Download - Kinetic Theory of Gases (1)
Kinetic Theory of Gases (1)
Chapter 11
Min Soo Kim
Seoul National University
Advanced Thermodynamics (M2794.007900)
2/17
11.1 Basic Assumption
Basic assumptions of the kinetic theory
1) Large number of molecules (Avogadroโs number)
๐๐ด = 6.02 ร 1026 molecules per kilomole
2) Identical molecules which behave like hard spheres
3) No intermolecular forces except when in collision
4) Collisions are perfectly elastic
5) Uniform distribution throughout the container
n =N
Vd๐ = ndV
n: The average number of molecules per unit volume
3/17
11.1 Basic Assumption
6) Equal probability on the direction of molecular velocity average number of
intersections of velocity vectors per unit area;
ฮธ
ฯ
r d ฮธr sin ฮธd๐
r
N
4ฯr2
Where dA = r2 sin ฮธ dฮธdฯd2Nฮธฯ =N
4ฯr2d๐ด =
N sin ฮธ dฮธdฯ
4ฯ
d2nฮธฯ =nsin ฮธ dฮธdฯ
4๐
the number of intersections in dA
Nฮธฯ: The number of molecules having velocities
in a direction (ฮธ ~ ฮธ + dฮธ) and (ฯ ~ ฯ + dฯ)
4/17
11.1 Basic Assumption
7) Magnitude of molecular velocity : 0 ~ โ
c (speed of light)
dN๐ฃ : The number of molecules with specified speed (v< <v+dv)
5/17
11.1 Basic Assumption
โข Let dN๐ฃ as the number of molecules with specified speed (v ~ v+dv)
โข 0โdN๐ฃ = N
โข Mean speed is าง๐ฃ =1
N0โ๐ฃdN๐ฃ
โข Mean square speed is ๐ฃ2 =1
N0โ๐ฃ2dN๐ฃ
โข Square root of ๐ฃ2 is called the root mean square or rms speed:
๐ฃ๐๐๐ = ๐ฃ2 =1
N0โ๐ฃ2dN๐ฃ
โข The n-th moment of distribution is defined as
๐ฃ๐ =1
N0โ๐ฃ๐dN๐ฃ
6/17
11.2 Molecular Flux
โข The number of gas molecules that strike a surface per unit area and unit time
โข Molecules coming from particular direction ฮธ, ฯ with specified speed v in time dt
โ ฮธฯ๐ฃ collision ฮธ ~ ฮธ + dฮธฯ ~ ฯ + dฯv ~ v+dv
โข The number of ฮธฯ๐ฃ collisions with dA dt
= ฮธฯ๐ฃ molecules in
= ฮธฯ molecules with speed v
Fig. Slant cylinder geometry used to calculate
the number of molecules that strike the area dA in time dt.
7/17
11.2 Molecular Flux
โข How many molecules in unit volume
โข The number of ฮธฯ๐ฃ molecules in the cylinder toward dA
dn๐ฃ : Density between speed (v ~ v+dv)
dA : Surface of spherical shell of radius v and thickness dv (i.e., ฮธ, ฯ molecules)
Volume of cylinder: dV = dA (vdt cosฮธ )
d3nฮธฯ๐ฃ๐๐ = (๐๐ด๐ฃdt cosฮธ )dn๐ฃsin ฮธ dฮธdฯ
4๐
d3nฮธฯ๐ฃ = dn๐ฃ โ๐๐ด
๐ด= dn๐ฃ
๐ฃ2 sin ฮธ dฮธdฯ
4๐๐ฃ2
8/17
11.2 Molecular Flux
โข The number of collisions per unit area and time (i.e., particle flux)
โข Total number of collisions per unit area and time by molecules having all speed
d3nฮธฯ๐ฃdV
dA dt= 0
2๐dฯ0
๐/2sinฮธcosฮธdฮธ โ
1
4ฯ0โ๐ฃdn๐ฃ =
๐
๐๐เดฅ๐
Cf. average speed าง๐ฃ =ฯ เดค๐ฃ
๐=
ฯ๐๐๐ฃ๐
๐=ฯ ๐๐๐ฃ๐ฯ ๐๐
= ๐ฃd๐๐ฃ
๐
0)โ๐ฃdn๐ฃ = ๐ าง๐ฃ)
d3nฮธฯ๐ฃdV
dA dt=
1
4ฯ๐ฃdn๐ฃsinฮธcosฮธdฮธdฯ
9/17
11.3 Gas Pressure and Ideal Gas Law
โข Gas pressure in Kinetic theory
Gas pressure is interpreted as impulse flux of particles striking a surface
๐ ๐
๐๐ฃ1 ๐๐ฃ2
10/17
11.3 Gas Pressure and Ideal Gas Law
โข Perfect elastic ๐ฃ = ๐ฃโฒ
โข Average force exerted by molecules
โข Momentum change of one molecule (normal component only)
๐๐ฃcos๐ โ โ๐๐ฃcos๐ = 2๐๐ฃcos๐
โข The number of ฮธฯ๐ฃ collisions for dA, dt
๐ฃ sinฮธ
๐ฃ cos ฮธ
ฮธ ฮธ
ฯdA
๐ฃ
d3nฮธฯ๐ฃdV
dA dt=
1
4ฯ๐ฃdn๐ฃsinฮธcosฮธdฮธdฯ
F =d(๐ ิฆ๐ฃ)
dt= ๐ ิฆ๐ + แถ๐ ิฆ๐ฃ
11/17
11.3 Gas Pressure and Ideal Gas Law
โข Change in momentum due to ฮธฯ๐ฃ collisions in time dt
โข Change in momentum p in all v collisions 0 < ฮธ โค๐
2, 0 < ฯ โค 2๐ at all speed
โข Change in momentum from collisions of molecules with unit time
โข Average pressure เดค๐ =๐ ิฆ๐น
๐๐ด
dp = เถฑ0
โ
เถฑ0
ฯ/2
เถฑ0
2ฯ 1
2ฯmv2dnvsinฮธcos
2ฮธdฮธdฯ โ dAdt =1
3๐๐๐ฃ2dAdt
2๐๐ฃcos๐ ร1
4ฯ๐ฃdn๐ฃsinฮธcosฮธdฮธdฯ =
1
2ฯ๐๐ฃ2dn๐ฃsinฮธcos
2ฮธdฮธdฯdAdt
dp
dt= dF =
1
3๐๐๐ฃ2dA
เดค๐ =1
3๐๐๐ฃ2
cf. ๐ฃ2 =ฯ ๐ฃ2
๐=
๐ฃ2d๐๐ฃ
๐
12/17
11.3 Gas Pressure and Ideal Gas Law
Since ๐ =๐
๐then pressure ๐ =
1
3
๐
๐๐๐ฃ2 โด ๐๐ =
1
3๐๐๐ฃ2
EOS of an ideal gas: PV = n เดค๐ ๐ = mRT =N
๐๐ด
เดค๐ ๐ = ๐๐๐
๐๐ด : Avogadroโs number : 6.02 ร 1026 ๐๐๐๐๐๐ข๐๐๐ /๐๐๐๐๐
๐๐ต : Boltzmann constant : ๐๐ต =เดค๐
๐๐ด= 1.38 ร 10โ23๐ฝ/๐พ
๐๐ =1
3๐๐๐ฃ2 = ๐๐๐
โด๐
๐๐๐๐ =
๐
๐๐๐ป
The temperature is proportional to the average kinetic energy of molecule
13/17
11.4 Equipartition of Energy
โข Equipartition of energy
Because of even distribution of velocity of particles,
By assumption, no preferred direction
It can be interpreted that a degree of freedom allocate energy of 1
2๐๐
๐ฃ2 = ๐ฃ๐ฅ2 + ๐ฃ๐ฆ
2 + ๐ฃ๐ง2,
๐ฃ๐ฅ2 = ๐ฃ๐ฆ
2 = ๐ฃ๐ง2 =
1
3๐ฃ2 โ
1
2๐๐ฃ๐ฅ
2 =1
6๐๐ฃ2 =
1
2๐๐
14/17
11.5 Specific Heat
Total energy of a molecule in Cartesian coordinate
General expression of total energy of molecules for f โDOF (Degree of Freedom)
๐๐ฃ = แ๐๐ข
๐๐ ๐ฃ=
f
2๐ from the above equation
๐๐ =๐โ
๐๐ ๐=
๐
2๐ + ๐ =
(๐+2)
2๐ cf) cp = cv + R
The ratio of specific heat: ฮณ =cp
cv=
f+2
f
เดคฮต = เดคฮตx + เดคฮตy + เดคฮตz =1
2๐๐ฃ๐ฅ
2 +1
2๐๐ฃ๐ฆ
2 +1
2๐๐ฃz
2 =๐๐
2+
๐๐
2+
๐๐
2=
3
2๐๐
U = Nเดคฮต =f
2NkT =
f
2nRT โ u =
U
๐=๐
2๐ ๐
15/17
Monatomic
gas
Diatomic
gas
11.5 Specific Heat
negligible
3
5
Near room temperature, rotational or vibrational DOF are excited,
but not both. DOF: 7 โ 5
1
2๐๐ฃ๐ฅ
2, 1
2๐๐ฃ๐ฆ
2, 1
2๐๐ฃ๐ง
2
1
2๐๐ฃ๐ฅ
2, 1
2๐๐ฃ๐ฆ
2, 1
2๐๐ฃ๐ง
2
1
2๐ผ๐ค๐ฅ
2, 1
2๐ผ๐ค๐ฆ
2, 1
2๐ผ๐ค๐ง
2
1
2๐๐ฅ2,
1
2๐ แถ๐ฅ2 no y,z vibration
๐๐
๐๐ฃ=5 + 2
5= 1.4
๐๐๐๐ฃ
=3 + 2
3= 1.67
Translational
Rotational
Vibrational
DOF
DOF
16/17
Triatomic
gas
11.5 Specific Heat
9CO2 translational 3
rotational 2
vibrational 4
๐๐๐๐ฃ
=7 + 2
7= 1.28
โข Vibration modes of CO2
Bending
Stretch
DOF
17/17
Solid
11.5 Specific Heat
x,y,z direction๐๐
2(kinetic)
๐๐
2(potential)
U =3๐๐
2+3๐๐
2= 3๐๐๐
cv = 3๐ (Dulong-Petit Law)