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Imperfect Gases

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    78477
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    In the limit of low density p = r kT ( ideal gas law). By assumption, ideal particles do not interact, U = 0 (the intermolecular potential is zero).

    The classical partition function is (Chapter 7)

    Since H has the form

    we can immediately integrate over the momenta to get

    where ZN is the configuration integral

    If we can neglect UN in the configuration integral, then

    ZN=VN and Q = qN/N!.

    This result holds true for monatomic and polyatomic gases (proof now shown). As the density is increased the interactions between gas molecules become non-negligible. Deviations from the ideal gas law have been described in a large number of equations of state. The virial equation of state expresses the deviation from ideality in terms of a power series in the density.

    B2(T) is called the second virial coefficient, B3(T) is called the third virial coefficient, etc. The jth virial coefficient can be calculated in terms of the interaction of j molecules in a volume V. The second and third virial coefficients give most of the deviation from ideal (p/r kT ) up to 100 atm.

    The virial equation of state from the grand partition function

    A grand canonical ensemble can be considered to be a collection of canonical ensembles in thermal equilibrium with each other but with all possible values of N. The grand canonical ensemble turns out to be very useful for calculating systems that have a fluctuating number of particles or for open systems. The ensemble is at constant temperature and constant chemical potential, but the pressure P and number of particles N can vary as can the internal energy EN. For each value of N there is a set of energy state {ENj(V)}. We let aNj be the number of systems in the ensemble that contain N molecules and are in the state j. The set of occupation numbers {aNj} is a distribution.

    The partition function is

    where l = exp(bm ).

    When N = 0, the system has only one state with E = 0, and Q = 1.

    The characteristic thermodynamic function associated with theta is pV

    The average number of molecules in the system is given by

    The variables P, V, and N are now given in terms of theta. We can eliminate theta in these two expressions to obtain a power series for ln(theta). We have ln(theta) as a power series in l above. The strategy employed by McQuarrie is to define an activity z, proportional to l , such that z à r as r à 0. By taking the limit l à 0 we find that

    The derivative is

    As l à 0, the Q1 term dominates.

    The density is r = N/V so as l à 0, r à l Q1/V. Our required activity is thus z = l Q1/V. In terms of this activity

    The classical configuration integral ZN can be recast as

    Using this definition we can express the grand partition function as

    This gives theta as a power series in z.

    Assuming that the pressure can be expanded in powers of z according to

    We substitute this equation into

    We expand the exponential

    and substitute in for P as a power series in z to obtain theta as a power series in z.

    or

    Comparing like powers of z in the two series we find

    b1 = (1!V)-1Z1 = 1

    b2 = (2!V)-1(Z2 - Z12)

    b3 = (3!V)-1(Z3 - 3Z2Z12 + 2Z13)

    …..

    Calculation of b2 involves Z1 and Z2 which are partition functions of one and two particles. Calculation of b3 involves partition functions one, two, and three particles etc.

    We want an expression for the pressure in terms of the density r rather than the activity z.

    The justification for the above points is first that z is proportional to l and secondly that we can substitute ln X = PV/kT. At this point both P and r have been expressed as power series in z. To eliminate z we write the expansion

    z = a1r + a2r 2 + a3r 3 + …

    Substitute this into the above expression for the density and equate like powers on both sides of the equation to obtain

    a1 = 1

    a2 = -2b2

    a3 = -3b3 + 8b22

    ….

    We can z as a power series in r to obtain P as a power series in r (the virial expansion).

    where

    Calculation of virial coefficients

    The configuration integrals for Z1, Z2, and Z3 are

    The series method allows the calculation of a number of virial coefficients. Recall that the second and third virial coefficient can account for the properties of gases up the hundreds of atmospheres. We will discuss the calculation of the second virial coefficient for a monatomic gas to illustrate the procedure. Higher virial coefficients are calculated in McQuarrie, Chapter 12. To calculation B2(T) we need U2. For monatomic particles it is reasonable to assume that the potential depends only on the separation of the two particles so U2 = u(r12), where r12 = |r2 - r1|. We have

    Using a change of variables we can write this integral r12 = r2 - r1 and after integration over r1 we can transform variables from dr12 to 4p r2dr. The result is

    As discussed in Allen & Tildesley's book this expression can be used to obtain parameters from experiment. The second virial coefficient is tabulated for a number of gases.

    For a hard-sphere potential

    there is an infinite repulsive wall at a particle radius s . There is no attractive part.

    The Lennard-Jones potential cannot be calculated analytically, but the integral can be computed numerically. The second virial coefficient was a useful starting point for obtaining Lennard-Jones parameters that were used in simulations.


    Imperfect Gases is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by LibreTexts.

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