A gas consists of a mixture of two types of molecules,...

A gas consists of a mixture of two types of molecules, having molecular masses $M_1$ and $M_2$ grams, and number densities $N_1$ and $N_2$ molecules per cubic centimeter, respectively. The cross-section for collisions between the two different kinds of molecules is given by $A\left|V_{12}\right|$, where $A$ is a constant, and $V_{12}$ is the relative velocity of the pair. (a) Derive the average, over all pairs of dissimilar molecules, of the center-of-mass kinetic energy per pair. (b) How many collisions take place per cubic centimeter per second between dissimilar molecules?

A gas consists of a mixture of two types of molecules, having molecular masses $M_1$ and $M_2$ grams, and number densities $N_1$ and $N_2$ molecules per cubic centimeter, respectively.

The cross-section for collisions between the two different kinds of molecules is given by $A\left|V_{12}\right|$, where $A$ is a constant, and $V_{12}$ is the relative velocity of the pair.
(a) Derive the average, over all pairs of dissimilar molecules, of the center-of-mass kinetic energy per pair.
(b) How many collisions take place per cubic centimeter per second between dissimilar molecules?

Key Concepts

Kinetic Theory of Gases

This theory provides the framework for understanding the motion, collisions, and energy distribution of molecules in a gas. It connects microscopic properties, such as molecular speeds and collisions, to macroscopic observables like pressure and temperature, and underpins the statistical averaging used when dealing with ensembles of particle collisions.

Center-of-Mass Frame

The center-of-mass frame is a coordinate system in which the total momentum of a system is zero. Analyzing problems in this frame simplifies calculations, especially for collisions, by isolating the energy available in the relative motion of the colliding particles from the translational energy of the whole system.

Reduced Mass

In two-body problems, the concept of reduced mass (? = M?M?/(M?+M?)) is used to simplify the expression for the kinetic energy associated with the relative motion in the center-of-mass frame. This effective mass allows for the treatment of a two-body collision as a one-body problem with a single mass moving in the potential created by the other.

Relative Velocity Distribution

This concept involves the statistical distribution of the differences in velocities between pairs of molecules. It is crucial for averaging quantities like the kinetic energy in the center-of-mass frame and determines the likelihood and intensity of molecular collisions in a gas.

Collision Cross-Section

The collision cross-section represents an effective area that quantifies the likelihood of two molecules interacting or colliding. When combined with the relative velocity, it determines the rate at which collisions occur, acting as a bridge between microscopic encounter dynamics and macroscopic collision rates.

Collision Frequency

Collision frequency refers to the number of collisions per unit time per unit volume in a gas. It is determined by the number densities of interacting molecules, the relative velocity between them, and the collision cross-section. This concept is key to understanding energy transfer and reaction rates in gaseous systems.

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