1. 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

2. One mole is the number of atoms in a 12 g sample of carbon-12.
19.2 Avogadro’s Number
Italian scientist Amedeo Avogadro ( ) 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, NA.
If n is the number of moles contained in a sample of any substance, N is the number of molecules, Msam is the mass of the sample, m is the molecular mass, and M is the molar mass, then

3. The equation of state of a dilute gas is found to be
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.)

4. 19.3: Ideal Gases; Work Done by an Ideal Gas

5. Example, Ideal Gas Processes

6. Example, Work done by an Ideal Gas

7. 19.4: Pressure, Temperature, and RMS Speed
The momentum delivered to the wall is +2mvx
Considering , we have
Defining , we have
Comparing that with , we have
The temperature has a direct connection to the RMS speed squared.

8. Translational Kinetic Energy

9. 19.4: RMS Speed

10. Example:

11. 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 (a) The Maxwell speed distribution for oxygen molecules at T =300 K. The three characteristic speeds are marked.

13. Example, Speed Distribution in a Gas:

14. Example, Different Speeds

15. 19.8: Molar Specific Heat of Ideal Gases: Internal Energy
The internal energy Eint 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.

16. 19.8: Molar Specific Heat at Constant Volume
where CV is a constant called the molar specific heat at constant volume.
But,
Therefore,
With the volume held constant, the gas cannot expand and thus cannot do any work.
# When a confined ideal gas undergoes temperature change DT, the resulting change in its internal energy is
A change in the internal energy Eint of a confined ideal gas depends on only the
change in the temperature, not on what type of process produces the change.

17. 19.8: Molar Specific Heat at Constant Pressure
where Cp is a constant called the molar specific heat at constant pressure.
This Cp is greater than the molar specific heat at constant volume CV, since for the same internal energy change, more heat is needed to provide work.

18. Example, Monatomic Gas:

19. Molar specific heats at 1 atm, 300K
CV (J/mol/K)
CP-CV (J/mol/K)
g=CP/CV
monatomic
1.5R=12.5
R=8.3 He
12.5
8.3
1.67 Ar
diatomic
2.5R=20.8 H2
20.4
8.4
1.41 N2
20.8
1.40 O2
21.0
Cl2
25.2
8.8
1.35
polyatomic
3.0R=24.9
CO2
28.5
8.5
1.30
H2O(100°C)
27.0
1.31

20. 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

21. CV (J/mol/K)
CP-CV (J/mol/K)
g=CP/CV
monatomic
1.5R=12.5
R=8.3 He
12.5
8.3
1.67 Ar
diatomic
2.5R=20.8 H2
20.4
8.4
1.41 N2
20.8
1.40 O2
21.0
Cl2
25.2
8.8
1.35
polyatomic
3.0R=24.9
CO2
28.5
8.5
1.30
H2O(100°C)
27.0
1.31

22. Example, Diatomic Gas:

23. # Oscillations are excited with 2 degrees of freedom
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.

24. 19.11: The Adiabatic Expansion of an Ideal Gas
with Q=0 and dEint=nCVdT , we get:
From the ideal gas law,
and since CP-CV = R,
we get:
With g = CP/CV, and integrating, we get:
Finally we obtain:

25. 19.11: The Adiabatic Expansion of an Ideal Gas

26. 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 ΔEint =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,

27. Example, Adiabatic Expansion:

28. Four Gas Processes for an Ideal Gas

29. Homework:
Problems 13, 24, 38, 52, 59