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Physics 52 - Heat and Optics
Dr. Joseph F. Becker
Physics Department
San Jose State University
© 2005 J. F. Becker
Chapter 18
Thermal Properties of
Matter
© 2005 J. F. Becker
San Jose State University
Physics 52 Heat and Optics
Thermal Properties of Matter
2. Equation of state
-Ideal gas equation
-Van der Waals equation
3. Molecular properties of matter
4. Kinetic-molecular model of an ideal gas
5. Heat capacities (Cv) – theory
6. Molecular speeds
7. Phases of matter
© 2005 J. F. Becker
San Jose State University
Physics 52 Heat and Optics
A hypothetical
apparatus for
studying the behavior
of gases. The
pressure p,
volume V,
temperature T, and
number of moles n,
of a gas can be
varied and measured.
Cutaway of an automobile engine showing
the intake and exhaust valves.
Ideal gas model assumptions:
No molecular volume
No attractive forces
No potential energy
100% elastic collisions
Ideal gas equation: p V = n R T
P=
V=
n=
T=
R=
absolute pressure = pgauge + 1 atm
volume in m3
number of moles; mole = 6.022 (10)23
absolute temperature (Kelvin)
gas constant = 8.3145 J/mole K
= 0.0821 L atm / mole K
© 2005 J. F. Becker
San Jose State University
Physics 52 Heat and Optics
Van der Waals gas equation:
( p + a n 2/ V
2
) (V – n b ) = n R T
where a and b are empirical constants
and are different for different gasses.
b = volume on one mole of molecules, so
nb = the total volume of the molecules.
a = attractive intermolecular forces
(called “Van der Waals forces”) which
reduce the pressure of the gas for a
given n, V, and T.
Ideal gas p  p + a (n 2/ V 2)
© 200 J. F. Becker
San Jose State University
Physics 52 Heat and Optics
Isotherms, or constant-temperature
curves, for a constant amount of an
ideal gas.
p = nRT / V
HOT
COLD
pV diagram for a non-ideal gas
isotherms for temperatures
above and below the
critical temperature Tc.
The force between two molecules
(blue curve) is zero at a
separation r = ro, where the
potential energy (red curve)
is a minimum.
U = Uo[(Ro/r)12 - 2(Ro/r)6]
The force is attractive
when the separation
is greater than ro and
repulsive when the
separation is
less than ro
F= -dU/dr
Schematic
representation
of the cubic
crystal
structure of
sodium
chloride.
Kinetic-molecular model of an ideal gas
• Container of volume V contains a large
number N of identical molecules mass m.
• The molecules are point particles –
average distance between molecules and
walls is large.
• Molecules are in motion, obey Newton’s
laws, undergo 100% elastic collisions with
walls - no heat, friction, etc.
Container walls are perfectly rigid and
don’t move.
O
© 2005 J. F. Becker
San Jose State University
Physics 52 Heat and Optics
Elastic collision of a molecule with an
idealized container
wall. The velocity
component parallel
to the wall does
not change; the
component
perpendicular
to the wall
reverses direction.
The change in
momentum is
DP = m 2vx
A molecule moving
toward the wall with
speed vx collides with
the area A during the
time interval dt
if it is within a
distance vx dt
of the wall at the
beginning of the
interval. All such
molecules are
contained within a
volume = A vx dt.
The number of collisions with the area A of
the wall is ½ (N/V) (A vx dt). For all the
molecules in the cyl. D(momentum):
DPx = (# of collisions) x (DPx per collision)
DPx = ½ (N/V)A vx dt x (m 2vx ) =NAm vx2dt /V
Force on wall = DPx / dt = N A m vx2 /V, so the
pressure p = F/A = N m vx2 /V and pV = N m vx2
We need to take the average (or mean) speed
of all the molecules, so v2 = vx2 + vy2 + vz2 and
(v2)avg = (vx2)avg + (vy2)avg + (vz2)avg = 3 (vx2)avg
(vx2)avg = (v2)avg/3 and pV = N m (v2)avg/3.
Now multiply through by 2/2 to get
pV = (2/3) N [½ m (v2)avg] = (2/3) Ktr
© 2005 J. F. Becker
San Jose State University
Physics 52 Heat and Optics
pV = n R T = (2/3) Ktr
T a total translational KE of all N molecules.
We can also show that n R = N k where k is
Boltzmann’s constant k = 1.38 (10)-23 J/K so
pV = n R T = N k T = (2/3) Ktr
Ktr = (3/2) n R T = (3/2) N k T so
Average transl. KE / mole = (3/2) R T and
Average transl. KE / molecule = (3/2) k T or
[½ m (v2)avg] = (3/2) k T
v
rms
© 2005 J. F. Becker
= [(v2)avg] ½ = [3 kT/m] ½ = [3 RT/M] ½
where the molar mass M = NA n
San Jose State University
Physics 52 Heat and Optics
In a time dt a molecule with a radius r will
collide with any other molecule within a
cylindrical volume of
radius 2r and
length v dt.
COLLISIONS
BETWEEN MOLECULES
COLLISIONS BETWEEN MOLECULES
Now we can get an estimate of the mean free
time & the mean free path between collisions:
If the molecules are not points, but rather
rigid spheres of radius r, the number of
molecules inside a cylinder of radius 2r is
dN = (N/V) (p (2r)2 v dt)
(See Fig. 18.12)
Collisions / time = dN/dt = (N/V) p (2r)2 v
Now, if ALL the molecules are moving (not just
one) there are 1.41 times more collisions and:
time/collision = dt/dN = V / N 1.41 p (2r)2 v
and l = v tmean = V / N 1.41 p (2r)2
From pV = NkT  l = kT / p 1.41 p (2r)2
© 2005 J. F. Becker
San Jose State University
Physics 52 Heat and Optics
GASES
Heat dQ is added to
(a) a constant volume of
monatomic ideal gas
molecules.
(b) The total
translational kinetic C = molar heat capacity
v
energy increases by
at constant volume
dKtr = dQ, and the
temperature increases
by dT = dQ / n Cv
dQ
dQ = n Cv dT
MOLECULAR PROPERTIES OF MATTER
Heat Capacities of Gases
For now we assume constant volume so we can
avoid taking into account work done by the gas
on the atmosphere.
From pV = n R T = N k T = (2/3) Ktr we get
Ktr = (3/2) n R T or dKtr = (3/2) n R dT.
Comparison with dQ = n Cv dT
recall dKtr = dQ so
Cv = (3/2) R
ideal gas of point particles
Cv = (5/2) R
diatomic gas (3/2 + 2/2)
© 2005 J. F. Becker
San Jose State University
Physics 52 Heat and Optics
A diatomic molecule almost all the mass of each
atom is in its tiny nucleus.
(a) The center of mass has
3 independent velocity
components.
(b) The molecule has 2
independent axes of
rotation through its c.m.
(c) The atoms and “spring”
have additional kinetic and
potential energies of
vibration.
Experimental values of Cv for hydrogen
gas (H2). Appreciable rotational motion
begins to occur above 50 K, and above
600 K the molecule begins to
increase its vibrational
motion.
The forces
between
neighboring
particles in a
crystal may be
visualized by
imagining
every particle
as being
connected to
its neighbors
by springs.
SOLIDS
MOLECULAR PROPERTIES OF MATTER
Heat Capacities of Solids
The atoms in a crystal can vibrate in 3
directions with energy per degree of freedom
kT/2 per molecule or RT/2 per mole  3kT/2.
In addition to KE, each atom vibrating in a
solid has PE = kHx2/2 (and average KE =
average PE). So the total energy is
KE + PE = 3kT/2 + 3kT/2 = 3kT/molecule or
3RT/mole.
Cv = 3 R for a diatomic solid (3/2 + 3/2)
© 2005 J. F. Becker
San Jose State University
Physics 52 Heat and Optics
At high temperatures Cv for
each solid approaches approx.
3R, in agreement with the rule
of Dulong and Petite.
MOLECULAR PROPERTIES OF MATTER
Molecular Speeds
v rms = [3 kT/m] ½
The molecules don’t all have the same speed!
Maxwell-Boltzmann distribution function:
f(v) = 4p (m/2pkT) 3/2 v 2 exp {-mv2/kT}
The number of molecules dN having speeds in
the range between v and v+dv is given by
dN = N f(v) dv
v avg = Vv f(v) dv
(v 2)
= Vv 2 f(v) dv
© 2005 J. F. Becker
San Jose State University
Physics 52 Heat and Optics
Curves of the MaxwellBoltzmann distribution
function f(v) for various
temperatures. As the
temperature increases, the
maximum shifts to higher
speeds.
(b) At temperature T3 the
fraction of molecules having
speeds in the range v1 to v2
is shown by the shaded area
under the T3 curve. The
fraction with speeds greater
than vA is shown by the area
from vA to infinity.
MOLECULAR PROPERTIES OF MATTER
Phases of Matter
Now we consider phases (gas, liquid, solid) of
matter at various pTV. An ideal gas has no
phase transitions because there is no
interaction between the molecules, but real
matter does have these transitions.
Triple point – the point (p3T3 ) at which gas,
liquid and solid can coexist. (H2O: 0.01 atm, 273.16 K)
Critical point - the point (pCTC ) above which
liquid and vapor do not undergo a phase
transition, only continuous gradual changes
from one phase to the other. (H2O: 200 atm, 650K)
© 2005 J. F. Becker
San Jose State University
Physics 52 Heat and Optics
A typical pT phase diagram, showing regions of
temperature and pressure
at which the various
phases exist and
where phase
changes occur.
GAS
pVT-surface for substance that expands
on melting.
pVT-surface for an ideal gas.
A THERMODYNAMIC SYSTEM
is described by its STATE VARIABLES,
like pVT. Each state is described as a
point (pVT ) on the surface of a phase
diagram.
A PROCESS takes a system through
changes in its state variables.
© 2005 J. F. Becker
San Jose State University
Physics 52 Heat and Optics
A molecule with a speed v passes through the first
slit. When it reaches the second slit, the slits have
rotated rotated through offset angle q.
If v = w x / q, the molecule passes through the
second slit and reaches the detector.
Review