Molecular%20Speeds

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Molecular Speeds

• Maxwell-Boltzmann Distribution REVIEW
• For ideal gas at T, distribution of speeds
• The M-B distribution function has property
• This integral is the fraction of molecules with
  speeds lying between the limits v1 and v2
• From the mean value theorem of calculus

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Molecular Speeds

• Maxwell-Boltzmann Distribution
• Physical interpretation
• Velocity components of molecule (vx, vy, vz). N
  molecules represented by N points in velocity
  space. Volume of space between v and v ?v is
  4?v2 ?v. Because kinetic energy depends on v,
  the volume of this velocity space is proportional
  to the number of ways of obtaining a particular
  kinetic energy (within a small range), i.e., all
  points in this thin spherical shell of fixed
  thickness ?v correspond to the same kinetic
  energy. The greater the radius of the shell, the
  more points it encloses.

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Molecular Speeds

• Maxwell-Boltzmann Distribution
• Physical interpretation
• We have 4?v2 in the distribution function. This
  says that, all else being equal, we expect more
  molecules to have speeds with a range between v
  and v ?v, where ?v is fixed, the larger the
  value of v. However, all else is not equal. The
  factoraccounts for the decreased likelihood
  that a molecule will have a given speed. The
  most probable speed, vp, is that for which f is a
  maximum. By differentiating our distribution
  function and setting it 0

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Molecular Speeds

• Maxwell-Boltzmann Distribution
• Physical interpretation
• We have
• With this result, we can rewrite the distribution
  function as
• To determine how vp depends on T, differentiate
  with respect to T to get
• Then approximate the ratio of differences
• ?vp/?T to get

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Molecular Speeds

• Maxwell-Boltzmann Distribution
• Physical interpretation
• According to this, if the temperature drops from
  20C to -20C, vp changes by 7. On average air
  molecules in summer move only slightly faster
  than in winter. The shift in the distribution of
  speeds with T is shown in the figure.
• The most probable speed is only one of several
  possible speeds that can be used to characterize
  the distribution of molecular speeds. Lets look
  at some others

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Molecular Speeds

• Maxwell-Boltzmann Distribution
• Physical interpretation
• The mean speed is written as
• The root-mean-square speed is written as
• Where
• The three speeds are shown on the right. They
  are not all that different (ratio 11.131.22).
  Any one of them could be used to specify the
  average speed, you just need to specify which
  average you use.

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Intermolecular Separation

• Characteristic of gas mostly empty space
• From the ideal gas law
• where n is the number density (/m3) of
  molecules.
• Inverse of n is the average volume of space
  allocated to each molecule. Then
• where d is the average separation between
  molecules.

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Intermolecular Separation

• Take p 105 Pa, and T 293 K
• We calculate d 3.43 10-9 m (3.43 nm)
• While molecules have no sharp boundaries, the
  approximate diameter of common atmospheric
  molecules is d0 0.3 nm.
• Volume fraction of air occupied by matter is the
  ratio of the molecular volume to the volume
  allocated to the molecule, i.e., (d0/d)3 ? 10-3.
  About 1 part per 1000 (by volume) of air is
  occupied by matter.

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Mean Free Path

• Intermolecular separation is not the distance a
  molecule must travel before interacting with
  another.
• Consider cylindrical volume of cross section A,
  length x, and molecular number density n (? 3
  1019 cm-3), then the total number of molecules is
  N nAx.
• Assume molecules are spheres of diameter, d0.
• Two spheres interact when centers approach each
  other within d0.
• Consider projectile molecule a point and target
  molecule as sphere with collision cross section ?
  ?d02.
• ? ? 3 10-15 cm2

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Mean Free Path

• Consider a point molecule incident at one end of
  the cylinder traveling parallel to cylinder
  x-axis. Its position on A where it enters the
  cylinder is random.
• Question How far will it go before it collides
  with a target molecule?
• Some may travel a short distance and some a
  longer distance.
• The target area presented to point molecule over
  distance x is nAx?.
• Total target cross-sectional area per cylinder
  cross sectional area is nx?.
• This is, approximately, the probability that a
  point molecule will have a collision in distance
  x.

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Mean Free Path

• If nx? 1, the molecule is unlikely to not
  collide.
• Corresponding distance, x ?, at which n?? 1
  is the mean free path.
• As n increases, ? decreases.
• As ? decreases, ? increases.
• Because number density decreases exponentially
  with z, ? increases exponentially.
• A more exact derivation would lead to the same
  expression, but with a numerical constant that is
  close to 1.

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Mean Free Path

• Using ? 3 10-15 cm2, and n 3 1019 cm-3,
  we get a mean free path of 0.1 ?m, an order of
  magnitude greater than the average molecular
  separation (3.4 nm).
• Result makes physical sense.
• In order for ? ? d, after each collision the
  molecule would have to rebound at an angle
  exactly toward its nearest neighbor. Not likely.

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Intermolecular Collision Rate

• Assume all molecules are fixed, except one.
• All fixed molecules are points, the moving
  molecule a sphere with projected area ?.
• In time ?t the molecule moving at speed v sweeps
  out a volume v?t?.
• All fixed molecules collide with moving molecule.
• Total number of collisions in time ?t is nv?t?.

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Intermolecular Collision Rate

• Collision rate obtained by dividing by ?t.
• This yields nv?, the collision rate per molecule.
• To obtain the collision rate per unit volume,
  multiply by the number density of molecules, n.
• This yields the volume collision rate n2v?.
• Because molecules are distributed in speed, we
  should interpret v as a mean relative speed.
• We know that mean speeds are proportional to T ½.
• Using this result and the ideal gas law, we can
  write the rate of intermolecular collisions/unit
  volume in a gas in the form
• where C is a constant of order 1.

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Intermolecular Collision Rate

• Consider 2 different gases, both at the same T
  and p, but with different collision cross
  sections and masses.
• According to our result, the intermolecular
  collision rates are different because m and ? are
  different.
• This tells us that neither p nor T is determined
  by intermolecular collision rates.
• None of the thermodynamic variables (p, T, ?)
  depends explicitly on collisions. Collisions
  provide for energy and momentum transfer, however
  (randomization).
• For n 1019 cm-3, ? 10-15 cm2, and v 400
  m/s, the collision rate 4 1027 cm-3 s-1!!
• High collision rate at the heart of thermodynamic
  equilibrium.

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