The Probability Distribution for a Dilute Gas

The equation for the molecular probability distribution for a dilute gas can now be written in the form... 

Consider a dense gas of hard spheres, all with mass m and diameter a. Since the collisions of hard-sphere molecules are instantaneous, the probability is zero that any particle will collide with more than one particle at a time. Hence we still suppose that the dynamical events taking place in the gas are made up of binary collisions, and that to derive an equation for the single-particle distribution function /(r, v, /) we need only take binary collisions into account. However, the Stosszahlansatz used in deriving the Boltzmann equation for a dilute gas should be modified to take into account any spatial and velocity correlations that may exist between the colliding spheres. The Enskog theory continues to ignore the possibility of correlations in the velocities before collision, but attempts to take into account the spatial correlations. In addition, the Enskog theory takes into account the variation of the distribution function over distances of the order of the molecular diameter, which also leads to corrections to the Boltzmann equation. 

In Chapter 9 there is a derivation of the Boltzmann probability distribution for classical dilute gases. There is a derivation of this probability distribution for a quantum dilute gas in Part 4. For now, we introduce it without derivation. The important fact about the Boltzmann probability distribution is that states of energy much larger than k T are... 

For a dilute gas, we can average over molecular states with a molecular probability distribution. The most probable distribution was found as an approximation to the average distribution. It is the Boltzmann distribution... 

In the previous chapter we obtained the probability distribution for molecular states in a dilute gas and obtained a formula for the thermodynamic energy of a dilute gas in terms of the partition function. Statistical mechanics will not be very useful until we obtain formulas for the other thermodynamic functions, which is the topic of this chapter. 

In Chapters 25 and 26 we applied statistical mechanics to dilute gases, based on the probability distribution for states of molecules in a dilute gas with fixed energy, fixed volume, and fixed number of molecules. This chapter presents an alternative approach to statistical mechanics, which is in principle applicable to any kind of a system. It is based on the concept of an ensemble, which is an imaginary collection of very many replicas of the real system. Each replica occupies the same macrostate as the real system, but different members of the ensemble occupy different system microstates. 

We have derived this distribution for particles without rotation or vibration, but we now assert that rotation, vibration, and electtonic motion occur independently of translation (the only motion of structureless particles) so that we can use this distribution for the translational motion of any molecules in a dilute gas. The normalized probability distribution is represented in Figure 9.7 for a velocity component of oxygen molecules at 298 K. The most probable value of the velocity component is zero, and most of the oxygen molecules have values of the velocity component between —400ms and 400ms ... 

The first model system designed to represent a dilute gas consists of noninteracting point-mass molecules that obey classical mechanics. We obtained the Maxwell-Boltzmann probability distribution for molecular velocities ... 

In the derivation of the most probable distribution there was no restriction to a particular dilute gas. We showed that p = l/(k T) for a dilute monatomic gas with negligible electronic excitation, but this relation must be valid for all dilute gases. With this identification the probability of a molecular energy level of any dilute gas is... 

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