kinetic theory of gas气体动力学理论

2 be the average value of v Let v 2 x x 2 the average value of v y v
2 y
2 the average value of vz2 vz
z vz
2 v ∑ ix
r v
v + v + L+ v = i v = N N 2 2 v v ∑ iy ∑ iz 2 2 vy = i = i vz N N
N
+
N
2 2 v2 = v2 + v + v x y z
v
2 x
dN v v n= dV are statistical average values.
2 y
2 z
1 2 v = vy = v = v 3
2 x 2 2 z
2. Calculating the Pressure Exerted by a Gas
ni = Ni V
The number of molecules per unit volume equals
n = ∑ ni
v =
2 x
∑
N
k =1
2 v kx
N
Ni 2 N1 2 N2 2 v1x + v2 x + L vix N v + N v + LN v V V = = V N N /V
2 1 1x 2 2 2x 2 i ix
dA x
v v i dt
v ix dt
ni vix dt dA
Volume of the box
The impulse exerted by molecules r r r on dA that have velocity between vi − vi + dvi equals
ni vix dt dA（ 2m vix）
v2 x =
2 n v ∑ i ix i
n
Suppose dA is perpendicular to x axis ，the collision time is dt.
For each (elastic) collision, the impulse on dA
2m vix
The number of molecules that collide with dA in the time interval dt equals the number in the box.
F root mean square speed vrms
vrms = v =
for a gas, m T ↑
2
3kT = m
v
2
3 RT
µ
1 3 2 ε t = mv = kT 2 2
R k R = = µ m N Am
at a temperature T
µ↑
v2
↓
This is a particular type of average for a statistical process.
The impulse exerted by all molecules on dA in the time interval dt
dI =
( v ix > 0 )
∑
2 2 mn i v ix dAdt
The impulse exerted by all molecules on dA in the time interval dt
− vix
− mvix
v mvi
As a result of a collision the momentum changes by
− m vix − mvix = −2m vix
The impulse exerted on the wall by the collision = 2 m vix
x
v ix
16-1 The Ideal Gas Law and the Molecular Interpretation of Temperature
Our first model of a many-particle system: the Ideal Gas
Models of matter: gas models (random motion of particles)
mvix
dA
Consider a container of volume V containing N molecules, each of mass m moving with speed v. The number of molecules per unit volume is n .
Let ni be the number of molecules that have velocity between r r r vi − vi + dvi per unit volume.
2 x 2
2 z
For a single molecule
v =v +v +v
2 i 2 ix 2 iy
2 v ∑ i 2 v ∑ ix
2 iz
v = vy = v
2 x 2
2 v ∑ iz
2 z
Summing over all molecules in the gas, we have
N
=
N
+
2 v ∑ iy
1. The ideal gas model
1) Assumptions about the molecules in an ideal gas
The ideal gas model - works well at low densities (diluted gases) • all the molecules are identical, N is huge; • the molecules are tiny compared to their average separation (point masses); • the molecules do not interact with each other; • the molecules obey Newton’s laws of motion, their motion is random; • collisions between the molecules and the container walls are elastic.
= 6.42 × 10
−21
(J)
Example (a) What is the rms speed of hydrogen molecules when temperature is 0 ℃ , and (b) the rms speed of oxygen molecules at the same temperature. For H2 molecules
i
2 1 2 1 2 2 P = mn v = n( m v ) = n ε t 3 3 2 3
1 2 ε t = mv 2
─ Average kinetic energy of the translational motion of molecules
The pressure of a gas is of statistical.
2 x 2 1x 2 2x 2 ix
vx x
vy
y
Hale Waihona Puke Baidu
Since the velocity of the molecules in the gas are assumed to be random, there is no preference to one direction or another.
v = vy = v
Air at normal conditions: ~ 2.7×1019 molecules in 1 cm3 of air Size of the molecules ~ (2-3) × 10-10 m, Distance between the molecules ~ 3 × 10-9 m The average speed - 500 m/s The mean free path - 10-7 m (0.1 micron) The number of collisions in 1 second - 5 × 109
1 2 d I = ∑ 2 mn i v d A d t = ∑ 2 mn i v ix d A d t 2 ( v ix > 0 ) dI dI 2 dF = P= = ∑ mn i v ix d A d t dt dAdt
2 ix
P=
2 2 mn v = m n v ∑ i ix ∑ i ix = mn v 2 x i
dN N n= = dV V
)
Velocities of molecules are different. Each molecule has its velocity, which may be changed due to collisions. In an equilibrium state, velocity of each molecule has the same probability to point to any directions. That is, the distribution of velocity of molecules is uniform in direction, which leads to the mean-square speeds of all components of velocity are same.
3. Temperature of an Ideal gas
3 ε t = kT 2
2 P = nε t 3
P = n kT
Temperature in Kelvin
The average translational kinetic energy of molecules in an ideal gas is directly proportional to the absolute temperature. The molecular interpretation of temperature: The temperature of a gas is a direct measure of the average translational kinetic energy of its molecules! The higher the temperature, the faster the molecules are moving on the average. It applies reasonably accurately to liquids and solids.
2) Statistical assumptions about an ideal gas
) In an equilibrium state, the distribution of molecules on the
position is uniform, which means that the density of number of molecules is the same everywhere,
Example What is the average translational kinetic energy of molecules in an ideal gas at 37 °C? Solution:
3 ε t = kT 2
3 = × 1.38 × 10 − 23 × ( 37 + 273) 2
The pressure that a gas exerted on its container is due to collision between gas molecules and the container walls.
Example：consider a collision between the molecule of mass m and the wall of a container.Calculate the impulse exerted by the molecule on the wall. Solution： Before collision after collision
Chapter 16
Kinetic Theory of Gases
16-1 The Ideal Gas Law and the Molecular Interpretation of Temperature
16-2 Distribution of Molecular Speeds
16-6 Mean Free Path