecular reaction is proportional to the number of collisions between O3 and NO. The more such collisions there are, the faster the reaction occurs. 

The mechanism of this reaction requires a collision between an NO molecule and an O3 molecule, as shown in Figure 15-7. The rate of the reaction therefore depends on the number of collisions that occur, and the collision frequency is proportional to the number of molecules. 

When a reaction proceeds in a single elementary step, its rate law will mirror its stoichiometry. An example is the rate law for O3 reacting with NO. Experiments show that this reaction is first order in each of the starting materials and second order overall NO + 03- NO2 + O2 Experimental rate = i [N0][03 J This rate law is fully consistent with the molecular view of the mechanism shown in Figure 15-7. If the concentration of either O3 or NO is doubled, the number of collisions between starting material molecules doubles too, and so does the rate of reaction. If the concentrations of both starting materials are doubled, the collision rate and the reaction rate increase by a factor of four. 

The rate-determining step of Mechanism II is a bimolecular collision between two identical molecules. A bimolecular reaction has a constant rate on a per collision basis. Thus, if the number of collisions between NO2 molecules increases, the rate of decomposition increases accordingly. Doubling the concentration of NO2 doubles the number of molecules present, and it also doubles the number of collisions for each molecule. Each of these factors doubles the rate of reaction, so doubling the concentration of NO2 increases the rate for this mechanism by a factor offour. Consequently, if NO2 decomposes by Mechanism II, the rate law will be Predicted rate (Mechanism n) = < [N02][N02] = J [N02] ... 

The effect of double counting is most easily seen in the following calculation. Suppose that the density of molecules is Pa = Pb = 10 and that A and B are identical. Consequently, the number of collisions between A and B is... 

If we now take B equal to A, we have Pa = 20. The total number of collisions in the volume does not change and becomes ... 

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