Transcript Lecture 34.KineticTh..

Kinetic Theory of Gases
Lecturer:
Professor Stephen T. Thornton
Reading Quiz
A gas cylinder is used to fill dozens of balloons
with helium gas over a weekend. Which of the
following statements is most true on Monday?
A) The remaining helium atoms in the cylinder have less
kinetic energy.
B) The pressure in the cylinder and the rms speed of the
remaining helium atoms increases.
C) The pressure in the cylinder and the rms speed of the
remaining helium atoms decreases.
D) The pressure in the cylinder decreases, but the rms
speed of the remaining helium atoms remains constant.
Reading Quiz
Answer: D
The pressure in the cylinder goes down,
but the rms speed of the helium atoms only
depends on the ambient temperature,
which remains constant.
Last Time
Thermal expansion - more
Brownian motion
Ideal gas law
Moles, Avogadro's number, etc.
Today
Kinetic energy of molecules
Maxwell distribution
Phase changes
Vapor pressure and humidity
Van de Waals Equation
Mean free path
Diffusion
Kinetic Theory of Gases
• James Clerk Maxwell, a great Scottish
mathematical physicist, did much of the early
work in statistical theory, but died at age 48.
• Ludwig Boltzmann despaired so much about
his work in statistical theory that he
committed suicide in 1906.
• Paul Ehrenfest, who picked up Boltzmann’s
work, committed suicide in 1933.
• Now it is our turn to study the subject.
• Perhaps we should proceed cautiously!
Kinetic theory of gases
Assumptions:
• Container has large number N of
identical molecules, each of mass m.
• Molecules bounce around and are far apart.
• Molecules collide elastically with wall
and each other.
Force Exerted by a Molecule
on the Wall of a Container
px  2mvx
px  2mvx
We have cube of sides L
Round trip time t  2 L / vx
px
2mvx
mv
F


t 2 L / vx
L
2
x
2
2
x
2
x
2
x
mv
F mv / L mv
P 
 3 
A
L
L
V
2
PV  mvx
Assumed only motion in x-direction.
Note connection between kinetic energy
of molecule and macroscopic variables.
In a real gas, the molecules are moving
around with different speeds. Maxwell did
a more complete calculation (see later)
using the average velocity and included
molecules moving in all 3 directions.
He found
2 1 2
2
PV  N  mv   NK av
3 2
av 3
The pressure is proportional to the
average kinetic energy.
We have found a clear connection
between microscopic behavior
and macroscopic variables.
2 1 2
PV  NkT  N  mv 
3 2
av
2 1 2 
So kT   mv 
3 2
av
3
1 2
 mv   K av  kT
2
2
av
This indicates that the average kinetic energy
of molecules in a gas is proportional to T !
Conceptual Quiz:
Imagine that that the temperature of a fixedvolume container of ideal gas changes from 1000C
to 2000C. What happens to the pressure and the
average kinetic energy of the molecules?
A) Pressure goes up; average kinetic energy goes
up by factor of 2.
B) Pressure goes up; average kinetic energy goes
up, but less than a factor of 2.
C) Pressure goes down; average kinetic energy
goes up by some unknown amount.
D) Pressure goes down; average kinetic energy
goes up by a factor of 2.
Answer: B
Pressure goes up according to ideal
gas law. Kinetic energy is
proportional to Kelvin temperature,
not Celsius temperature, so K
increases, but not by a factor of 2.
373 K to 473 K
3
3kT
1 2
2
From  mv   kT , we have  v  
av
m
2
av 2
vrms  v  3kT / m called root mean square
2
We can show this is also
vrms  3RT / M where M is molecular weight
3
If K av  kT for one molecule, then
2
for N molecules,
3
U  internal energy = NkT for monotonic gas
2
The Maxwell Speed Distribution
The molecules in a gas will not all have the same speed; their
distribution of speeds is called the Maxwell distribution:
3
2
1 mv 2
2
2 kT
æ m ÷
ö
f (v) = 4p N çç
ve
÷
÷
çè 2p kT ø
Most probable speed
Sometimes we want to know
Most probable speed vp
Average speed v
Root mean square vrms
kT  1.41 kT
Most probable speed vp : set d f (v)  0; vp  2m
m
dv

v f (v) dv

Average speed v : v  0

N
8 kT 
 m
kT 
Root mean square vrms  v2  3m
1.60 kT
m
1.73 kT
m
Hydrogen Gas Molecules. A 1.0mol sample of hydrogen gas has a
temperature of 27°C. (a) What is
the total kinetic energy of all the
gas molecules in the sample? (b)
How fast would a 65-kg person
have to run to have the same
kinetic energy?
Uranium Isotopes. Two isotopes of
uranium,
235U and 238U
(the superscripts refer to their atomic
masses), can be separated by a gas diffusion
process by combining them with fluorine to
make the gaseous compound UF6 .
Calculate the ratio of the rms speeds of
these molecules for the two isotopes, at
constant T. Find masses in Appendix or on
Internet.
Real Gases and Changes of Phase
The curves here represent the behavior of the
gas at different temperatures. The cooler it gets,
the further the gas is from ideal.
In curve D, the gas
becomes liquid; it begins
condensing at (b) and is
entirely liquid at (a). The
point (c) is called the
critical point.
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A PT diagram is called a phase diagram; it shows
all three phases of matter. The solid-liquid
transition is melting or freezing; the liquid-vapor
one is boiling or condensing; and the solid-vapor
one is sublimation.
Phase diagram
of water.
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The triple point is the only point where all
three phases can coexist in equilibrium.
Phase diagram of
carbon dioxide.
The solid form is
called “dry ice”.
Dry ice at atmospheric pressure, -78.5 0C
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Normal
substance
Water
The Solid-Liquid
Phase Boundary
Look at point A. If we
increase the pressure,
the ice melts. This is
why ice skate blades
glide so easily.
A Liquid in Equilibrium with its Vapor.
In equilibrium the number of molecules
evaporating into gas is equal to number from
gas condensing into liquid.
Vapor
pressure
The Vapor-Pressure Curve for Water.
A liquid boils at the temperature at which its
vapor pressure equals the external pressure.
Pressure Cooker. A pressure
cooker is a sealed pot designed
to cook food with the steam
produced by boiling water
somewhat above 100°C. The
pressure cooker in the figure
uses a weight of mass m to allow
steam to escape at a certain
pressure through a small hole
(diameter d) in the cooker’s lid.
If d = 3.0 mm, what should m be
in order to cook food at 120°C?
Assume that atmospheric
pressure outside the cooker is
1.013 x 105 Pa.
Vapor Pressure and Humidity
An open container of water can
evaporate, rather than boil, away.
The fastest molecules are escaping
from the water’s surface, so
evaporation is a cooling process as
well.
The inverse process is called
condensation.
When the evaporation and
condensation processes are in
equilibrium, the vapor just above the
liquid is said to be saturated, and its
pressure is the saturated vapor
pressure.
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Partial pressure is the pressure each
component of a mixture of gases would exert if
it were the only gas present. The partial
pressure of water in the air can be as low as
zero, and as high as the saturated vapor
pressure at that temperature.
Relative humidity is a measure of the saturation
of the air.
partial pressure of H 2 0
Relative humidity =
´ 100%
saturated vapor pressure of H 2 0
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When the humidity is high, it feels muggy; it is
hard for any more water to evaporate.
The dew point is the
temperature at which the air
would be saturated with
water.
If the temperature goes
below the dew point, dew,
fog, or even rain may occur.
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Vapor Pressure and Humidity
The saturated vapor
pressure increases with
temperature.
Fog tends to occur
during winter mornings
when it is cold, and the
air is saturated with
water vapor.
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The molecules having higher speed are able to
leave a sweat drop on the skin, resulting in a lower
temperature and drawing heat from the skin. This
in turn heats the drop and the process occurs again.
Speed Distribution for O2 and H2 at 20 ºC
3
1 2
 mv   K av  kT
2
2
av
Much easier for H 2 molecules
to leave Earth than O2
Do demos
Boiling by cooling
Boiling by reducing pressure
v kT
m
Think about this!
Van der Waals Equation of State
To get a more realistic model of a gas, we
include the finite size of the molecules
and the range of the intermolecular force
beyond the size of the molecule.
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We assume that some fraction b of the
volume is unavailable due to the finite size
of the molecules. We also expect that the
pressure will be reduced by a factor
proportional to the square of the density,
due to interactions near the walls. This
gives the Van der Waals equation of state;
the constants a and b are found
experimentally for each gas:
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The PV diagram for a Van der Waals gas fits
most experimental data quite well.
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Mean Free Path
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.
The average distance the molecule travels
between collisions, called the mean free
path, can be calculated.
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The mean free path can be calculated, given the
average speed, the density of the gas, the size of
the molecules, and the relative speed of the
colliding molecules. The result:
M
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=
1
2
4p 2r (N / V )
Diffusion
Even without stirring, a few drops of dye in
water will gradually spread throughout. This
process is called diffusion.
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Diffusion occurs from a region of high
concentration to a region of lower concentration.
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The rate of diffusion is given by:
dC
J = DA
dx
If concentration C is mol/m3, J
is mol/s passing a given point.
In this equation, D is the diffusion
constant.
If concentration C
is kg/m3, J is kg/s
passing a given point.
Paper
chromotograhpy
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Diffusion simulations
http://comp.uark.edu/~jgeabana/mol_dyn/
KinThI.html
http://www.biosci.ohiou.edu/introbioslab/
Bios170/diffusion/Diffusion.html
Best one:
http://serendip.brynmawr.edu/exchange/d
iffusion/applet
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Relative Humidity. Air that is at its
dew point of 5°C is drawn into a
building where it is heated to 20°C.
What will be the relative humidity at
this temperature? Assume constant
pressure of 1.0 atm. Take into account
the expansion of the air.
Mean Free Path. The mean free path
of molecules at STP is measured to be
about 5.6 x 10-8 m. Estimate the
diameter of a CO2 molecule. (b) Do the
same for He gas for which the mean free
path is ~25 x 10-8 m at STP.