Chapter 13 The Molecular Basis of

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Chapter 13 The Molecular Basis of
Thermal Physics
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Main Points of Chapter 13

• Microscopic view of gases
• Pressure
• Temperature
• Probability distributions
• Maxwell-Boltzmann velocity distribution
• Collisions and transport

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13-1 A Microscopic View of Gases
GOAL relate temperature and pressure to the
motion of the molecules
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Microscopic description of an ideal gas

• On average, molecules are separated by about 20
  times their radii(10-10m). They are far enough
  apart that they do not interact with one another

• Molecules interact only by elastic collisions
  with the other molecules and with the walls

• The individual molecular actions are random.
  Molecules move with random velocities
• every direction equally probable
• a distribution of speeds from 0 to

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13-2 Pressure and Molecular Motion
1. Some Average Values in a Gas
Average velocity in any direction is zero
The brackets are used to indicate average value.
The average squares of velocities in all three
directions are the same
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Internal energy U comes mostly from the kinetic
energies of the molecules. For ideal gases
m the mass of a molecule
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2. The origin of Pressure
Pressure arises from the multiple collisions the
molecules of a gas with the walls that contain
the gas.
If we look at a wall in the y-z plane, consider a
molecule having velocity vi, only the x-component
of the molecules momentum transfers to it
(because of elastic collisions)
m is the mass of a molecule
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Next look at the number of collisions
Take imaginary cylindrical volume whose base is
against the wall.
If the density of molecules having velocity vi is
Ni/V, then the number of collisions in time dt is
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The momentum transferred in time interval dt
To find the total momentum , we must add up the
contributions of all the molecules that strike
the wall,
The pressure is the total momentum
transferred per unit time per unit area.
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These are important! Microscopic properties have
been used to find a relation between macroscopic
thermodynamic variables.
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13-3 The Meaning of Temperature
The temperature of an ideal gas is a measurement
of the average kinetic energy of the constituents
the rms (root mean square) value
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Macroscopic and microscopic views of the work
done by an insulated gas
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• ACT Can we assign a temperature to a single
  molecule? Explain.

Solution
No. Temperature is a statistical quantity that
depends on the average behavior of a large number
of molecules.
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• Act Consider a fixed volume of an ideal gas.

If you double either T or N, p goes up a factor
of 2. ( pV NkT )
1.If you double T, how many times as often will a
particular atom hit the container walls? (A)
1 (B) 1.4 (C) 2 (D) 4
2.If you double N, how many times as often will a
particular atom hit the container walls? (A)
1 (B) 1.4 (C) 2 (D) 4
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Interpretation of the van der Waals Gas
F
R
1mol ideal
r
d
Correction of repulsive force
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Correction of attractive force
It includes the effects of intermolecular forces,
which act over distances larger than the size of
the molecule, are repulsive at short range, and
slightly attractive at longer distances.
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13-4 The Velocity Distribution of Gases
Not all molecules have the average velocity, of
course for a more complete description we need a
velocity distribution function, which tells us
how many molecules have any particular velocity.
The position distribution function does the same
thing for the molecules location in space.
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1. The Distribution of Molecule Speeds
There are enough molecules in a macroscopic gas
sample that the distributions can be considered
to be continuous from 0 to .
We define the distribution function
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f(v)dv is the ratio of the number of molecules in
the speed interval vvdv to the total number of
molecules
f(v)dv is the probability of one molecule in the
speed interval vvdv
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f(v)
v
o
The probabilities are normalized
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To find the average of any functions of v
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• Act find the meanings of the formulas below

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Maxwell speed distribution function
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The most probable speed
The average speed
The average value of the velocity squared
The root-mean-square speed
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(a) The Maxwell speed distribution for oxygen
molecules at T 300 K. The three characteristic
speeds are marked. (b) The curves for 300 K and
80 K. Note that the molecules move more slowly at
the lower temperature. Because these are
probability distributions, the area under each
curve has a numerical value of unity.
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• ACT At temperature T, the rms speed of gas
  particles of mass M is v. At temperature 2T,
  what is the mass of particles with rms speed v?
• (a) M/2
• (b) 2M
• (c) 4M
• (d) M
• (e) M/4

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• Act The graphs below are Maxwell speed
  distributions
• for gas He and O2
• Which is for He?

A
B
v
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• ACT Explain the fact that very few hydrogen
  molecules are present in Earths atmosphere
  today. In explaining your answer, ignore the
  possibility that hydrogen is removed by chemical
  reactions.

Solution
Hydrogen molecules are the least massive of all
gas molecules. So their average speed is the
highest at a given temperature. With high enough
speed, most of the hydrogen molecules in the
atmosphere have escaped Earths gravitational
field and gone into space.
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• Example Find the ratio of number of molecules
  in the speed interval vm1.01vmto the total
  number of molecules for ideal gas.

Solution
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• Example Suppose that the distribution of speed
  of
• N particles is shown in the figure below.
  Find a in
• terms of N,v0. Calculate

Solution
a
v0
o
2v0
v
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a
v0
o
2v0
v
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2. The Velocity Distribution of Gases
We define
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The relation between f (v) and F (vx,vy,vz)
is the ratio of number of molecules in the speed
between v and vdv to the total number of
molecules
in the volume of velocity space between the two
neighboring spheres, both centered at the origin,
the inner one with radius v, the outer one
infinitesimally bigger, with radius vdv
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Maxwell velocity distribution
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How to find the fraction of molecules having
velocity in the x-direction between vx and vx
dvx ?
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• Example derive the pressure of ideal gas using
  Maxwell velocity distribution.

Solution
The impulse on the wall per second is
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Pressure
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• ACT Find the number of molecules of ideal gas
  colliding with the wall in 1 second pea area.

Solution
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13-5 The Maxwell-Boltzmann Distribution
The velocity distribution for an ideal gas can be
written
Z is a factor that ensures proper normalization.
Maxwell-Boltzmann distribution
E total energy of the molecular
For monatomic ideal gas
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The Energy Distribution for Diatomic Molecules
A diatomic molecule can do more than just move
it can rotate or vibrate
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Every motion contributes to its total kinetic
energy.
Each quadratic term in the energy of a particle
in the gas is found to have the average energy
kT/2
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Consider vibration, it contributes two more
factors of ½ kT, one for the motion of the atoms
and one for the energy in the spring
So, each possible independent motion is seen to
add ½ kT to the energy of the molecule these are
called degrees of freedom.
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Equipartition theorem
Each degree of freedom contributes ½ kT to the
average energy of a molecule.
For s degrees of freedom
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• ACT What is ratio of the total mean energy of gas
  particles with 7 degrees of freedom to the total
  mean energy of monatomic gas particles at
  temperature T?
• (a) 7/3
• (b) 7/5
• (c) 5/7
• (d) 1
• (e) 1/7

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• ACT The mean internal energy per molecule of an
  ideal gas depends on ________
• (a) the moment of inertia of the molecules.
• (b) the mass of the molecules.
• (c) the distance between the two atoms in the
  molecules.
• (d) the number of degrees of freedom of the
  molecules.
• (e) all of the above

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The internal Energy of ideal gas (rigid molecules)
non-linear molecules (H2O, NH3, ) 3
translational modes (x, y, z) 3 rotational
modes (wx, wy, wz)
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• Example Internal Energy of a Gas

A pressurized gas bottle (V 0.05 m3), contains
1mol helium gas (an ideal monatomic gas) at a
pressure p 107 Pa and temperature T. What is
the internal thermal energy of this gas? How
would the answer change if the container were
filled with a diatomic gas?
Solution
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• Example A small room at room temperature
  (T300 K) and atmospheric pressure measures 3.0 m
  x 2.4 m x 5.2 m contains 1mol ideal gas.
• 1) Estimate the total translational kinetic
  energy associated with the molecules in the room.
  
• 2) If we assume these molecules are primarily
  nitrogen (N2) molecules, what is their
  root-mean-square speed?

Solution
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• Example A container has m kg diatomic ideal gas
  .
• It is moving with velocity v. Suddenly it stops
  and the system finally in equilibrium. What kind
  of energy is the kinetic energy of the gas
  transformed into ? Calculate the change of the
  average velocity squared of the molecule.

The macroscopic kinetic energy of the gas is
transformed into internal energy
Solution
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• ACT A certain amount of energy is to be
  transferred as heat to 1 mol of a monatomic gas
  (a) at constant pressure and (b) at constant
  volume, and to 1 mol of a diatomic gas (c) at
  constant pressure and (d) at constant volume.
  Figure below shows four paths from an initial
  point to four final points on a p-V diagram.
  Which path goes with which process? (e) Are the
  molecules of the diatomic gas rotating?

(a) 3 (b) 1 (c) 4 (d) 2 (e) yes
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Failure of the Equipartition theorem
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The energy for molecules vibration and rotation
can only have certain discrete values.
The probability that the particles in a
particular state with energy En
The Boltzmann Factor
The probability of finding a particle in a
certain state increases as the energy of the
state decreases.
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• Example A particular molecule has three states,
  with energy spacing e 10-20 J, as shown the
  molecule is in equilibrium with a temperature of
  1000K.

a) What is P1, the probability that the molecule
is in the middle energy state?
b) What is P2, the probability that it is in the
highest energy state?
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Solution
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13-6 Collisions and Transport Phenomena
Molecules in ideal gases have many collisions per
second billions in the case of air at STP but
between collisions the molecules interact very
little. Therefore, the average distance the
molecule travels between collisions, called the
mean free path, can be calculated.
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Collisions and Molecular Movement in a Gas
Molecules will collide when they overlap
Here, D is the diameter of molecules. s is
called the collision cross section.
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Mean collision time t the average time between
collisions
consider a gas molecule moving with average speed
through a region of stationary molecules
The correct expression
n the number density of molecules
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• Example What is the mean free path in air at sea
  level when the temperature is 300K? Estimate the
  collision frequency. Take r10-10 m as a typical
  molecular radius, and treat the atmosphere as
  ideal.

Solution
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• ACT Why, from the point of view of kinetic
  theory, does the air near a hot stove become
  heated?

Solution
The stoves thermal energy is transferred to
molecules by collisions between molecules and the
stove walls, as well as by molecular absorption
of electromagnetic radiation emitted by the stove.
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The Random Walk and Diffusion

• Consider the molecules in a gas. They bounce
  around randomly, colliding with other molecules
  and the walls.

• How far on average does a single molecule go in
  time t??

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Let the successive displacements
After N steps the net displacement
random walk
For molecule, after time t, it will move on
average a distanced squared
The displacement is proportional to the square
root of time
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• ACT If someone across the room from you opens a
  perfume bottle, you detect the order after
  several minutes, is it by the diffusion of the
  perfume molecules? (Hint estimate the time for
  the perfume molecules to diffuse across the room)

No. You smell the perfume is due to air current(
convection)
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Summary of Chapter 13

• Assumption Gases have molecules that are
  relatively sparse and interact only during
  collisions.
• Internal energy

• Pressure

• Temperature is a measure of the average kinetic
  energy of molecules

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Summary of Chapter 13, cont.

• Probability distribution for molecules in a gas

• Maxwell-Boltzmann distribution

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Summary of Chapter 13, cont.

• Each degree of freedom contributes ½ kT to the
  average energy

• Mean collision time and mean free path