The Hard-Sphere Gas

The simplest representation of the pair potential function is the hard-sphere potential  

We take as our approximate equation of state that of an ideal gas with a volume equal to the volume in which each particle can actually move. 

9 Gas Kinetic Theory The Moiecuiar Theory of Diiute Gases at Equiiibrium 

Assume that the parameter b in Eq. (9.8-6) can be identified with the van der Waals parameter b. Calculate the radius of an argon atom from the value of the van der Waals parameter b in Table A.3. 

From Table A.15, d = 3.61 x 10 m. This value is calculated from viscosity data taken at 293 K. The values differ because atoms and molecules are not actually hard spheres, and different kinds of measurements give different effective hard-sphere sizes. 

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

A second model system was the hard-sphere gas. We derived an approximate equation of state for this system and discussed molecular collisions using this model system. We obtained formulas for the mean free paths between collisions and for collision rates, for both one-component and multicomponent systems. An important result was that the total rate of collisions in a one-component gas was proportional to the square of the number density and to the squareroot of the temperature. In a multicomponent gas, therateof collisions between molecules of two different substances was found to be proportional to the number densities of both substances and to the square root of the temperature. 

The virial coefficient of the hard-sphere gas is equal to a constant... 

The second virial coefficient of a hard-sphere gas is positive, illustrating the fact that repulsive forces correspond to a raising of the pressure of the gas over that of an ideal gas at the same molar volume and temperature. The second virial coefficient of the square-well gas has a constant positive part that is identical with that of the hard-sphere gas, and a temperature-dependent negative part due to the attractive part of the potential, illustrating the fact that attractive forces contribute to lowering the pressure of the gas at fixed volume and temperature. 

Now let us add the possibility of collisions. Before we proceed, we make the following two assumptions (1) only binary collisions occur, i.e. we rule out situations in which three or more hard-spheres simultaneously come together (which is a physically reasonable assumption provided that the gas is sufficiently dilute), and (2) Boltzman s Stosszahlansatz, or his molecular chaos assumption that the motion of the hard-spheres is effectively pairwise uncorrelated i.e. that the pair-distribution function is the product of individual distribution functions ... 

When bounding walls exist, the particles confined within them not only collide with each other, but also collide with the walls. With the decrease of wall spacing, the frequency of particle-particle collisions will decrease, while the particle-wall collision frequency will increase. This can be demonstrated by calculation of collisions of particles in two parallel plates with the DSMC method. In Fig. 5 the result of such a simulation is shown. In the simulation [18], 2,000 representative nitrogen gas molecules with 50 cells were employed. Other parameters used here were viscosity /r= 1.656 X 10 Pa-s, molecular mass m =4.65 X 10 kg, and the ambient temperature 7 ref=273 K. Instead of the hard-sphere (HS) model, the variable hard-sphere (VHS) model was adopted in the simulation, which gives a better prediction of the viscosity-temperature dependence than the HS model. For the VHS model, the mean free path becomes ... 

Figure 3.10. Phase diagrams of attractive monodisperse dispersions. Uc is the contact pair potential and (j) is the particle volume fraction. For udk T = 0, the only accessible one-phase transition is the hard sphere transition. If Uc/hgT 0, two distinct scenarios are possible according to the value of the ratio (range of the pair potential over particle radius). For < 0.3 (a), only fluid-solid equilibrium is predicted. For % > 0.3 (b), in addition to fluid-solid equilibrium, a fluid-fluid (liquid-gas) coexistence is predicted with a critical point (C) and a triple point (T).

The molecular dynamics method is based on the time evolution of the path (p (t), for each particle to feel the attractions and repulsions from all other particles, following Newton s law of motion. The simplest case is a dilute gas following the hard sphere force field, where there is no interaction between molecules except during brief moments of collision. The particles move in straight lines at constant velocities, until collisions take place. For a more advanced model, the force fields between two particles may follow the Lennard-Jones 6-12 potential, or any other potential, which exerts forces between molecules even between collisions. 

Air at room temperature and pressure consists of 99.9% void and 0.1% molecules of nitrogen and oxygen. In such a dilute gas, each individual molecule is free to travel at great speed without interference, except during brief moments when it undertakes a collision with another molecule or with the container walls. The intermolecular attractive and repulsive forces are assumed in the hard sphere model to be zero when two molecules are not in contact, but they rise to infinite repulsion upon contact. This model is applicable when the gas density is low, encountered at low pressure and high temperature. This model predicts that, even at very low temperature and high pressure, the ideal gas does not condense into a liquid and eventually a solid. 

The ideal gas law is consistent with the assumption that molecules are point masses with no volume. However, the hard sphere model assumes that molecules are spheres with a finite diameters a and introduces a central concept of the mean free path k, which is the mean distance traveled by a molecule before collision with another molecule. It is given by... 

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... 

FIGURE 7.9 B T) for a Lennard-Jones 6-12 potential. B is the ratio between B(T ) and the hard-sphere value 2n ALvogadroff3/3. T is the ratio of the thermal energy kB T to the well depth . B and T make Figure 7.10 a universal curve, valid for any gas. This curve agrees well with experiments on many molecules. 

The behaviour of the repulsive term of the lattice EOS is more complicated and will not be discussed in detail. At liquid-like densities this repulsion term is a better approximation to the hard spheres repulsion than the van der Waals repulsion term. At gas-like densities, the repulsion term of the lattice model and the van der Waals EOS have the same functional form. 

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