Explain the teremr collision diametere, collision Cross-
Section, collision numbere, collision frequency and
mearn freee path.
Answers
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Explanation:
Collision theory is a theory proposed independently by Max Trautz in 1916 and William Lewis in 1918, that qualitatively explains how chemical reactions occur and why reaction rates differ for different reactions. The collision theory states that when suitable particles of the ractant hit each other, only a certain percentage of the collisions cause any noticeable or significant chemical change; these successful changes are called successful collisions. The successful collisions have enough energy, also known as activation energy, at the moment of impact to break the preexisting bonds and form all new bonds. This results in the products of the reaction. Increasing the concentration of the reactant particles or raising the temperature, thus bringing about more collisions and therefore many more successful collisions, increases the rate of reaction.
Collision Energy
Consider two particles and in a system. The kinetic energy of these two particles is
Let us change to center-of-mass and relative momenta, which are given by
where is the total mass of the two particles. Substituting this into the kinetic energy, we find
where
is called the reduced mass of the two particles. Note that the kinetic energy separates into a sum of a center-of-mass term and a relative term.
Now the relative position is so that the relative velocity is or . Thus, if the two particles are approaching each other such that , then . However, by equipartitioning the relative kinetic energy, being mass independent, is
which is called the collision energy.
Collision cross section
Consider two molecules in a system. The probability that they will collide increases with the effective “size” of each particle. However, the size measure that is relevant is the apparent cross-section area of each particle. For simplicity, suppose the particles are spherical, which is not a bad approximation for small molecules. If we are looking at a sphere, what we perceive as the size of the sphere is the cross section area of a great circle. Recall that each spherical particle has an associated “collision sphere” that just encloses two particles at closest contact, i.e., at the moment of a collision, and that this sphere is a radius , where is the diameter of each spherical particle (see lecture 5). The cross-section of this collision sphere represents an effective cross section for each particle inside which a collision is imminent. The cross-section of the collision sphere is the area of a great circle, which is . We denote this apparent cross section area . Thus, for spherical particles and with diameters and , the individual cross sections are
The collision cross section, is determined by an effective diameter characteristic of both particles. The collision probability increases of both particles have large diameters and decreases if one of them has a smaller diameter than the other. Hence, a simple measure sensitive to this is the arithmetic average
and the resulting collision cross section becomes
which, interestingly, is an average of the two types of averages of the two individual cross sections, the arithmetic and geometric averages!
Average collision Frequency
Consider a system of particles with individual cross sections . A particle of cross section that moves a distance in a time will sweep out a cylindrical volume (ignoring the spherical caps) of volume (Figure ). If the system has a number density , then the number of collisions that will occur is
Figure : Collision cylinder. Any particle that partially overlaps with this volume will experience a collision with a test particle tracing out this volume.
We define the average collision rate as , i.e.,
where is the average relative speed. If all of the particles are of the same type (say, type ), then performing the average over a Maxwell-Boltzmann speed distribution gives
where is the reduced mass. The average speed of a particle is
so that
Mean Free Path
The mean free path is defined as the distance a particle will travel, on average, before experiencing a collision event. This is defined as the product of the speed of a particle and the time between collisions. The former is , while the latter is . Hence, we have
Random Walks
In any system, a particle undergoing frequent collisions will have the direction of its motion changed with each collision and will trace out a path that appears to be random. In fact, if we treat the process as statistical, then, we are, in fact, treating each collision event as a random event, and the particle will change its direction at random times in random ways! Such a path might appear as shown in Figure \(\PageIndex{2. Such a path is often referred to as a random walk path.
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