VIRIAL EQUATION FOR REAL GASES

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VIRIAL EQUATION FOR REAL GASES

• Statistical Mechanics
• Final Project
• CHEM-2440
• Leonardo Alvarez

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Let us consider a monatomic gas in which the
atoms may interact with one another. Assuming the
interaction between the gas atoms being weak, the
kinetic energy of the gas atoms using the
particle in a box model, and the interaction
between the gas atoms with a potential energy of
interaction V(r). One common form for this
potential energy of interaction is the Lennard
Jones (or 6-12 potential)1, see figure 1.
Equation (1) gives the explicit form of this
potential energy as
REAL GASES
Fig 1 A Lennard Jones Potential
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the kinetic energy operator and the second term
is the potential energy term. The first term
depends only on the momenta of the gas atoms and
the second term depends on the distances between
all of the gas atoms. In general, this potential
energy term is quite complicated.It may be
shown that Q is given by2

• For distances between the atoms which are shorter
  than s, the repulsive interaction dominates. If
  we use this potential energy to model the
  interaction between the gas atoms, we can write
  the Hamiltonian for the gas as equation 2 where
  the first term is

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The configuration integral2 depends in a
complicated way on the potential energy of
interaction between all N gas atoms. In the case
in which V(r)0 i.e., an ideal gas, the integral
over space gives the volume i.e., ZN VN. In
this limit the previous form for the partition
function is obtained, namely

• Since the atoms' kinetic energy terms are
  separable, the integral over momenta is
  straightforward and we find that
• Where equation (5) is
• The Configuration Integral.

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THE VIRIAL EQUATION

• the treatment of more realistic systems is
  usually concerned with how to better model this
  function ZN or V(r). Clearly the ideal gas model
  is the simplest possible model for ZN.
• Assuming that we can express the pressure of the
  gas in powers of the density of the gas, so that
• where the coefficients B2(T), B3(T) are taken to
  be functions of temperature only. At very low
  densities the quadratic, cubic and higher terms
  are not important and the ideal gas law is
  obtained. As the density of the gas becomes
  higher, B2 and B3 must be included to accurately
  model the equation of state for the gas. Equation
  (7) is the Virial Expansion2,6.

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Table 1 B2(T) for some gases in cm3/mol

• The coefficients in the virial expansion are
  related to the configuration integral in a very
  direct manner. It may be shown that
• for the second virial2,6, and

for the third virial coefficient2,6. Similar, but
more complicated, expressions can be derived for
the virial coefficient of arbitrary order.
Fortunately, even for dense gases only the first
few terms in the viral expansion are needed.
Some representative values for the second virial
parameter3.
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• For the case of a monatomic gas Z2 has a somewhat
  simple form, namely
• the integral Z2 treats the potential of
  interaction between two atoms. For this case the
  second virial coefficient corresponds to
• Given equation (11), it is clear that we require
  a model for the interatomic potential.

For molecules we possess rotational and
vibrational degrees of freedom which also need to
be included in our partition function. The
intermolecular potential energy and the
evaluation of the integral in equation (11) will
be much more complicated.
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SOME MODELS FOR V(r)

• HARD SPHERE POTENTIAL
• A hard sphere potential corresponds to the case
  in which the interaction between particles is
  zero when they are greater than some distance s,
  but is infinitely repulsive for that distance.
  This model is that of ping pong balls zooming
  around in a box. In this case the potential
  energy is written as
• where s is the diameter of the hard sphere. In
  this case it is straightforward to evaluate the
  integral for the second virial coefficient it is

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• This virial coefficient is temperature
  independent and it has a value which is four
  times larger than the volume of the sphere. This
  volume represents the excluded volume in the gas
  (see the figure 2)4. When the atoms collide it
  occurs at their surfaces, hence a spherical
  volume of radius 2s is excluded to one atom by
  the other. Using this result we can write an
  equation of state for the hard sphere gas which
  is correct to second order in the density, namely

• Fig 2 A diagram of a hard sphere potential is
  shown on the left, and the excluded volume in a
  gas.

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• SQUARE WELL POTENTIAL2
• In this case the potential energy has an
  attractive part added to it. See figure 3. The
  potential energy may be written as
• the solution for the virial coefficient with this
  potential is
• where ? is the well depth and ? is the extent of
  the attractive well. Clearly, now that the
  potential has an attractive term, the virial
  coefficient has a temperature dependence.

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• Fig 3 A potential energy diagram is sketched for
  a square well.

• Figure 4 shows the B2(T) for argon, and it is fit
  to this model. The best fit parameters give a ?
  value of 1.68,
• ? 1.36 x 10-21 J (68.5 cm-1) and s 3.04 Å.
  Clearly a simple model can perform quite well.

• Fig 4 The second virial data of argon is plotted
  and fit to a square well potential (dotted line)
  and a Lennard-Jones potential (solid line)3.

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• LENNARD JONES POTENTIAL
• A Lennard Jones potential1 may also be used to
  describe the intermolecular interactions. For
  such a potential the virial coefficient may be
  written as
• this equation is also fit to the Ar data figure
  4. If we express the Lennard Jones potential in
  reduced units i.e., let T kT/? and x r/s,
  we can write the second virial coefficient as

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• Values of B are tabulated. Figure 5 shows a plot
  of this reduced virial coefficient for 4 gases5,
  one of which is polyatomic but spherical. Clearly
  this resealing of the data by use of the
  potential parameters works quite well for these
  nonpolar gases. The agreement between the data
  suggests that the Lennard Jones potential
  captures the main features of the interaction
  between molecules of this type7.

• Fig 5 This figure shows the reduced viral
  coefficient for five gases. Data was taken from
  Intermolecular Forces by Maitland, et al.

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CONCLUSIONS

• The empirical observations about real gases can
  be understood by consideration of the second
  virial coefficient given in equations (17) and
  (18) for the Lennard Jones potential.
• T is a parameter that compares the available
  thermal energy kT to the depth ? of the
  attractive well. When kT is large, the attractive
  well does not matter so much and the repulsive
  part of the potential energy determines the
  interactions.

• In contrast, the attractive well will matter
  more when the temperature is low. It is in this
  regime where the second virial coefficient can
  have a negative value, leading to a
  compressibility factor value less than 1 (which
  happens just at standard temperatures and
  relative high pressures). See Fig 6.

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REFERENCES

• Huheey J. Inorganic chemistry principles of
  structure and reactivity. New York, Harper Row
  1972.
• Mc Quarrie D., Statistical Mechanics. University
  Science Books 2000.
• Herzberg G., Infrared and Raman Spectra of
  Polyatomic Molecules, New York Van Nostrand
  1945.
• Rice, O. Statistical Mechanics. Thermodynamics
  and Kinetics. New York, Freeman 1967.
• Maitland, et al. Intermolecular Forces.
• Davidson N., Statistical Mechanics. New York,
  McGraw Hill 1962.
• Hirschfelder et al. Molecular Theory of Gases and
  Liquids.