Transcript Chapter 19 - The Kinetic Theory of Gases
Chapter 19 The Kinetic Theory of Gases
From the macro-world to the micro-world Key contents: Ideal gases Pressure, temperature, and the RMS speed Molar specific heats Adiabatic expansion of ideal gases
19.2 Avogadro’s Number
Italian scientist Amedeo Avogadro (1776-1856) suggested that all gases occupy the same volume under the condition of the same temperature, the same pressure, and the same number of atoms or molecules. => So, what matters is the ‘number’ .
One mole
is the number of atoms in a 12 g sample of carbon-12.
The number of atoms or molecules in a mole is called
Avogadro’s Number
,
N A
.
If
n
is the number of moles contained in a sample of any substance,
N
is the number of molecules,
M sam
the sample,
m
is the molecular mass, and
M
is the mass of is the molar mass, then
19.3: Ideal Gases
The equation of state of a dilute gas is found to be Here
p
is the pressure,
n
is the number of moles of gas present, and
T
is its temperature in kelvins.
R
is the gas constant that has the same value for all gases.
Or equivalently, Here,
k
is the Boltzmann constant , and
N
the number of molecules.
(# The ideal gas law can be derived from the Maxwell distribution; see slides below.)
19.3: Ideal Gases; Work Done by an Ideal Gas
Example, Ideal Gas Processes
Example, Work done by an Ideal Gas
19.4: Pressure, Temperature, and RMS Speed
The momentum delivered to the wall is +2
mv x
Considering , we have Defining , we have
nRT V
= 3
kT m
The temperature has a direct connection to the RMS speed squared.
Translational Kinetic Energy
19.4: RMS Speed
Example:
19.7: The Distribution of Molecular Speeds
Maxwell’s law of speed distribution is: The quantity
P(v)
is a probability distribution function: For any speed
v
, the product
P(v) dv
is the fraction of molecules with speeds in the interval
dv
centered on speed
v
.
Fig. 19-8 (a) The Maxwell speed distribution for oxygen molecules at T =300 K. The three characteristic speeds are marked.
= 1.41
kT m
= 1.59
kT m
= 1.73
kT m
ò 0 ¥
x
2
e
-
ax
2
dx
= p 16
a
3 ò 0 ¥
x
3
e
-
ax
2 ò 0 ¥
x
2
e
-
ax
2
dx
= 1
dx
= 2
a
2 9 p 64
a
5
Example, Speed Distribution in a Gas:
Example, Different Speeds
19.8: Molar Specific Heat of Ideal Gases: Internal Energy
The internal energy E
int
of an ideal gas is a function of the gas temperature only; it does not depend on any other variable.
For a monatomic ideal gas, only translational kinetic energy is involved.
19.8: Molar Specific Heat at Constant Volume
where
C V
is a constant called
heat at constant volume
.
the molar specific
But, Therefore, With the volume held constant, the gas cannot expand and thus cannot do any work. Therefore, # When a confined ideal gas undergoes temperature change D
T,
the resulting change in its internal energy is
A change in the internal energy E
int
of a confined ideal gas depends on only the change in the temperature, not on what type of process produces the change.
19.8: Molar Specific Heat at Constant Pressure
D
E
int =
nC V
D
T
Example, Monatomic Gas:
Molar specific heats at 1 atm, 300K
C V
(J/mol/K)
C P -C V
(J/mol/K) g
=C P /C V
monatomic He Ar diatomic H 2 N 2 O 2 Cl 2 polyatomic CO 2 H 2 O(100 ° C) 1.5R=12.5
12.5
12.5
2.5R=20.8
20.4
20.8
21.0
25.2
3.0R=24.9
28.5
27.0
R=8.3
8.3
8.3
8.4
8.3
8.4
8.8
8.5
8.4
1.67
1.67
1.41
1.40
1.40
1.35
1.30
1.31
19.9: Degrees of Freedom and Molar Specific Heats Every kind of molecule has a certain number f of degrees of freedom , which are independent ways in which the molecule can store energy. Each such degree of freedom has associated with it — on average — an energy of ½ kT per molecule (or ½ RT per mole). This is equipartition of energy.
Recall that
D
E
int =
nC V
D
T
monatomic He Ar diatomic H 2 N 2 O 2 Cl 2 polyatomic CO 2 H 2 O(100 ° C)
C V
(J/mol/K) 1.5R=12.5
12.5
12.5
2.5R=20.8
20.4
20.8
21.0
25.2
3.0R=24.9
28.5
27.0
C P -C V
(J/mol/K) R=8.3
8.3
8.3
8.4
8.3
8.4
8.8
8.5
8.4
g
=C P /C V
1.67
1.67
1.41
1.40
1.40
1.35
1.30
1.31
Example, Diatomic Gas:
19.10: A Hint of Quantum Theory
A crystalline solid has 6 degrees of freedom for oscillations in the lattice. These degrees of freedom are frozen (hidden) at low temperatures.
# Oscillations are excited with 2 degrees of freedom (kinetic and potential energy) for each dimension.
# Hidden degrees of freedom; minimum amount of energy # Quantum Mechanics is needed.
19.11: The Adiabatic Expansion of an Ideal Gas
with
Q=0 and dE int =nC V dT ,
we get: From the ideal gas law, and since
C P -C V
=
R
, we get: With g
= C P /C V
, and integrating, we get: Finally we obtain:
19.11: The Adiabatic Expansion of an Ideal Gas
19.11: The Adiabatic Expansion of an Ideal Gas, Free Expansion
A free expansion of a gas is an adiabatic process with no work or change in internal energy. Thus, a free expansion differs from the adiabatic process described earlier, in which work is done and the internal energy changes.
In a free expansion, a gas is in equilibrium only at its initial and final points; thus, we can plot only those points, but not the expansion itself, on a
p-V
diagram. Since Δ
E int =0
, the temperature of the final state must be that of the initial state. Thus, the initial and final points on a
p-V
diagram must be on the same isotherm, and we have Also, if the gas is ideal,
Example, Adiabatic Expansion:
Four Gas Processes for an Ideal Gas