Hard-sphere gas

The ergodic hypothesis assumes Eq. (3.9) to be valid and independent of choice of to. It has been proven for a hard-sphere gas that Eqs. (3.5) and (3.9) are indeed equivalent (Ford 1973). No such proof is available for more realistic systems, but a large body of empirical evidence suggests that the ergodic hypothesis is valid in most molecular simulations. 

According to the kinetic theory of gases, the self-diffusivity of a hard-sphere gas is given by DG = (2/5)(u)L, where (u) is the average velocity and L is the mean free path [4]. Because the mean free path of a confined particle in the liquid is about equal to the diameter of its confining volume, the contribution of the confined particle to the self-diffusivity of the liquid may be written... 

In Chap. VII we investigated the properties of a hard sphere gas molecule under conditions of low density and equilibrium. In the present chapter we shall extend this method to the calculation of the properties of gases under nonequilibrium conditions. As we shall see, this is a treatment which breaks down when the departure from equilibrium is great. However, it is the most fruitful method from the point of view of mathematical simplicity and the insight which it gives into the behavior of gases. 

While the equilibrium kinetic theory permits us to develop in fairly simple manner the properties of a dilute hard sphere gas, it becomes progressively more complicated and difficult to apply to both dense systems and systems in which there are forces acting between particles. To deal with such systems, we shall here outline briefly the very powerful statistical methods of Gibbs. ... 

Kirkwood and Poirier (1954) had earUer used a result equivalent to the PDT in a specialized context, and Stell (1985) discusses Boltzmann s use of an equivalent proach for die hard-sphere gas. 

The Boltzman probability distribution function P may be written either in a discrete hypothesis (which can be proven rigorously only for a hard-sphere gas) this is assumed ... 

A study on a dilute hard-sphere gas in the transition regime using the DSMC was conducted by [44]. The simulation is for 0.02 < Kn < 2 and unity Em and Fj. They found a weak dependence of the Nusselt number on the Peclet number, which explains the weak dependence on the axial heat conduction. In the case of constant wall heat flux, a positive thermal creep, which occurs when the exit temperature is higher than the inlet temperature, tends to increase the Nusselt number while negative thermal creep tends to decrease the Nusselt number. 

We conclude, based on the insight that we have drawn from the hard-sphere gas model and our general understanding of the molecular origins of T, that... 

The pressure of the hard-sphere gas exceeds that of the perfect gas at the same temperature and density. To a first approximation, this can be thought to be a result of a reduction in the volume available to the molecules because of the volume occupied by the molecules themselves. The hard spheres can be said to have less free volume than the perfect gas. 

The hard-sphere gas cannot be liquified. Liquification requires attractive forces. Attractive forces can also cause the pressure to be less than the perfect gas result. Interestingly, attractive forces are not required for the existence of a solid phase. If the hard sphere gas is compressed. 

The liquid-gas phase can be referred to by the single term fluid. Thus, a theory of the liquid state is of necessity also a theory of an imperfect gas. The earliest theory of the liquid state is that of van der Waals. Although more than a century old, with slight modifications it is viable today. The idea of van der Waals was that a liquid behaved as a hard sphere gas except that the pressure must include the internal pressure due to the attractive forces of the molecules in the liquid. It is reasonable to assume that the contribution of the internal pressure to the free energy is proportional to the density. Thus,... 

Simplified Transport Theory Dilute Hard-Sphere Gas... 

The second model of a dilute gas is the hard-sphere gas, which allows analysis of molecular collisions. 

Obtain a formula for the second virial coefficient of a hard-sphere gas. Solution... 

We now want to study the rate of molecular collisions in our model hard-sphere gas of a single substance. In this model system a molecule moves at a constant velocity between collisions. We first pretend that only particle number 1 is moving while the others are stationary and distributed uniformly throughout the container. The mean number of stationary particles per unit volume is given by... 

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