Kinetic Theory of Gases
Molecular Velocity
Velocity of the nth molecule
„
Want to connect microscopic properties
of molecules such as velocity
momentum kinetic energy, etc. to
macroscopic “state” properties of the
gas such as pressure and temperature
This will require some statistics
„
Assumptions
„
‰
G
vn = vnx iˆ + vn y ˆj + vnz kˆ
Average velocity of all molecules
G
v =0
Average speed of all molecules
v ≠0
There a large number of molecules, N
„
„
Each has mass m
Moving in random directions with a variety of speeds
‰
Molecules are far apart from each other on average
‰
The molecules obey the laws of classical mechanics
‰
Collisions are perfectly elastic
„
„
„
Root-Mean-Square (rms) speed of all molecules
vrms ≠ 0
Average separation >> molecular diameter
Newton’s laws
No permanent deformation
Molecular Basis of Pressure & Temperature
Molecular BasisG of Pressure
vi = v x iˆ + v y ˆj + vz kˆ
G
Reflected velocity of a molecule vr = −v x iˆ + v y ˆj + v z kˆ
So
Change in momentum
But pressure is
force/area
Incident velocity of a molecule
F=
m v2
N
Lx 3
Impulse and momentum (Newton’s 2nd Law) states
Rearrange
For N molecules
But
Also
(
m 2
F=
v x + v x22 + v x23 + ... + v x2N
Lx 1
and
)
But
„
„
„
1
K = mv 2
2
and
PV = NkT
1
3
∴ K = mv 2 = kT
2
2
Temperature is a measure of the average kinetic energy (internal
energy?) of the gas.
For constant volume, pressure increases directly proportional to an
increase in average kinetic energy (temperature) AND an increase in
the number of molecules.
We can directly connect macroscopic state variables (temperature
and pressure) to microscopic molecular properties (kinetic energy and
momentum).
1
Molecular Speeds
1
3
K = mv 2 = kT
2
2
Since
⇒
Molecular Speeds
3kT
v2 =
= vrms
m
Distribution of molecular speeds,
For a given gas, the MaxwellBoltzmann distribution only
depends on temperature
The Maxwell-Boltzmann distribution
3
mv 2
⎛ m ⎞ 2 2 − 2 kt
f (v) = 4πN ⎜
⎟ v e
⎝ 2πkT ⎠
∞
∫0
Note
v=
f (v)dv = N
1 ∞
8kT
∫ vf (v)dv = πm
N 0
vrms =
vp ⇒
2kT
df (v)
= 0 ⇒ vp =
m
dv
However, for different gases
at the same temperature, the
Maxwell-Boltzmann
distributions will be different.
1 ∞ 2
3kT
v f (v)dv =
∫
0
N
m
One mol of a monatomic ideal gas is placed in a chamber under 5.00
atm pressure. The volume of the chamber is 5000 cm3. (a) What is the
internal energy of the gas? (b) What is the temperature of the gas? (c)
Assuming that the mass of a molecule of the gas is 3.36 x 10-26 kg,
what is the root-mean-square (rms) velocity?
If the rms speed of molecules of gaseous H2O is 200 m/s what will be
the rms speed of CO2 molecules at the same temperature? Assume
that both of these are an ideal gas.
2