Download Thermal Properties of Matter

Thermal Properties of Matter
Dr. Gary Grim
7/24/02
State Equations
The conditions under which matter finds itself are
described by equations called “state equations,” or
“equations of state.”
These equations are comprised of “state” variables.
Example state variable include pressure, temperature,
volume, mass, etc...
Ideal Gas Law
The state equation for an ideal gas is given by, “the ideal
gas law,”
PV = nRT
The ideal gas law gives the relationship between pressure,
temperature, volume and mass, all state variables.
The quantity, n, is the mole count, I.e. the number of
moles present in the gas, and therefore the mass of the
gas is given by:
m= nM
Where M is the molar mass.
Ideal Gas Law
R is the gas constant, or ideal gas constant and is
given by:
R = 8.31 J/(mol K)
In units of liters and atmospheres, R is:
R = 0.082 L atm/(mol K)
For a constant amount of mass, the ratio
pV
T
must be constant if the pressure, volume or temp. change.
Van Der Waals EOS
The ideal gas law is just that, an ideal law. The van der
Waals equation of state is an empirical improvement.
2

an
 p + 2 (V − nb) = nRT

V 
The constants a & b, must be determined by experiment
with each gas.
pV Diagrams
Molecular Properties
As stated previously, all matter is made of atoms and
molecules.
For our purposes, we’ll not distinguish between the two for
now.
We will idealize the molecules and atoms as point objects
for most of our discussion, but their structure gives rise to
important thermodynamic properties.
Molecules in all matter are always in motion and have a
kinetic energy associate with them. This is true
regardless of the state of the matter.
Molecular Properties
For a gas of molecules, the force between molecules is
generally small and attractive at large distances, and large
and repulsive at very small distances.
At a distance which corresponds to the distance between
molecules in a solid or liquid, the inter-molecular forces are
roughly zero, and any deviations from this distance results
in a restoring force.
The molecules are said to be resting in a potential well.
Molecular Properties
For molecules with a kinetic energy which is larger than
the height of the “potential well,” the molecules do not
remain “bound” to each other, resulting in a “gas”
phase of matter.
For molecules with a kinetic energy which is smaller
than the potential well, the molecules oscillate with
some amplitude around the average inter-molecular
spacing. The matter is in either a liquid or solid phase
for this situation.
A liquid has a higher energy than a solid, thereby
allowing a molecule to escape the potential well with
one molecule and be captured by the well of another.
Molecular Properties
For a solid the probability for a molecule to escape the
potential well with another molecule is very small.
At low temperatures, matter is in the solid phase.
As the temperature rises, the matter melts and then
vaporizes.
From a molecular perspective, temperature and kinetic
energy are closely related!
Moles & Gophers
The book defines one mole as the amount of substance
that contains as many elementary entities as there are
atoms in 12 g of carbon 12.
Translation: there are NA, molecules in a mole, where NA, is
given by Avogadro’s number, 6.02 x 1023 molecules/mole.
Thus, molar mass, the mass of 1 mole of a compound, is
given by: M = NAm, where m is the mass of one molecule.
Kinetic Theory of Gas
We’d like to use our molecular model of a gas to form a
theory on the macroscopic properties of the gas.
Our assumptions include:
1) A container of volume V, with a very large number, N,
of identical gas molecules, each with mass m.
2) The molecules behave as point objects
3) The molecules are in constant motion, obeying the
laws of mechanics.
4) The container walls are rigid and immovable.
Kinetic Theory of Gas
Now, let’s calculate what happens to a molecule when
it elastically collides with the wall of the container.
In the direction parallel to the wall, (y), the velocity
remains constant after collision.
In the direction normal to the wall, the velocity
reverses direction.
Thus the momentum change in the collision is
directed along the normal to the wall is equal to:
mvx − − mvx = 2mvx
Kinetic Theory of Gas
Now, for the collision to take place in an amount of time,
dt, the molecule will have to be a distance no further
than vxdt from the wall.
If we now count the number of molecules that strike an
area A on the surface of the wall, erroneously assuming,
all molecules have an x-component of velocity vx, then the
number striking the wall in a time dt, is given by:
1 N 
Avx dt
2 V 
The 1/2 comes from the fact that half the molecules are
moving away from the wall at any time, while half are
moving towards the wall.
Kinetic Theory of Gas
The N/V term is just the gas density in the volume.
1 N 
Avx dt
2 V 
The total momentum change in a time dt, then is just 2mvx
times this quantity:
NAmvx2 dt
dPx NAmvx2
dPx =
⇒
=
V
dt
V
Employing Newton’s second law, this is just the force
being exerted on the wall by the gas and therefore pressure
is:
F Nmv x2
p= =
A
V
Kinetic Theory of Gas
The gas is made up of a distribution of velocities, and
what we’re interested in describing is the average
pressure exerted by the gas on the walls not the
pressure from one particular range of velocities,
therefore:
v → (v )
2
x
2
x avg
1 2
= (v )avg
3
This gives:
1 2
2 1
2

pV = Nm (v )avg = N m(v )avg

3
3 2
Kinetic Theory of Gas
The in square brackets represents the average
translational kinetic energy of the individual gas
molecules. We can rewrite the equation as:
2
pV = K tr
3
where K is the total translational kinetic energy of
the gas. Comparing this with the ideal gas law,
pV=nRT,
3
K tr = nRT
2
Kinetic energy is proportional to temperature!
Kinetic Theory of Gas
If we divide this result by N, the total number of
molecules, and remember that n, the number of
moles in the gas, is given by N/NA, then we get:
K tr 1
3  R 
2
= m(v )avg =
T
N
2
2  NA 
Thus, the average kinetic energy of a molecule is given
by:
1
3
2
m(v )avg = kT
2
2
Where k = R/NA is Boltzmann’s constant and is equal to
1.381 x 10-23 J/(molecule K)
Kinetic Theory of Gas
The average velocity of a gas molecule, may now be
determined by the temperature and mass.
The root mean square of a velocity is given by:
vrms =
( )
v
2
avg
=
3kT
=
m
3RT
M
The reason for using rms, is that (vavg)2 will be zero for
a gas confined to a container.
Kinetic Theory
The molecules in our container don’t just collide with
walls. They also collide with themselves.
Clearly, the more dense the gas, the more frequent the
collisions, and the larger the molecules are, the more
frequent the collision.
The “mean free path” is the average distance a
molecule travels between collisions and is given by the
velocity of the molecule times the mean time between
collisions:
λ = vtmean =
V
4π 2r 2 N
Kinetic Theory
N/V is the number of molecules per unit volume
(number density), and r is the radius of a molecule.
Substituting in for macroscopic quantities, we can recast
the formula as:
kT
λ=
4π 2r 2 p
Heat Capacity
From our kinetic theory, we just calculated that the
translational kinetic energy of the gas was given by:
K tr =
3
3
nRT ⇒ dK tr = nRdT
2
2
where the last form represents the “change in energy” as
the temperature changes, i.e., “heat,” or energy in
motion. Thus,
3
nRdT ⇒
2
3
Cv = R
2
nCv dT =
Heat Capacity
The last result was for “point” masses, meaning the only
degrees of freedom for motion are translational, as in the
case of “monatomic” gasses.
If the molecules are “diatomic” or “polyatomic,” then the
along with translational degrees of freedom the gasses
have rotational degrees of freedom, at least along axes not
parallel to the axis through the diatom.
This means kinetic energy of the translational motion and
the rotational motion must be considered, along with any
vibrational motion associated with the atomic bonding.
Heat Capacity
For a diatomic molecule with rotational degrees of
freedom, the heat capacity is given by:
5
Cv = R = 20.79J /(mol ⋅ K )
2
Adding vibrational degrees of freedom to a diatom gives:
7
Cv = R
2
Heat Capacity
For solids, we can carry out the same analysis as above.
Details are in the text.
The punchline is that heat capacity is given by:
Cv = 3R
This is known as the rule of Dulong and Petit, and pretty
much fails miserably at low temperatures, but works for
higher temps. Need solid state theory!
Phases
Phase equilibrium -- a locus of (p,V,T) point where
two phases of matter may coexist.
Phase diagram -- a 2-D plot of p vs. T showing the
phases of matter at a given volume.
Triple point -- the point on the phase diagram where
all three phases may coexist.
Critical Point -- the maximum pressure and
temperature at which the material may exist in liquidgas phase equilibrium.
Phases
T c, pc -- The critical point temperature and pressure.
A gas at a pressure above the critical pressure will not
separate into two phases when cooled at constant
pressure.
Instead the gas will gradually and continuously from
the gas state to the liquid state without a phase
transition.