Chapter 16 Thermal properties of matter

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Chapter 16 Thermal properties of matter

• Equations of state
• Molecular properties of matter
• Kinetic-molecular model of an ideal gas
• Heat capacity of gas
• Molecular speeds
• Phase of matter

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Equation of state

• The variables which can describe the states of
  material are called state variables
• The relationship among state variables is called
  equations of state
• Simplified by equation
• If complicated, we can describe by graph or
  tables.

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The ideal gas equation

• important state variables of gas are P, V, T, n,
  r, etc.
• The relationship arises from gas laws such as
  Boyles law, Charles law and Guy Lussacs law.

P1 V1 n1 T1
state 1
P2 V2 n2 T2
state 2
2 states
PV nRT NkBT
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Other equation of states
2 states
m total mass m nM M Molar mass
P1 V1 m1 N1, r1, T1
state 1
N total number of molecules
state 2
P2 V2 m2 N2, r2, T2
r density (kg/m3)
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The atmospheric pressure at different height

• From
• Ideal gas equation

M Molar mass (kg/mol) y height from sea level
(m)
If y1 0, then P1 P0 1.013 x 105 Pa
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The van der Waals Equation

• Ideal gas ignores the volume of the molecules and
  the attractive forces between them.
• van der Waals equation take these two omissions
  into account.
• The interaction between gas atoms is called van
  der Waals Interaction.

The van der Waals Equation
a and b are empirical constants
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The van der Waals Equation

• Term V-nb is the net volume available for the
  molecules to move around.
• The constant a depends on the attractive
  intermolecular forces.
• If gas is diluted, n/V is not significant, the
  van der Waals equation becomes the ideal gas
  equation,

If n/V is very small
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The PV-diagram of a non-ideal gas

• Below TC, the isoterm shows the flat regions
  in which we can compress the material
  without an increasing in pressure.
• This shaded region is called
  Liquid-vapor phase equilibrium region.
• At point a, gas begin to liquefy, the volume is
  then decreased.
• At point b , all gas change phase to liquid.
• All temperature over TC, no phase transition
  occur.

TC Critical temperature
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Molecular propertied of matter

• All familiar matter is made up of identical
  molecules behaving in the same manner.
• The interaction of each molecule describes by a
  force and potential.
• Molecules are always in motion, either vibrate
  or move, causing the kinetic energy. The kinetic
  energy is generally much less than the potential
  energy, which mean the molecules are bounded.

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Molecular Properties of matter

• solid

• molecules vibrate about the fixed point.
• arranges in periodic called crystal lattice.
• the vibration may be nearly simple harmonic.

• liquid

• intermolecular distance are slightly greater
  than in solid phase.
• molecules have greater freedom of movement.
• liquid shows regularity of structure in the
  short range.

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Molecular Properties of matter

• gas

• molecules are widely separated.
• molecules have only small attractive forces.
• molecules move in a straight line until they
  collides with another molecule or a wall of
  container.
• ideal gas has no attractive force and has no
  potential energy.

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Kinetic molecular model of an ideal gas
We try to understand the macroscopic properties
of gas in terms of its atomic or molecular
structure and behavior.
Assumptions

• A container with contains a very large number N
  of identical molecules with mass m.
• The molecules behave as point particles, their
  size is very small comparing to distance between
  particles.
• Molecules are in constant motions and collides
  perfectly elastic.
• The container wall is rigid and do not move.

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Kinetic molecular model of an ideal gas
number of molecules
v
vy
vx
v
vy
A
vx
-vx
vxdt
I 2mvx
number of molecules moving toward
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Kinetic molecular model of an ideal gas
The force acting on the wall is,
v
vy
vx
v
vy
-vx
I 2mvx
where
Pressure
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Kinetic molecular model of an ideal gas
Other relations
Ktr average translational kinetic energy of
gas.
Ek kinetic energy of one molecule.
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Molecular speeds
We define the root mean square speed, vrms
M Molar mass

• vrms relates to the kinetic energy of the gas
  system.

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Collisions between molecules
In the time dt a molecule with radius r will
collide with any other molecule within a
cylindrical volume of radius 2r and length vdt.
2r
The number dN with centers in this cylinder is,
vdt
number of collision per unit time
assume one molecule is moving
If all molecules are moving
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Collisions between molecules
The mean free time, tmean
Time that molecule can move freely
The mean free path, l
The distance that molecule can move freely.
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Heat capacity of gases
T
By adding energy dQ into the system of gas, the
temperature increases by dT and the kinetic
energy increase by dKtr ,
Ktr
V
and,
TdT
KtrdKtr
CV heat capacity at constant volume
dQ
V
J/mol K
For ideal gas point particle
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Degree of Freedom
3 translational df
2 rotational df
2 vibrational df
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Heat capacity of gases
one df energy
Equipartition of energy

• Each molecule has 3 degrees of freedom (df) of
  translations, give

• Diatomic with 2 rotations df

• Diatomic with 2 rotations df and 2 vibrations df

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Heat capacity of solid

• Each atom has 3 df.
• solid has both kinetic and potential energy.

CV
3R
CV 3R
Rule of Dulong and Petit
T
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Phases of matter
Triple point the only condition under which
all three phases can coexist. Critical point
the top of the liquid-vapor equilibrium region.
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Phases of matter