Download Chapter 6 - January 17 Slides

M. Burt
Chemistry 1050
Winter 2011
January 17, 2011 – Class 6 Overview
Chapter 6
• Section 6.7 (Page 216 – 223): Kinetic-Molecular Theory of
Gases
- Postulates of the kinetic-molecular theory, molecular speed,
translational kinetic energy, Maxwell-Boltzmann distribution.
- Suggested end-of-chapter exercises: 73, 77
• Section 6.8 (Page 223 – 226): Gas Properties Related to the
Kinetic-Molecular Theory
- Diffusion, effusion, Graham’s Law.
- Suggested end-of-chapter exercises: 81
M. Burt
Chemistry 1050
The Kinetic Theory of Gases
• The macroscopic properties
(P, V, and T) of gases can be
determined by considering the
molecular composition and
motion of the component
particles.
• At low pressures however, all
gases obey the ideal gas law,
therefore this relationship is
independent of the nature of
the gas itself.
Winter 2011
M. Burt
Chemistry 1050
The Kinetic Theory of Gases
• The kinetic-molecular theory of
gases describes the behaviour
of gases by considering the
motion (kinetics) of the atoms
or molecules that make up the
system.
Winter 2011
M. Burt
Chemistry 1050
Winter 2011
Postulates of the Kinetic Theory
• Postulate 1: Particle Volume.
- A gas is composed of many individual particles.
- The volume of this collection of particles is negligible
compared to the total volume of the gas container.
- The average distance between particles is therefore much
longer than the particles themselves.
- A gas is primarily empty space.
M. Burt
Chemistry 1050
Winter 2011
Postulates of the Kinetic Theory
• Postulate 2: Particle Motion.
- Particles are constantly moving in straight lines in all
directions.
- Their motion is random.
• Postulate 3: Particle Collisions.
- Molecules are approximated as hard spheres.
- Any particle collisions, either with the container walls or
other particles, are assumed to be perfectly elastic.
- Collisions between particles can lead to a transfer of kinetic
energy.
- Attractive forces between molecules are negligible except
during collisions. They exert no force on one another.
M. Burt
Chemistry 1050
Winter 2011
Postulates of the Kinetic Theory
• Postulate 4: Temperature.
- At a fixed temperature, the total amount of energy in an
isolated gas container is fixed.
- Particles may transfer energy through collisions, but the
total amount of energy in the system remains unchanged.
- Temperature is directly proportional to the amount of
energy in a system.
- Increasing the temperature raises the total kinetic energy of
a system, resulting in faster moving particles.
M. Burt
Chemistry 1050
Winter 2011
The Distribution of Molecular Speeds
• Although the total energy of a system is fixed, particles have
different amounts of kinetic energy due to intermolecular
collisions.
• As a result, the particles in a gas have a broad distribution of
observed speeds.
um = most probable (modal) speed
uav (ū) = average speed
urms = root-mean-square speed
M. Burt
Chemistry 1050
Winter 2011
The Average and Most Probable Speeds
• The average speed, ū, represents the average speed of all the
molecules comprising the gas.
n
u=
∑u
i =1
i
u = speed
n = number of particles in the gas
n
• The most probable (or modal) speed, um, is the speed
characteristic of the largest number of gas particles.
2 RT
um =
M
M. Burt
Chemistry 1050
Winter 2011
Translational Kinetic Energy
• Translational kinetic energy, EK, is the energy carried by an
object as it moves through space.
1
1
2
2
Ek = mu = murms
2
2
m = particle mass (g/molecule)
ū = average speed (m/s)
• The root-mean-square speed, urms, is the speed at which a
molecule possesses the average kinetic energy of the system.
1
1
2
2
Ek = mu = murms
2
2
urms = u 2
M. Burt
Chemistry 1050
Winter 2011
Root-Mean Square Speed
• Boyle’s Law:
PV = a
• The kinetic theory of gases tells us that pressure is
proportional to the product of collision frequency (v) and
momentum transfer (I). Therefore, in one dimension (x):
N
ν = ux
V
• In three dimensions:
1N
P=
mu 2
3V
N
P = mu x2
V
I = mu
1 2
u =u =u = u
3
2
x
2
y
2
z
M. Burt
Chemistry 1050
Winter 2011
Root-Mean Square Speed
• Assume 1 mol of gas, therefore N = Na.
• Using the ideal gas law, PV = RT.
3RT = N a mu 2
3RT = M u 2
• Nam is molar mass (g/mol).
• Using the definition of urms:
1 Na 2
P=
mu
3V
urms
3RT
= u =
M
2
Temperature is proportional to speed, as expected
from the postulates of the kinetic-molecular theory!
M. Burt
Chemistry 1050
Molecular Speed Distributions
• Problem: Calculate the root-mean-square speed for a
molecule of oxygen gas at 25°C.
Winter 2011
M. Burt
Chemistry 1050
Winter 2011
Molecular Speed Distributions
• Problem: The average speed of O2(g) at 25°C is 444 m/s. Why
is this different than the root-mean-square speed?
M. Burt
Chemistry 1050
Winter 2011
Molecular Speed Distributions
• Problem: The average speed of O2(g) at 25°C is 444 m/s. Why
is this different than the root-mean-square speed?
Gas
ū (m/s)
urms (m/s)
NH3
609
661
CO2
379
411
He
1260
1360
H2
1770
1920
CH4
627
681
N2
475
515
O2
444
482
SF6
208
226
M. Burt
Chemistry 1050
Winter 2011
Molecular Speed Distributions
• Problem: Calculate the most probable speed of O2(g) at 25°C.
M. Burt
Chemistry 1050
Winter 2011
The Maxwell-Boltzmann Distribution
• The large amount of particles in a gas makes it impossible to
measure the speed of each individual particle.
• The Maxwell-Boltzmann distribution can be used to
statistically predict this distribution of speeds.
M 3/ 2 2 − ( Mu 2 / 2 RT )
F (u ) = 4π (
) ue
2π RT
• The relationship determines the fraction of molecules with
molecular speed u (F(u)); it is dependent on molar mass (M)
and temperature (T).
M. Burt
Chemistry 1050
Winter 2011
The Maxwell-Boltzmann Distribution
M 3/ 2 2 − ( Mu 2 / 2 RT )
) ue
F (u ) = 4π (
2π RT
M. Burt
Chemistry 1050
Winter 2011
The Effect of Temperature on Molecular Speed
• The average kinetic
energy of a gas is
dependent on the Kelvin
temperature of the
system.
• Increasing the
temperature of a gas
results in a higher
average molecular speed.
M. Burt
Chemistry 1050
The Effect of Mass on Molecular Speed
• The lighter the gas, the
broader the
distribution of speeds.
• This is because the
average kinetic energy
of a particle is directly
proportional to its
mass.
Winter 2011
M. Burt
Chemistry 1050
Winter 2011
Gas Properties: Diffusion
• Diffusion occurs when two or more gases intermingle; it is the
result of random molecular motion.
M. Burt
Chemistry 1050
Gas Properties: Effusion
• Effusion is the flow of gas through a tiny opening.
Winter 2011
M. Burt
Chemistry 1050
Winter 2011
The Rate of Effusion
• Molecules with high speeds will effuse faster than slower
molecules. The relative rates of effusion (k) for multiple gases
can therefore be determined using molecular speeds.
ka (urms ) a
3RT / M a
=
=
=
kb (urms )b
3RT / M b
Mb
Ma
• The above is Graham’s Law, which states that the rate of
effusion of a gas is inversely proportional to the square root
of its molar mass.
M. Burt
Chemistry 1050
Winter 2011
Graham’s Law
• Q: Are there any limitations to Graham’s Law? Does it also
apply to diffusion?
• A: Graham’s Law does not consider intermolecular collisions;
it is only valid at low pressures since this allows the effusing
molecules to flow freely through the orifice instead of being
ejected as a high pressure jet of gas.
During diffusion, the intermingling gases frequently collide
with each other. This causes many particles to be scattered in
the opposite direction.
M. Burt
Chemistry 1050
Winter 2011
The Rate of Effusion
• Problem: If 2.2x10-4 mol N2(g) effuses through a tiny hole in
105 s, then how much H2(g) would effuse through the same
orifice in 105 s?
M. Burt
Chemistry 1050
Winter 2011
Looking Ahead - January 18, 2011
Chapter 6
• Section 6.9 (Page 226 - 229): Nonideal (Real) Gases
– Qualitative comparison to ideal gases. A student must understand the
significance of the correction terms in the van der Waals equation, e.g.
the excluded volume term, nb, and the correction for intermolecular
forces, n2a/V2.